The #1 AI tutor AI Learning Service

AI tutor
AI tutor
Join our Discord

Connect With Us on Social Media

AI tutorAI tutorAI tutorAI tutor

AI tutor | The No.1 Homework Finishing Free App

Functions and Analysis

Advanced Functions - Fundamentals of Differentiation and Integration

Q.01

"What is the purpose of the first-year university course on 'Calculus'?"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.02

'The midpoint is the average of 2 points, the centroid is the average of 3 points. S can be considered as the point that divides line segment AB in a 1:2 ratio.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.03

'Exercise 82\nFind functions p(x) and q(x) that satisfy the following conditions.\n- The derivative of p(x) is 3\n- p(0) = 3\n- The derivative of q(x) is 4x + k\n- q(0) = 2\nAlso, find functions f(x) and g(x) that satisfy q(x) = f(x)g(x) as a quadratic function, and p(x) = f(x) + g(x) as a linear function. Determine the value of k corresponding to them.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.04

'Explain the Cauchy-Schwarz inequality.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.05

'[2] Given that 2x<x<2+x 2-x<x<2+x , i.e., when x>1 x>1 ,'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.06

'For two polynomials f(x) and g(x) that satisfy f(0)=1, g(0)=2, let p(x)=f(x)+g(x), q(x)=f(x)g(x).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.07

'Investigate the extremum of the function. Determine the maximum or minimum value according to the following definitions: 1. If f(x) reaches a maximum value around x=a, and 2. If f(x) reaches a minimum value around x=a.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.08

'Find the value of the following expression.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.09

"(2) From y = x^{3} - 2 x^{2} - x + 2, we get y' = 3 x^{2} - 4 x - 1. When x = 1, y' = -2. Therefore, the equation of the tangent line l is y = -2(x - 1). Given x^{3} - 2 x^{2} - x + 2 = -2(x - 1), we have x(x - 1)^{2}=0, hence x=0,1"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.10

'Find the maximum and minimum values of the function f(x)=int01leftt2x2rightdtf(x)=\\int_{0}^{1}\\left|t^{2}-x^{2}\\right| d t for 0leqqxleqq20 \\leqq x \\leqq 2.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.11

'Express the definite integral \\( \\int_{-1}^{1}\\left(9 x t^{2}+2 x^{2} t-x^{3}\\right) d t \\) in terms of x.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.12

'Find the first derivative of the function h(x) = 2x^2 - x + 3.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.13

'Definite integral and volume\nLet V be the volume of a solid enclosed between two parallel planes \\\alpha, \eta\. Take a line perpendicular to \\\alpha, \eta\ as the \x\ axis, with coordinates of intersection with \\\alpha, \eta\ being \a, b\ respectively. Also, consider \a \\leqq x \\leqq b\, and when this solid is sliced by a plane perpendicular to the \x\ axis and having intersection coordinate \x\ with the \x\ axis, the cross-sectional area is denoted by \\(S(x)\\). Then, the volume \V\ is given by the following definite integral.\n\n\\[ V=\\int_{a}^{b} S(x) d x \\quad \\text{where} a < b \\]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.14

"There is a spherical rubber balloon that is being inflated at a rate of 0.1 cm per second in terms of its radius r. Starting from a radius of 1 cm, find the rate of change of the balloon's volume V with respect to time t when the radius reaches 3 cm."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.15

'Determining function from extreme value conditions'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.16

'Exercise 85 | II | \ \\Rightarrow \ Book \ p.340 \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.17

'Find the rate of change of the surface area of the sphere. Let the radius of the sphere after t minutes be r cm. From the condition, r=t+10, so S=4πr^2=4π(t+10)^2. Therefore, dS/dt=4π×2(t+10)×1=8π(t+10)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.18

'(2) f(x)=2x+int01(x+t)f(t)dt f(x)=2 x+\\int_{0}^{1}(x+t) f(t) d t '

A. ...

Ask AI tutor for answer!
Join our Discord

Q.19

'Find the equations of the tangent and normal of the curve at the given points on the curve. \n(1) y = x^2 - 3x + 2, (1,0) \n(2) y = x^3 - 3x^2 + 6, (2,2)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.20

'(2) Let V be the volume of a cone with radius r and height h. Consider V as a function of r, find the differential coefficient at r=3.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.21

'By increasing or decreasing functions or using graphs, we can find maximum and minimum values, or determine the number of real solutions of equations.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.22

'Find the function f(x) f(x) and the value of the constant a a that satisfy the equation intaxf(t)dt=3x22x1 \\int_{a}^{x} f(t) d t=3 x^{2}-2 x-1 .'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.23

'When calculating the area of the region enclosed by a parabola and a straight line, or two parabolas, it is useful to use the formula for definite integral ∫_(α)^(β) (x-α)(x-β) dx. Through the proof of the formula and examples, you will understand how to use the formula for calculations.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.24

'34 Derivative and its calculation Standard 174 Determination of function from differentiation coefficient conditions'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.25

'Let a and b be constants. Prove the following inequality:\n\n\\[\n\\int_{0}^{1}(ax+b)^{2}dx \\geqq\\left\\{\\int_{0}^{1}(ax+b)dx\\right\\}^{2}\n\\]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.26

'Differentiate the following functions with respect to the variables provided.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.27

'Example: Find the maximum and minimum values of x + y, as well as the values of x, y that satisfy the four inequalities x ≥ 0, y ≥ 0, x + 2y ≤ 6, 3x + 2y ≤ 10 simultaneously.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.28

'Development learning Development 187 Extremum and Graph of a 4th-degree function'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.29

'When it is difficult to determine the pattern of a given sequence in order to find its general term, one should try different approaches such as looking at the differences between consecutive terms, considering ratios between terms, or using the method of finite differences. One common technique is to observe the sequence for a few initial terms and then try to identify any arithmetic progression, geometric progression, or any other known sequence pattern.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.30

'Find the derivative of the function at the given value of x according to the definition.\n(1) f(x)=2 x-3 (x=1)\n(2) f(x)=2 x^{2}-x+1 (x=-2)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.31

'Differentiate the following functions with respect to the variables given in []:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.32

"Let's deepen our understanding of expected value, variance, and standard deviation."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.33

'Differential calculus'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.34

'Find the derivative of the function f(x)=x^{2}-6 x+7 at x=a using the definition. Also, determine the value of a such that the derivative is 2.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.35

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.36

'Find the function \\( f(x) \\) and the value of the constant \ a \ that satisfy the equation \\( \\int_{a}^{x} f(t) d t=3 x^{2}-2 x-1 \\).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.37

'Find the derivatives of the following functions. (1) y=(2 x^{2}-3)(x+5) (2) y=(x+2)^{3}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.38

'Find the quadratic function f(x) that satisfies the following conditions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.39

'Find the following indefinite integrals. In (3), α is a constant.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.40

'Tangent Equation: Find the tangent equation. What is the equation of the tangent line at point A(a, f(a)) on the curve y=f(x)?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.41

'According to the definition, find the derivatives of the following functions. (1) f(x)=-5x (2) f(x)=2x^{2}+5 (3) f(x)=x^{3}-x'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.42

'Differentiate the following functions and find the derivative at x=2.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.43

'Determining the coefficients of a cubic function based on conditions of maximum and minimum values'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.44

"It neatly summarizes how to distinguish and use theorems, formulas, etc. based on the type of problem in 'STEP into Sorting here'. It can be used for confirming and organizing formulas."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.45

"Explains in detail the approach to problem solving that requires more thinking power called 'Zoom UP'."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.46

'Explain how to determine the increasing and decreasing behavior, maximum, and minimum values of a function.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.47

'Define the function S(t) as S(t)=\\int_{0}^{1}\\left|x^{2}-t^{2}\\right| d x. Find the maximum and minimum values of S(t) for 0 ≤ t ≤ 1, and the corresponding t values.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.48

'Find the following definite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.49

'Integration method'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.50

'Development Study Development 185 The tangent of the graph of a quadratic function'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.51

'Let V be the volume of a cone with radius r and height h. Considering V as a function of h, find the derivative at h=3.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.52

'Differentiate the following functions with respect to the variables indicated within [ ].'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.53

'After solving basic and standard example problems, how should one deepen their understanding?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.54

"Let the polynomial f(x) satisfies the equation f(x)f'(x)=∫[0,x]f(t)dt+49⋯⋯(1). Answer the following questions."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.55

'Differentiation of functions and its calculation basic 173 differentiated with variables other than x'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.56

'Development study Development 188 Conditions for a cubic function to have extreme values'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.57

'Find the function f(x) that satisfies the equation f(x)=1+2 \\int_{0}^{1}(x t+1) f(t) d t.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.58

'Application of Increase and Decrease of Functions, Standard 184, Proof of Inequalities (Using Differentiation)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.59

'Calculate the area between two curves.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.60

'Find the following definite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.61

'Find the following indefinite integrals: (1) ∫√(1+√x) dx (2) ∫(cos x)/(cos^2 x + 2sin x -2) dx'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.62

"Let F(x) be the primitive function of f(x). The following conditions [1], [2] hold true. Find f'(x) and f(x) under the condition that x > 0. [1] F(x) = x f(x) - 1/x [2] F(1/√2) = √2"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.63

'Domain and Analysis of Increase and Decrease of a Function'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.64

'Find the maximum and minimum of the function\n(299) The function f(x)=a x + x cos(x) - 2 sin(x) has exactly one extremum between π/2 and π. Here, -1<a<1.\n[Similar to Maebashi Institute of Technology]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.65

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.66

"The function f(x) has a continuous second derivative f''(x) for x > -2. Also, for x > 0, f(x) > 0 and f'(x) > 0 hold, and for any positive number t, the x-coordinate of the intersection point P between the tangent line at point (t, f(t)) of the curve y=f(x) and the x-axis is equal to -∫0^t f(x) dx."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.67

'Determine the coefficients of the function from the area.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.68

'Evaluate the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.69

'State the definition of function f(x) being differentiable at x=a.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.70

"Let f(x) = x^(1/3) (x>0). Find the derivative f'(x) using the following two methods."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.71

"When the function f(x) satisfies f(0)=0, f'(x)=x cos x, answer the following questions: (1) Find f(x). (2) Find the maximum value of f(x) for 0 <= x <= π."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.72

'Find the maximum and minimum values of the function f(x)=∫₀ˣ (1-t²)eᵗ dt within the range -2≤x≤2, and determine the corresponding values of x.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.73

'Find the following indefinite integrals.\\n(1) \ \\int \\frac{x^{3}+x}{x^{2}-1} d x \\\n(2) \ \\int \\frac{x+5}{x^{2}+x-2} d x \\\n(3) \\( \\int \\frac{x}{(2 x-1)^{4}} d x \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.74

"(1) Prove by contradiction that for all real numbers x, f(x)>0. (2) Show that for all real numbers x, f'(x)=f(x) f'(0). (3) Express f(x) using k when f'(0) = k."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.75

'Find the general term of the following recurrence relation.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.76

'When a real number t moves in the range 1 ≤ t ≤ e, find the maximum and minimum values of S(t) = ∫_{0}^{1} |e^{x} - t| dx.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.77

'To deal with the inequality of two variables a, b, the following methods can be considered.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.78

'Find the derivative of the function y=x/√(4+3x^2). [Miyazaki University]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.79

'In a cone with a volume of √2/3, find the minimum lateral surface area of the cone. Also, determine the radius of the circular base and the height of the cone at this minimum.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.80

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.81

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.82

'Express the coordinate x of point P moving along a straight line as a function of time t, x = f(t). Then answer the following questions:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.83

'Find the following definite integrals. In (1), let \ a \ be a constant. (1) \ \\int_{-a}^{a} \\frac{x^{3}}{\\sqrt{a^{2}+x^{2}} dx } \ (2) \\( \\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}}(2 \\sin x+\\cos x)^{3} dx \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.84

"Basic technique for solving differential equations\n(1) Let y be a function of x. Solve the following differential equation. Solve it under the initial conditions in brackets.\n(1) 2yy'=1 [When x=1, y=1]\n(2) y=xy'+1"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.85

'Calculate the volume of a solid of revolution in coordinate space (1).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.86

'For a function y=f(x) that is differentiable in a certain interval, when the value of x increases, if the slope of the tangent line increases, the curve y=f(x) in that interval is defined as concave downwards; when the slope of the tangent line decreases, the curve y=f(x) in that interval is defined as concave upwards; but the actual definition is as follows: a function f(x) is concave downwards if for any different real numbers x_{1}, x_{2} contained in a certain interval and any real numbers s, t where s+t=1, s ≥ 0, t ≥ 0, the inequality f(s x_{1}+t x_{2}) ≤ s f(x_{1})+t f(x_{2}) holds, and f(x) is concave upwards if the inequality f(s x_{1}+t x_{2}) ≥ s f(x_{1})+t f(x_{2}) holds. A function that is concave downwards in its domain is called a convex function, and a function that is concave upwards is called a concave function.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.87

'Calculate the amount and integral of water discharge.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.88

'Differentiate the following function. (1) y = (x^2 + 1)^3'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.89

'Explain the method of calculating derivatives by definition.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.90

'Please explain the product and quotient rules of differentiation.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.91

'Using the method of integration by parts, find the definite integral below.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.92

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.93

'(1) ∫ 1/√(x^2+1) dx'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.94

'Find the tangent of the curve and the volume of the solid of revolution.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.95

''

A. ...

Ask AI tutor for answer!
Join our Discord

Q.96

'Differentiate the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.97

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.98

"Prove that f'(x) is divisible by x=1 when f(x)=(x-1)^2 Q(x) where Q(x) is a polynomial."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.99

"Explain the definition and characteristics of derivatives. Also, provide the formula for the derivative f'(x) of the function f(x)."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.00

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.01

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.02

'Differentiate the following functions according to the definition of derivatives. (Additional question)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.03

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.04

'For real numbers x and y, find the minimum value of x^{2}-4 x y+7 y^{2}-4 y+3, and determine the values of x and y at that time.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.05

'Circular permutations and permutations with repetition of the same item'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.06

'Determine the value of the constant a so that the parabola y=x^{2}-ax+a+1 is tangent to the x-axis. Also, find the coordinates of the point of tangency.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.07

'Review of necessary and sufficient conditions'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.08

'How should the parabola y=3x^2-6x+5 be translated to overlap with the parabola y=3x^2+9x?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.09

'Maximum and minimum of a function when the graph moves'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.10

'What are the maximum and minimum values of a function when the entire domain is considered?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.11

'Maxima and Minima of a Two-Variable Function'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.12

'Max value is 0 at x=0, min value is 8(a+1) at x=2'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.13

'Let a be a constant, find the minimum value m(a) of the function f(x)=(1+2a)(1-x)+(2-a)x.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.14

'[3] The graph passes through 3 points (1,3), (2,5), (3,9)\nLet the 2nd degree function be y=ax^{2}+bx+c\nSince it passes through (1,3), we have 3=a*1^{2}+b*1+c\nSince it passes through (2,5), we have 5=a*2^{2}+b*2+c\nSince it passes through (3,9), we have 9=a*3^{2}+b*3+c\nBy solving this system of equations, we can find the values of a, b, c, and determine the 2nd degree function.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.15

"Explain the relationships between a proposition, its converse, inverse, and contrapositive, and find the converse, inverse, and contrapositive of the following proposition S: Proposition S: 'If x is even, then x is divisible by 2.'"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.16

'Find the extreme values of the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.17

"(2) Let y' = -4x - 8 = -4(x + 2), y' = 0, then when x = -2, the table of increase and decrease of y is as follows on the right. Therefore, y takes the maximum value of -4 at x = -2."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.18

'For the function f(x)=x3+\x0crac32x26x f(x)=x^{3}+\x0crac{3}{2} x^{2}-6 x , answer the following questions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.19

'Range of values for the constant k when f(x)=x^{4}-8 x^{3}+18 k x^{2} does not have a maximum value.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.20

'Let f(x) = x^{3} - 6x^{2} + 9x + 1. Prove that the curve y = f(x) is symmetric with respect to the point A(2,3) on the curve.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.21

'Find the value of the constant a such that the areas enclosed by the curve y=x^{3}+x^{2} and the line y=a^{2}(x+1) are equal. It is given that 0<a<1.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.22

"Equation of tangent\nAt point A(a, f(a)) on the curve y=f(x)\n- Equation of tangent: y-f(a)=f'(a)(x-a)\n- Equation of normal: y-f(a)=-1/f'(a)(x-a)"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.23

"(2) y' = 3x^2 + 2x - 1 = (x + 1)(3x - 1) when y' = 0, x = -1, 1/3, the table of increase and decrease of y is as follows. Therefore, y reaches a maximum at x = -1 and a minimum at x = 1/3."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.24

'According to the definition of the derivative, find the derivative of the following functions: (1) y=x^{2}-3 x+9 (2) y=-2 x^{3}+3 x^{2}-1'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.25

'Find the equation of the circle passing through the points (4,-1), (6,3), (-3,0).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.26

'A cube with a side length of 1 cm is increasing in size at a rate of 1 mm per second. Find the rates of change of its surface area and volume after 10 seconds (cm²/s, cm³/s).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.27

'The x-coordinate of the intersection points of the curve y = 2x ^ 3-5x ^ 2 + x + 2 and the x-axis are the solutions to the equation 2x ^ 3-5x ^ 2 + x + 2 = 0. Let P(x) = 2x ^ 3-5x ^ 2 + x + 2; thus, P(1) = 2-5 + 1 + 2 = 0. Therefore, P(x) = (x-1)(2x ^ 2-3x-2) = (x-1)(x-2)(2x + 1). The solutions to P(x) = 0 are x = 1, 2, -1/2. Hence, the curve looks as shown in the right figure, and the area S to find is S = ∫(-1/2 to 1)(2x ^ 3-5x ^ 2 + x + 2) dx + ∫(1 to 2)(-(2x ^ 3-5x ^ 2 + x + 2)) dx = [x ^ 4/2 - 5/3 x ^ 3 + x ^ 2/2 + 2x](-1/2 to 1) - [x ^ 4/2 - 5/3 x ^ 3 + x ^ 2/2 + 2x](1 to 2) = 2(1/2 - 5/3 + 1/2 + 2) - (2 ^ 4/2 - 5/3 * 2 ^ 3 + 2 ^ 2/2 + 2 * 2) - (1/2(-1/2) ^ 4 - 5/3(-1/2) ^ 3 + 1/2(-1/2) ^ 2 + 2 * (-1/2)) = 8/3 - 2/3 - (- 61/96) = 253/96'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.28

'Find the function f(x) that satisfies the following equations:\n(1) f(x) = 3x^2 - x + ∫[\u200b-1,1] f(t) dt\n(2) f(x) = 2x^2 + 1 + ∫[0,1] xf(t) dt'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.29

'Mathematics II'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.30

'Find the extreme values of the given function. Also, plot its graph. (1) y=x^{4}-2 x^{3}-2 x^{2}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.31

'65\n\\[\n\egin{array}{l}\n\\text { (1) } \oldsymbol{y}^{\\prime}=2(x)^{\\prime}+(1)^{\\prime}=2 \\cdot 1=2 \\\\\n\oldsymbol{y}^{\\prime}=3\\left(x^{2}\\right)^{\\prime}-6(x)^{\\prime}+(2)^{\\prime}=3 \\cdot 2 x-6 \\cdot 1 \\\\\n=6 \oldsymbol{x}-6\n\\end{array}\n\\]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.32

'Find the functions f(x) and g(x) that satisfy the conditions f(0)-g(0)=1, the derivative of the integral of {f(t)+g(t)} from 0 to x equals 5 x^{2}+11 x+13, and the integral of the derivative of {f(t)-g(t)} from 0 to x equals x^{2}+x.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.33

'Find the range of possible values of the function f(x)=∫(t^2-2t-3)dt when -3≤x≤3.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.34

'Find the tangent lines of curve y=x^{3}-4x with a slope of -1.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.35

'For the example problem 190 involving the movement of one end of an interval, assuming a > 0. For the function y=-x³+3x² with 0≤x≤a, find:\n(1) The maximum value.\n(2) The minimum value.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.36

'In the given function, determine the value of h so that the average rate of change is 4 as x varies from 1 to 1+h in the function f(x)=x^3-x^2.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.37

"Supplementary Example 178: Calculate Derivatives (2)\nUsing the formula on page 278, find the derivatives of the following functions:\n(1) y=(2x-1)(x+1)\n(2) y=(x^2+2x+3)(x-1)\n(3) y=(2x-1)^3\n(4) y=(x-2)^2(x-3)\nPage 278 'STEP UP'"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.38

'Using the formula on page 278, differentiate the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.39

'Find the range of values that the function f(x) = ∫[−3, x](t^2−2t−3)dt can take when x is in the interval [-3, 3].'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.40

'Find the derivatives of the following functions and calculate the derivative at x=0,1 for each.(1) y=5 x^{2}-6 x+4 (2) y=x^{3}-3 x^{2}-1 (3) y=x^{2}(2 x+1) (4) y=(x-1)(x^{2}+x+1)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.41

'Given the sequence {an}, where a1=2 and an+1=3an-n^2+2n. By considering a quadratic function g(n) such that the sequence {an}-g(n) forms a geometric sequence with a common ratio of 3, find an expression for an in terms of n.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.42

'Determine the values of constants a, b, c such that the function f(x) = ax^2 + bx + c satisfies the following 3 conditions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.43

'Find a function f(n) that satisfies the following condition: b_{n+1}+f(n+1)=-2(b_{n}+f(n))'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.44

'297 Basic Example 189 Determine Coefficients from Maximum and Minimum Values'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.45

'Please explain why it is not necessary to tackle important examples when establishing the basics.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.46

'Problem of seeking common solutions'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.47

'Find the minimum value of the function f(x, y) = x^2 - 4xy + 5y^2 + 2y + 2 when x ≥ 0, y ≥ 0. Also, determine the values of x and y at that point.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.48

'Find the following indefinite integral: \\[\\int (x-\\sin x) \\cos x \\,dx\\]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.49

'Calculate the following definite integral:\n\n\\[\n\\int_{\\frac{a}{2}}^{a} \\frac{f(x)}{f(x)+f(a-x)} d x \n\\]\nLet x = a - t, then -dx = dt. The correspondence between x and t is as follows:\n \ x \\frac{a}{2} \\longrightarrow a \\n \ t \\frac{a}{2} \\longrightarrow 0 \\n Therefore,\n\\[ I = \\int_{\\frac{a}{2}}^{a} \\frac{f(a-t)}{f(a-t)+f(t)} (-1) dt = \\int_{0}^{\\frac{a}{2}} \\frac{f(a-t)}{f(t)+f(a-t)} dt = \\int_{0}^{\\frac{a}{2}}\\left\\{1 - \\frac{f(t)}{f(t)+f(a-t)}\\right\\} dt = [t]_{0}^{\\frac{a}{2}} - \\int_{0}^{\\frac{a}{2}} \\frac{f(t)}{f(t)+f(a-t)} dt = \\frac{a}{2} - b \\]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.50

'Find the following definite integrals. (1) int0frac12sqrt12x2dx\\int_{0}^{\\frac{1}{2}} \\sqrt{1-2 x^{2}} d x (2) int0sqrt3fracx2sqrt4x2dx\\int_{0}^{\\sqrt{3}} \\frac{x^{2}}{\\sqrt{4-x^{2}}} d x'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.51

'Find the maximum and minimum of the function represented by a definite integral (1)\n For real numbers a, b, find the minimum value of the definite integral intpipi(xasinxbcosx)2dx\\int_{-\\pi}^{\\pi} (x-a \\sin x - b \\cos x)^{2} dx. Also, determine the values of a and b at that time.\n[Shinshu University]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.52

'Practice 101 ⇒ Book p.452 (1) ∫ 1 / (√(x + 2) - √(x)) dx = ∫ (√(x + 2) + √(x)) / (x + 2 - x) dx (2) ∫ 2x / (√(x^2 + 1) + x) dx = ∫ 2x (√(x^2 + 1) - x) / ((x^2 + 1) - x^2) dx'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.53

'Find the extreme values of the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.54

'What to learn in this chapter》 In general, functions other than those expressed by polynomials are often not easy to integrate, even though they can be differentiated. In this chapter, we will learn more about integration methods for a wider variety of functions based on the differentiation formulas from Chapter 3. In the integration methods, even the indefinite integrals of rational functions may go beyond the scope of high school mathematics, so not all functions can always be integrated. However, the range of integrable functions is much broader than in Mathematics II.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.55

'Compute the following indefinite integral.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.56

'Differentiate the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.57

'\\[\egin{array}{l}\\int \\sqrt{x^{2}+1} d x=x \\sqrt{x^{2}+1}-\\left(\\int \\sqrt{x^{2}+1} d x-\\int \\frac{1}{\\sqrt{x^{2}+1}} d x\\right) \\\\\\text { Therefore, } \\quad 2 \\int \\sqrt{x^{2}+1} d x=x \\sqrt{x^{2}+1}+\\int \\frac{1}{\\sqrt{x^{2}+1}} d x\\end{array}\\]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.58

'Evaluate the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.59

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.60

'(2) Calculate \\int_{0}^{1} \\frac{x^{2}}{1+x^{2}} dx = \\int_{0}^{1}(1-\\frac{1}{1+x^{2}}) dx = \\int_{0}^{1} dx - \\int_{0}^{1} \\frac{1}{1+x^{2}} dx = 1 - \\int_{0}^{1} \\frac{1}{1+x^{2}} dx \\quad \\cdots \\cdots \\text{ (3) } \egin{\overlineray}{rl||l}x & 0 \\longrightarrow 1 \\hline\\= \\tan \\theta \\text{ Let} \\\\dx & = \\frac{1}{\\cos ^{2} \\theta} d \\theta \\quad 0 \\longrightarrow \\frac{\\pi}{4} \\end{\overlineray} \\text{ Therefore, } \\int_{0}^{1} \\frac{1}{1+x^{2}} dx = \\int_{0}^{\\frac{\\pi}{4}} \\frac{1}{1+\\tan ^{2} \\theta} \\cdot \\frac{1}{\\cos ^{2} \\theta} d \\theta = \\int_{0}^{\\frac{\\pi}{4}} \\cos ^{2} \\theta \\cdot \\frac{1}{\\cos ^{2} \\theta} d \\theta = \\int_{0}^{\\frac{\\pi}{4}} d \\theta = \\frac{\\pi}{4}.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.61

"Schwarz's inequality"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.62

"(x is a variable unrelated to t) ∫_(h(x))^(g(x)) f(t) dt = f(g(x)) g'(x) - f(h(x)) h'(x)"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.63

"Chapter 4 Applications of Differentiation\n19 Velocity and Acceleration, Approximation\nStudy/Maclaurin Expansion, Euler's Formula\nExercises"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.64

'Determine the minimum value of the definite integral I=intpipi(xasinxbsin2x)2dxI = \\int_{-\\pi}^{\\pi}(x-a \\sin x-b \\sin 2x)^{2} dx as the real numbers aa and bb vary, and find the values of aa and bb at that point.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.65

'Find the following indefinite integral.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.66

'Assuming the point P moves along the number line and its velocity at time t is 12-6t. Calculate the distance traveled by point P from t=0 to t=5.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.67

'Practice differentiating the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.68

'Practice 97 \\Rightarrow This book p.447\n(1) \\\int x \\sin 2 x dx\'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.69

'Calculate the definite integrals in column 40'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.70

"Increase and Decrease of Functions\nIn Mathematics II, we intuitively consider a curve y=f(x) by approximating it with its tangent line. In Mathematics II, we can theoretically prove it using the mean value theorem.\nThe function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b).\n1. If f'(x) > 0 for all x in the open interval (a, b), then f(x) is monotonically increasing on the closed interval [a, b].\n2. If f'(x) < 0 for all x in the open interval (a, b), then f(x) is monotonically decreasing on the closed interval [a, b].\n3. If f'(x) = 0 for all x in the open interval (a, b), then f(x) is constant on the closed interval [a, b]."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.71

'Translate the given text into multiple languages.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.72

'Explain how to find the derivative of a function represented by parametric variables.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.73

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.74

'Chapter 6: Applications of Integration'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.75

'Find the following indefinite integral'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.76

"When the coordinates (x, y) of point P are given as a function of time t, drawing perpendicular lines PQ and PR from P to the x-axis and y-axis respectively, the velocity of Q at time t is dx/dt=f'(t), and the velocity of R is dy/dt=g'(t). A vector v composed of these velocities is called the velocity or velocity vector of point P at time t. The magnitude of v, |v|, is called speed."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.77

"Since the cubic function f(x) has local extrema at x=1 and x=2, it can be expressed as f'(x)=a(x-1)(x-2)(a≠0). Also, given g(x)=3x/(2√(x^2+1))+1, we have g'(x)=3/2 * ((x^2+1)-x^2) / ((x^2+1)√(x^2+1)) = 3 / (2(x^2+1)√(x^2+1)). The condition for the curves y=f(x) and y=g(x) to have a common tangent at the point (0,1) is f'(0)=g'(0)."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.78

'Find the definite integral of a function f(x) that satisfies the following conditions:\n\n1. f(x) is an odd function y where f(-x)=-f(x) always holds.\n2. The range of the definite integral is [-a, a].'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.79

'Let ( ) be a constant greater than 1. Show that there exists a tangent to the curve ( ) passing through the origin ( ) when a differentiable function ( ) satisfies ( ).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.80

'Calculate the following definite integral. \\[ \\int_{0}^{1} x^{2}(x-1)^{2} e^{2 x} \\,dx \\]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.81

'Practice 38 ↠ This book p.341 Let h(x)=f(x)-g(x). Consider continuous functions f(x), g(x) on the interval [a, b]. For example, suppose f(x) is maximum at x=x and minimum at x=x^2, and g(x) is maximum at x=x^3 and minimum at x=x^4.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.82

'Find the value of the following definite integral \\ (2) \\\ \\int_ {0} ^ {9} \\ frac {1} {\\ sqrt {x +16} + \\ sqrt {x}} dx \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.83

'Math II\n\n[Question 1]\nLet I = ∫[0, π] sin(mx) cos(nx) dx.\n(1) When m - n ≠ 0, i.e., m ≠ n\n\nSolve the problem by transforming sin(mx)cos(nx):\n\nCalculate I = ∫[0, π] (1/2) {sin((m+n)x) + sin((m-n)x)} dx.\n\nWhat is I when m+n is even?\nWhat is I when m+n is odd?\n\n(2) When m - n = 0, i.e., m = n\n\nWhat is I in this case?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.84

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.85

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.86

'Consider the variable transformation of x, y and show the following equation: \\( \\left(\\frac{d x}{d \\theta}\\right)^{2} + \\left(\\frac{d y}{d \\theta}\\right)^{2} \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.87

'Given \\( y=(x-1)^{2}(x-2)^{3}(x-3)^{-5} \\), we have\n\\[\n\egin{aligned}\ny^{\\prime}= & 2(x-1)(x-2)^{3}(x-3)^{-5}+(x-1)^{2} \\cdot 3(x-2)^{2}(x-3)^{-5} \\\\\n& +(x-1)^{2}(x-2)^{3} \\cdot(-5)(x-3)^{-6} \\\\\n= & (x-1)(x-2)^{2}(x-3)^{-6} \\\\\n& \\times\\{2(x-2)(x-3)+3(x-1)(x-3)-5(x-1)(x-2)\\} \\\\\n= & (x-1)(x-2)^{2}(x-3)^{-6}(-7 x+11)\n\\end{aligned}\n\\]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.88

'Examine the concavity of the curve y=\\frac{4x}{x^{2}+1} and find the inflection points.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.89

'(2) \ \\int \\sin \\theta \\cos \\theta d \\theta = \\int \\frac{1}{2} \\sin 2 \\theta d \\theta\'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.90

'Find the following indefinite integrals: (1) \ \\int x^{2} \\cos x dx \ (2) \ \\int x^{2} e^{x} dx \ (3) \ \\int x \\tan ^{2} x dx \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.91

'Let V(t) be the volume of the solid obtained by rotating the region represented by the system of inequalities {0 ≤ y ≤ sin x, 0 ≤ x ≤ t - y} around the x-axis when 660<t<3. Find the value of t where dV(t)/dt=π/4 and the corresponding value of V(t).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.92

'Find the following integral.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.93

"Taylor's theorem"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.94

'Using formula (5), find the following integral.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.95

'Mathematics I'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.96

'Practice book 65 p.394'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.97

'Find the following definite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.98

'Calculate the area enclosed by the normal distribution curve (Gaussian integral)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.99

'Provide the definition of the derivative of the function f(x) at any point x=a.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.00

'Evaluate the following indefinite integrals. Here, a is a constant.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.01

'Find the following indefinite integrals. Note that the x in (4) is independent of t.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.02

'At (1) Maximum value of 7 at x = 0; minimum value of -9 at x = ±2. (2) Minimum value of -26 at x = 3.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.03

'In the interval c ≤ y ≤ d, the function f(y) is always greater than or equal to 0.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.04

'Find the range of values for the constant a such that the function f(x)=x^{3}+a x^{2}+(3 a-6) x+5 has extreme values.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.05

'Find the extreme values of the following functions and sketch the general shape of their graphs. (1) y=3 x^{4}-16 x^{3}+18 x^{2}+5 (2) y=x^{4}-8 x^{3}+18 x^{2}-11'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.06

'Find the values of the constants alpha,\eta(alpha<\eta) \\alpha, \eta(\\alpha<\eta) for which the equation int01f(x)dx=frac12f(alpha)+f(\eta) \\int_{0}^{1} f(x) dx=\\frac{1}{2}\\{f(\\alpha)+f(\eta)\\} holds for any quadratic function f(x) f(x) .'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.07

"Schwarz's inequality"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.08

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.09

'Explain about the increase and decrease of a function and the concept of maximum and minimum.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.10

'Let a be a positive constant. Prove that the area enclosed by the tangent line at any point P on the parabola y=x^{2}+a and the parabola y=x^{2} is constant regardless of the position of point P, and find the constant value.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.11

'Find the derivative of the following functions and determine the extreme values: (1) y=x^{3}+2x^{2}+x+1 (2) y=6x^{2}-x^{3} (3) y=x^{3}-12x^{2}+48x+5'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.12

'Find the extremum of the following functions and sketch their graphs.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.13

'Starting from the mathematical equation \ \\Pi \, we have (2) \\( \\int_{x}^{a} f(t) d t=-x^{3}+2 x-1 \\) which gives \\[ \\int_{a}^{x} f(t) d t=x^{3}-2 x+1 \\], and differentiating both sides of (2) with respect to \ x \ gives us \\( f(x)=3 x^{2}-2 \\). Furthermore, by setting \ x=a \ in (2), the left side becomes 0, hence \ 0=a^{3}-2 a+1 \, therefore \\( (a-1)\\left(a^{2}+a-1\\right)=0 \\), so \ a=1, \\frac{-1 \\pm \\sqrt{5}}{2} \, thus \\( f(x)=3 x^{2}-2 ; a=1, \\frac{-1 \\pm \\sqrt{5}}{2} \\) . \\[ \egin{array}{l}\\leftarrow \\int_{x}^{a} f(t) d t=-\\int_{a}^{x} f(t) d t \\leftarrow \\frac{d}{d x} \\int_{a}^{x} f(t) d t=f(x) \\leftarrow \\int_{a}^{a} f(t) d t=0\\end{array} \\] \ \\leftarrow \ Using the factor theorem.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.14

"Indefinite integral of x^n. When n is a positive integer (x^n)'=n x^(n-1) (refer to p.314) Here, by replacing n with n+1, (x^(n+1))'=(n+1) x^n, hence (x^(n+1)/(n+1))'=x^n Therefore, (1) holds."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.15

'Find the function f(x) and the value of the constant a that satisfy the following equations: (1) ∫_{a}^{x} f(t) d t=2 x^{2}-9 x+4 (2) ∫_{x}^{a} f(t) d t=-x^{3}+2 x-1'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.16

'Find the function f(x) that satisfies the following equations:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.17

'Find the maximum and minimum values of the following functions. Also, find the corresponding values of x.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.18

'Explain the properties of indefinite integrals using constants k and l.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.19

'Determination of undetermined coefficients in set 16 (2) [Numerical Substitution Method]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.20

'Explain how to find extreme values.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.21

'Find the derivative of the function y=2x^{3}-3x^{2}-12x+5 at x=1.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.22

'Translate the given text into multiple languages.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.23

'Indefinite integral of 2x^n is ∫ x^n dx= (1/(n+1)) x^(n+1) + C (where n is 0 or a positive integer)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.24

'Find the area between the curves y = x^2 and y = 2x.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.25

'Therefore, when the line (2) passes through the point (10, 50), the value of the y-intercept l/3 of the line (2) will be maximum. At this point, l will also be maximum. Hence, the profit x+3y will be maximum at (x, y) = (10, 50).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.26

'Find f(x) and g(x) that satisfy the given functions f(x) and g(x)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.27

'Find the following definite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.28

'If the radius of a sphere increases at a rate of 10 cm per second starting from 1 meter, find the rate of change of the surface area of the sphere after 30 seconds.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.29

'Find the extreme values of the following functions and sketch the graph.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.30

"Find a cubic function f(x) that satisfies the following conditions: f'(1)=f'(-1)=1, f(1)=0, f(-1)=2."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.31

'Examine the concavity of the following curves and find the points of inflection: \n(1) y=x^{4}+2 x^{3}+2 \n(2) y=x+cos 2 x(0 ≤ x ≤ π) \n(3) y=x e^{x} \n(4) y=x^{2}+1/x'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.32

'Let a be a non-zero constant. Let A = ∫_{0}^{π} e^{-a x} sin 2x dx and B = ∫_{0}^{π} e^{-a x} cos 2x dx. Find the values of A and B.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.33

'Prove the inequality and limit (using the squeeze theorem)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.34

'Integration by substitution and integration by parts for indefinite integrals'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.35

'Find the differentiable function f(x) such that the slope of the tangent line at the point (x, y) on the curve passing through (1,0) is x√x.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.36

'For the function y of x defined by the following equation, express dy/dx and d^2y/dx^2 in terms of x and y, respectively.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.37

'Let a and b be real numbers. Find the minimum value of the integral ∫{0}{1}{cosπx-(ax+b)2}dx as the values of a and b vary, and determine the values of a and b at that time.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.38

'Differentiate the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.39

'Explain the following approximate formulas:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.40

'(2) Let Iₙ=∫0π/4 tanⁿxdx (where n is a natural number). Express Iₙ for n >= 3 in terms of n and Iₙ-2. Also, find the values of I₃, I₄. [Similar to Yokohama National University]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.41

'(3) Solve using the sum of products formula: intsin3xsin2xdx=frac12int(cos5xcosx)dx\\int \\sin 3x \\sin 2x dx = -\\frac{1}{2} \\int (\\cos 5x - \\cos x) dx.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.42

'Find the extremum of the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.43

'Find the second derivative and the third derivative of the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.44

'Find the indefinite integral of f(ax+b).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.45

'Find the antiderivative of basic functions'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.46

'Differentiate the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.47

'Please calculate the derivative.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.48

'Find the following indefinite integral: #2)∫√(x^2+a^2) dx'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.49

'Find the indefinite integrals of the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.50

'Find the indefinite integral ∫(sin x + x cos x) dx. Also, using this result, find the indefinite integral ∫(sin x + x cos x) log x dx.\n\n[Rikkyo University]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.51

"Let the inverse function of the function f(x) be g(x). When f(1)=2, f'(1)=2, f''(1)=3, find the value of g''(2)."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.52

'Examine the increase and decrease of the function. (2) y=\\frac{x^{3}}{x-2} '

A. ...

Ask AI tutor for answer!
Join our Discord

Q.53

'Differentiate the function y=x^{3}sqrt{1+x^{2}}.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.54

'Differentiate the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.55

"From mathematics (4) (1), we have f(x+y)-f(x)=f(y)+8xy. Therefore, f'(x)=lim _{y \\rightarrow 0} \\frac{f(x+y)-f(x)}{y}=lim _{y \\rightarrow 0} \\frac{f(y)+ 8xy}{y}=lim _{y \\rightarrow 0}\\left\\{\\frac{f(y)}{y}+8 x\\right\\}=3+8 x"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.56

'Find the inflection points of the curve y=x^{3}+3 x^{2}-24 x+1.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.57

'Define the derivative of a function.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.58

'Using the example above, calculate the following definite integrals:\n1. \ \\int_{0}^{\\frac{\\pi}{2}} \\sin^{6} x \\cos^{3} x d x \\n2. \ \\int_{0}^{\\frac{\\pi}{2}} \\sin^{5} x \\cos^{7} x d x \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.59

'Using the method of integration by parts, evaluate the following definite integral.\\\int_{0}^{1} x^n e^{-x} dx\ (where n is a non-negative integer)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.60

'Find the definite integral \ \\int_{0}^{1} \\frac{1}{x^{3}+1} dx \.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.61

'Find the indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.62

'For an even function f(x) f(x) , find intaaf(x)dx \\int_{-a}^{a} f(x) dx . Here, an even function is a function that satisfies f(x)=f(x) f(-x) = f(x) .'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.63

'For an odd function f(x) f(x) , find \\[ \\int_{-a}^{a} f(x) dx \\]. Here, an odd function is a function that satisfies f(x)=f(x) f(-x) = -f(x).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.64

'Explain the second-order approximation formula.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.65

'Differentiate the following functions according to the definition of derivatives: (1) y=\\frac{1}{x^{2}} (2) y=\\sqrt{4 x+3} (3) y=\\sqrt[4]{x}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.66

'Equations of Tangent and Normal\nAt a point \\( \\mathrm{A}(a, f(a)) \\) on the curve \\( y=f(x) \\)\n[1] The equation of the tangent is \\( y-f(a)=f^{\\prime}(a)(x-a) \\)\n[2] The equation of the normal is, when \\( f^{\\prime}(a) \\neq 0 \\)\n\\[ y-f(a)=-\\frac{1}{f^{\\prime}(a)}(x-a) \\]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.67

'Using mathematical induction, prove that the sequence ana_{n} satisfies 0<an<2(n=1,2,3,cdotscdots)0<a_{n}<2 (n=1,2,3, \\cdots \\cdots).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.68

'Differentiate the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.69

'Evaluate `∫_{0}^{π} e^{-x} sin x dx`.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.70

'Find the derivative of the following f(x) using each method: f(x)=x^{1/3} (x>0) (1) Find according to the definition of derivative. (2) f(x)・f(x)・f(x)=x, apply the formula for the derivative of a product.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.71

'Mean Value Theorem'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.72

'Differentiate the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.73

'Find the derivative of the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.74

'Find the maximum value of f(x) = ∫₀ˣ eˣᶜᵒˢᵗ dt (0 ≤ x ≤ 2π) and the corresponding value of x.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.75

'Please compare the growth rates of the functions \ x^{p} \ and \\( x^{q}(0<p<q) \\).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.76

'Using definite integration and recurrence relation, find the following definite integral.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.77

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.78

'For a curve with the equation F(x, y) = 0 or parameterized as x = f(t), y = g(t), the equation of the tangent line at a point (x1, y1) on the curve is y - y1 = m(x - x1), where m is the slope obtained by substituting x = x1, y = y1 into the derivative dy/dx.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.79

'(3) \\( V = \\pi \\int_{1}^{4}\\left(x+\\frac{1}{\\sqrt{x}}\\right)^{2} d x \\)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.80

"Prove that if the function f(x) is continuous on the interval [a, b] and differentiable on the interval (a, b), then there exists a real number c such that [f(b) - f(a)] / (b - a) = f'(c) with a < c < b."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.81

'Find the following definite integral. (2) \ \\int_{0}^{2} \\frac{d x}{\\sqrt{16-x^{2}}} \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.82

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.83

'Explain how to find integrals using the rules of differentiation.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.84

'Motion of points on a plane'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.85

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.86

'Differentiate the following functions. In this question, (6) where a is a constant.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.87

'Find the indefinite integral \ \\int e^{x} \\sin x \\, d x \.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.88

'Prove that when the continuous function f(x) satisfies f(π-x)=f(x) for all real numbers x, then the integral from 0 to π of (x-π/2)f(x)dx=0. Also, using this, find the definite integral ∫₀ᵠ of xsin³x / 4-cos²x dx.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.89

'Find the second derivative of the following function: (1) y=x^3−3x^2+2x−1'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.90

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.91

'Use substitution method to solve the following integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.92

'For a natural number n, let S_n=∫[0,1] (1-(-x)^n)/(1+x) dx, T_n=Σ[k=1,n] (-1)^(k-1)/k(k+1).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.93

'Find the area enclosed by the curve y=f(x) and y=g(x) between the lines x=a and x=b.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.94

'(1) Solve using the half-angle formula intcos22xdx=intfrac1+cos4x2dx \\int \\cos^{2} 2x dx=\\int \\frac{1+\\cos 4x}{2} dx .'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.95

'Examine the increase and decrease of the following function:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.96

'Assuming that one-third of the people living outside Tokyo move into Tokyo each year, and one-third of the people living in Tokyo move out of Tokyo. Let an be the population outside Tokyo and bn be the population inside Tokyo in the nth year, find lim(n→∞)an/bn. It is assumed that the total population of Tokyo inside and outside is constant regardless of the year.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.97

'(2) Find the definite integral \\( \\int_{0}^{1}\\{x(1-x)\\}^{\\frac{3}{2}} d x \\).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.98

'(2) \ \\int \\sin^{3} x dx=\\int \\frac{3 \\sin x-\\sin 3x}{4} dx \ Solve using the triple angle formula.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.99

'Find y^{\\prime \\prime}(0) when the function y(x) has a second derivative y^{\\prime \\prime}(x) and satisfies x^{3}+(x+1)\\{y(x)\\}^{3}=1.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.00

'(4) int1efraclogxxdx\\int_{1}^{e} \\frac{\\log x}{x} d x'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.01

'Differentiate the following functions according to the definition of derivatives.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.02

'Differentiate the function y=x^3√(1+x^2).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.03

'Practice: Find the volume V of the solid obtained by rotating the region enclosed by the following curves or lines around the y-axis.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.04

'The condition for the inverse function to be equal to the original function is f1(x)=f(x)f^{-1}(x)=f(x)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.05

'Evaluate the following definite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.06

'Find the second derivative of the following function. (1) y=√[3]{x}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.07

'The composite function \\( f(x) \\) satisfies the condition \\( f(x+y)+f(x) f(y)=f(x)+f(y) \\) for any real numbers \ x, y \, and is differentiable at \ x=0 \ with \\( f^{\\prime}(0)=1 \\).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.08

'Math (2) ∫ 1 / (x^2 - 4) dx = 1 / 4 ∫ [ 1 / (x - 2) - 1 / (x + 2) ] dx When the denominators are cancelled in this way...'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.09

'Find the extreme values of the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.10

'Find the definite integral \ \\int_{0}^{1} \\frac{x^2+2}{x+2} dx \.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.11

'Prove the following equation: \ \\int_{-1}^{0} \\frac{x^{2}}{1+e^{x}} d x=\\int_{0}^{1} \\frac{x^{2}}{1+e^{-x}} d x \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.12

'Find the extreme values of the function f(x) = ∫₀ˣ(1-t²) eˣᵗ dt. [Tokyo University of Mercantile Marine]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.13

'Find the definite integral \\( \\int_{0}^{1} \\frac{2 x+1}{(x+1)^{2}(x-2)} d x \\).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.14

"Considering F'(t) = 1/(t²+1) "

A. ...

Ask AI tutor for answer!
Join our Discord

Q.15

'Let the function f(x) be continuous on the interval [a, b] and differentiable on the interval (a, b).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.16

'Who discovered calculus?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.17

'Find the area of the region enclosed by two curves and a line: Calculate the area S enclosed by the two curves y=e^x, y=1/(x+1), and the line x=1.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.18

"Let's try to calculate the volume by cutting with a plane that is perpendicular to the y-axis. Take point Q on the y-axis, where OQ=y, let the cross-sectional area be S(y), and then calculate V= \\int_{0}^{a} S(y) dy."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.19

'Find the derivatives of the following functions according to the given definitions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.20

'Find the extreme values of the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.21

'(2) By considering the result obtained in (1) as a function of x and differentiating it, find the sum 1+2x+3x^2+...+nx^(n-1) when x is not equal to 1.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.22

'Find the indefinite integral of f(ax+b).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.23

'How to find the inverse function?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.24

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.25

'(1) Using (1), find the following definite integral.\n(a) \ \\int_{0}^{\\frac{\\pi}{2}} \\sin^{7} x dx \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.26

'Find the following indefinite integrals: \n(1) \\( \\int \\frac{x-1}{(2 x+1)^{2}} d x \\)\n(2) \ \\int \\frac{9 x}{\\sqrt{3 x-1}} d x \\n(3) \ \\int x \\sqrt{x-2} d x \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.27

"Find the maximum value, minimum value, and the corresponding values of x of the following functions:\n(1) y = \\frac{2(x-1)}{x^{2}-2x+2}\n[Tokyo Women's Medical University]\n(2) y = (x+1)\\sqrt{1-x^{2}}\n[University of Technology throughout Long Naga]\n"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.28

"(3) Find the function f(x) such that f'(0) = 2."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.29

'Find the derivative of the following functions according to the definition.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.30

'Find \ \\frac{d y}{d x} \ for the function \ y \ defined by the following equations: (1) \ y^{2}=2 x \, (2) \ 4 x^{2}-y^{2}-4 x+5=0 \, (3) \ \\sqrt{x}+\\sqrt{y}=1 \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.31

'Find the indefinite integral: \ \\int_{0}^{1} \\frac{dx}{2+3e^x+e^{2x}} \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.32

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.33

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.34

'Find the definite integral \ I=\\int_{0}^{2 \\pi} \\cos m x \\cos n x d x \. [Similar to Hokkaido University]'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.35

'Differentiate the following functions. Where a is a constant.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.36

"Find the definite integral ∫[a,b] f(g(x)) g'(x) dx using the substitution method for integrals."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.37

"Basic Example 53 Derivatives and Identities\nLet f(x) be a polynomial of degree 2 or higher.\n(1) Express the remainder when f(x) is divided by (x-a)^2 in terms of a, f(a), f'(a).\n(2) Find the condition for f(x) to be divisible by (x-a)^2."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.38

'Find the following definite integral: \ \\int_{-1}^{1} \\frac{x^{2}}{1+e^{x}} d x \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.39

'13 \\n(1)\\n\\\y^{\\prime} =3 \\cdot 4 x^{3}+2 \\cdot 3 x^{2}-1 \\\\ =12 x^{3}+6 x^{2}-1\\\'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.40

'Differentiate the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.41

'Find the derivative of the composite function h(x)=f(g(x)).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.42

'Find the extreme values of the following functions. (1), (3) Nihon Joshi Daigaku'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.43

"Start with the 'self you want to become' and work backwards."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.44

'Concave and Convex Curves, Inflection Points'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.45

'(1) Find the definite integral \ \\int_{0}^{\\frac{\\pi}{2}}\\left|\\cos x-\\frac{1}{2}\\right| dx \.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.46

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.47

'Using (1), find the following definite integral.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.48

'Calculate the following definite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.49

'Find the extreme values of the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.50

'Find the following indefinite integrals. (1) \ \\int \\sin ^{2} x \\cos x d x \ (2) \\( \\int x\\left(x^{2}+1\\right)^{3} d x \\) (3) \ \\int \\frac{2 x+4}{x^{2}+4 x+1} d x \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.51

'Find the following indefinite integral.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.52

'Find the real number \ k \ that minimizes the integral value \\( \\int_{0}^{\\frac{\\pi}{2}}(\\sin x-k x)^{2} d x \\) and the integral value at that point.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.53

'Calculate dy/dx using the method of parametric differentiation.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.54

''

A. ...

Ask AI tutor for answer!
Join our Discord

Q.55

'Differentiate the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.56

"Let f(x)=(a x^{2}+b x+c) e^{-x} for constants a, b, c. Find the values of a, b, c when f'(x)=f(x)+x e^{-x} holds for all real numbers x."

A. ...

Ask AI tutor for answer!
Join our Discord

Q.57

'When the infinite series converges, let its sum be denoted as f(x). Draw the graph of the function y=f(x) and examine its continuity.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.58

'Let 1+x^{2}=u, then 2xdx=du, so ∫x cos(1+x^{2})dx=1/2∫cosudu=1/2sinu+C=1/2sin(1+x^{2})+C'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.59

'Differentiate the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.60

"Why wasn't calculus discovered in ancient Greece?"

A. ...

Ask AI tutor for answer!
Join our Discord

Q.61

'Using the second derivative, find the extreme values of the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.62

'Prove that when y=cos x, the nth derivative of y is cos(x+nπ/2).'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.63

'When finding the indefinite integral of a rational function, how should we proceed if the degree of the numerator is higher than the denominator or if the denominator is in the form of a product of multiple factors?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.64

'Differentiate the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.65

'Differentiate the following function. (1) y=\\sqrt[3]{x^{2}(x+1)}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.66

'Using the derivative rules, calculate the derivatives of the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.67

'Solve the following problem: Find the definite integral of the absolute value of t^3 over the interval 0 to 2.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.68

'Differentiate the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.69

'Find the following indefinite integral: \n\ \\int_{1}^{4} \\frac{d x}{\\sqrt{3-\\sqrt{x}}} \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.70

'The sequences {an},{bn} converge, with lim{n→∞} an=α, lim{n→∞} bn=β. Explain the following properties: 1) Constant multiple lim{n→∞} k an=k α, where k is a constant 2) Sum lim{n→∞}(an+bn)=α+β; Difference lim{n→∞}(an-bn)=α-β 3) lim{n→∞}(k an+l bn)=k α+l β, where k, l are constants 4) Product lim{n→∞} αn bn=α β 5) Quotient lim{n→∞} an/bn=α/β, where β ≠ 0.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.71

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.72

'Find the derivative of the following functions.\n(1) y=\\left(x^{2}-2\\right)^{3}\n(2) y=(1+x)^{3}(3-2 x)^{4}\n(3) y=\\sqrt{\\frac{x+1}{x-3}}\n(4) y=\\frac{\\sqrt{x+1}-\\sqrt{x-1}}{\\sqrt{x+1}+\\sqrt{x-1}}'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.73

'Calculate Example 134 of definite integral (using equations)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.74

'Create a first-order approximation for the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.75

'Explain the conditions for a function to have extremum.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.76

'Differentiate the following functions with respect to x:'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.77

'Find the derivative of the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.78

'Find the definite integral.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.79

'The definite integral of an even function or an odd function f(x) is'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.80

'Differentiate the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.81

'Translate the given text into multiple languages.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.82

'Chapter 3 Differentiation EX Differentiate the following functions: (1) y=(x^2-2)^3 (2) y=(1+x)^3(3-2x)^4 (3) y=√((x+1)/(x-3)) (4) y=(√(x+1)-√(x-1))/(√(x+1)+√(x-1))'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.83

'Find the value of c for which the mean value theorem conditions are satisfied for the given function f(x) and interval.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.84

'Differentiate the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.85

'Find the following definite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.86

'Find the following indefinite integrals.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.87

'Examine the concavity of the following curves and find any points of inflection.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.88

'List the exercises of the applications of integration method in Chapter 6 in the following order: 28 Area 29 Volume 30 Length of a curve 31 Speed and distance 32 Advanced differential equations Use Gaussian integration to find the area enclosed by the normal distribution curve.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.89

'Let a be a real number. Determine the range of values for a such that the function f(x)=ax+cosx+12sin2x does not have any extremum.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.90

'Using the second derivative, find the extreme values of the function y=x^{3}-3 x+1.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.91

'Find the third derivative of the following functions:\n(1) y = sin 2x\n(2) y = sqrt(x)\n(3) y = e^(3x)'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.92

'How can you make use of explanatory videos?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.93

'Find the following indefinite integral.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.94

'Determine the values of constants a, b, and c, when the inverse function of f(x)=a+\\frac{b}{2x-1} is g(x)=c+\\frac{2}{x-1}.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.95

'(1) Using (1), find the following definite integral. (イ) \ \\int_{0}^{\\frac{\\pi}{2}} \\sin^{3} x \\cos^{2} x dx \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.96

'Find the extreme values of the following functions.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.97

'Calculate the following indefinite integral. \ \\int_{0}^{\\frac{\\pi}{2}} \\frac{\\sin 2x}{3+\\cos^2 x} dx \'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.98

'Express \ \\frac{d^{2} y}{d x^{2}} \ in terms of \ x \ and \ y \ when \ x^{2}-y^{2}=a^{2} \. Here, \ a \ is a constant.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.99

'Find the following indefinite integral.'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.00

'What are some helpful ways of thinking when solving difficult problems? How does it benefit learning?'

A. ...

Ask AI tutor for answer!
Join our Discord

Q.01

The proposition 'Living in Tokyo \Longrightarrow Living in Japan' is true, but in this case, how can the relationship between 'Living in Tokyo' and 'Living in Japan' be expressed in terms of sufficient and necessary conditions?

A. ...

Ask AI tutor for answer!
Join our Discord

Q.02

Advanced 85 | Conditional Maximum and Minimum (1)

A. ...

Ask AI tutor for answer!
Join our Discord

Q.03

Given x0,y0,3x+2y=1 x \geqq 0, y \geqq 0, 3x + 2y = 1 , find the maximum and minimum values of 3x2+4y2 3x^2 + 4y^2 .

A. ...

Ask AI tutor for answer!
Join our Discord

Q.04

When the parabola y=3x2+6x+2 y=3 x^{2}+6 x+2 is translated 2 units in the x x -axis direction and -1 unit in the y y -axis direction, express the equation of the translated parabola in the form y=ax2+bx+c y=a x^{2}+b x+c .

A. ...

Ask AI tutor for answer!
Join our Discord

Q.05

Standard example: Problems that require some applied skills, such as using multiple pieces of knowledge.

A. ...

Ask AI tutor for answer!
Join our Discord