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## Functions and Analysis

### Advanced Functions - Fundamentals of Differentiation and Integration

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

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'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.'

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'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.'

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'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).'

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'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.'

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"(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"

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

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'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.'

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

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'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 \\]'

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"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."

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'Determining function from extreme value conditions'

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

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'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)'

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'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)'

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'(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.'

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

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

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'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.'

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

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'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\\]'

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

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'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.'

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

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'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.'

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'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)'

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

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

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'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.'

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'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 \\).'

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

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

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

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'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)?'

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'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'

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

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

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"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."

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

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

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'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.'

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

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'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.'

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

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

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

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

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

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

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

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

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"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"

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

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'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]'

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"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."

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

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

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

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"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."

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'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.'

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'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 \\)'

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"(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."

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

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'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.'

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

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

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'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.'

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'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:'

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'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 \\)'

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"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"

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

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'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.'

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'Calculate the amount and integral of water discharge.'

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

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'Explain the method of calculating derivatives by definition.'

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

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

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

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

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

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

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'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.'

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

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'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.'

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

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

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

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

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

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'[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.'

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"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.'"

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'Find the extreme values of the following functions.'

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"(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."

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

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'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.'

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'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.'

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'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.'

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"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)"

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"(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."

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'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'

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

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'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).'

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'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'

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'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'

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

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'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\\]'

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'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.'

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

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

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'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.'

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'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.'

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"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'"

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

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'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].'

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'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)'

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'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.'

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

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

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

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

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'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.'

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

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'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 \\]'

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

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'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 $\\int_{-\\pi}^{\\pi} (x-a \\sin x - b \\cos x)^{2} dx$. Also, determine the values of a and b at that time.\n[Shinshu University]'

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'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'

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'Find the extreme values of the following functions.'

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'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.'

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'\\[\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}\\]'

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'(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}.'

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

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

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

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'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.'

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'Practice differentiating the following functions.'

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

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"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]."

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'Translate the given text into multiple languages.'

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

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"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."

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"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)."

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'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].'

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'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 ( ).'

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

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'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.'

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

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'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?'

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'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} \\)'

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'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\\]'

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

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

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'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 \'

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'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).'

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

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

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

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

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

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

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'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.'

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'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'

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

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

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'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.'

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'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'

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

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'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.'

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"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."

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'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'

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

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

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

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

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

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'Translate the given text into multiple languages.'

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

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

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'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).'

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

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'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.'

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

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

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'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'

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'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.'

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

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

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'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.'

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'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.'

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'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.'

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'(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]'

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

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

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

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'Find the indefinite integrals of the following functions.'

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'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]'

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"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)."

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

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

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"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"

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

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'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 \'

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'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)'

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

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

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

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'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}'

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'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) \\]'

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

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'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.'

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

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

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

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'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.'

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

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"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."

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

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

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

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

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'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.'

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

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'Use substitution method to solve the following integrals.'

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'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).'

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

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

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

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'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.'

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

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

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'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.'

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

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

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

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

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'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 \\).'

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

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'Find the extreme values of the following functions.'

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

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'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 \'

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

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

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

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'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.'

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"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."

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

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'Find the extreme values of the following functions.'

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'(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.'

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

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'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 \'

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"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"

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

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'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 \'

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

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

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

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

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"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."

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

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

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

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

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

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

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

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'Find the extreme values of the following functions.'

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'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 \'

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'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.'

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

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"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."

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'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.'

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'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'

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

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

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

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'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?'

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

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

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

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

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'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.'

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'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}}'

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

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

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

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'Differentiate the following functions with respect to x:'

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

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'Translate the given text into multiple languages.'

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'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))'

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

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

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'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.'

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'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.'

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

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

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'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}.'

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

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'Find the extreme values of the following functions.'

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

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'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.'

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

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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?

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

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

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