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## Functions and Analysis

### Analysis - Single Variable Calculus

#### Q.01

'Important Example 79 | Maximum and Minimum in a Region (3)\nFind the maximum and minimum values of x²+(y-3)² when real numbers x, y satisfy the three inequalities y ≥ 2x-5, y ≤ x-1, y ≥ 0.\n[Tokyo Keizai University]\nExample 76'

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'Calculate the definite integral of the following expression.'

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'Practice finding the following definite integrals.\\n(1) \\( \\int_{-1}^{2}(x+1)(x-2) d x \\)\\n(2) \\( \\int_{-\\frac{1}{2}}^{3}(2 x+1)(x-3) d x \\)\\n(3) \\( \\int_{2-\\sqrt{7}}^{2+\\sqrt{7}}\\left(x^{2}-4 x-3\\right) d x \\)'

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'Similarly, consider the minimum value. Find that minimum value.'

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'Find the area between the parabola and the x-axis. (1) Basic'

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'Find the value of the following definite integrals.'

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'Find the following definite integrals. (1) \\( \\int_{1}^{2}(2 x-1) d x \\) (2) \\( \\int_{0}^{-1}\\left(3 x^{2}+6 x+1\\right) d x \\) (3) \\( \\int_{-1}^{3}(x+1)(x-3) d x \\) (4) \\( \\int_{-1}^{2}\\left(x^{3}-6 x-4\\right) d x \\) (5) \\( \\int_{-2}^{1}(2 t+1)^{2} d t+\\int_{-2}^{1} 2(t-1)^{2} d t \\)'

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'TRAINING 197 (1) Find the following definite integral. (1) \\( \\int_{-1}^{2}\\left(2 x^{2}-x+3\\right) d x \\)'

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

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

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'Find the minimum value of S(m), the sum of the areas enclosed by the curve y=x^2 and the line y=mx for 0<m<1, where 0 ≤ x ≤ 1.'

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'Find the area S between the curve y = f(x) and the x-axis.'

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'Using the properties of definite integrals, find the results of the following definite integrals:'

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'Master the properties of definite integrals to conquer example 198!'

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'What should be done when basic examples are not clear?'

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'Please solve a problem related to the Law of Large Numbers.'

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'Find the following definite integrals. In (4), \ a, b \ are constants. (1) \ \\int_{0}^{\\frac{1}{3}} x e^{3 x} d x \ (2) \ \\int_{1}^{e} x^{2} \\log x d x \ (3) \\( \\int_{1}^{e}(\\log x)^{2} d x \\) (4) \\( \\int_{a}^{b}(x-a)^{2}(x-b) d x \\) (5) \ \\int_{0}^{2 \\pi}\\left|x \\cos \\frac{x}{3}\\right| d x \'

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'Find the following definite integrals. (1) \ \\int_{0}^{2} \\frac{2x+1}{\\sqrt{x^2+4}} dx \(2) \ \\int_{\\frac{1}{2} a}^{\\frac{\\sqrt{3}}{2} a} \\frac{ \\sqrt{a^2-x^2 }}{x} dx \ (a > 0)'

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'The term includes explanations of concepts, proofs of theorems and formulas, making it easy to understand even for topics not covered in textbooks.'

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'Define necessary and sufficient conditions, and explain using the following example.'

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'Find the area enclosed by the following curves, the line, and the x-axis.\n(1) y=x^{2}-x-2\n(2) y=-x^{2}+3 x\n(-1 ≤ x ≤ 2),\nx=-1, x=2'

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'Find the following definite integrals.\n(1) \\( \\int_{-1}^{1}\\left(2 x^{3}-4 x^{2}+7 x+5\\right) d x \\)\n(2) \\( \\int_{-2}^{2}(x-1)\\left(2 x^{2}-3 x+1\\right) d x \\)'

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'Prove the following about the solutions α, β of the quadratic equation\nax^2 + bx + c = 0\n1. Condition for having two distinct real number solutions.\n2. When the inequality at^2 + 2bt + c > 0 holds for all real numbers t, it implies having only positive solutions.'

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'Find the area enclosed by the given curves and lines.'

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"Derivative The definition of the derivative of a function f(x) f'(x) is f'(x)=lim_{h→0}frac{f(x+h)-f(x)}{h}"

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'Compute the following definite integrals using the formulas.'

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"Thus, for all real numbers x, y'>0, so the given function always increases"

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'Prove the following condition. Let the values of the variable x be x1, x2, ..., xn. For a certain value t, consider the sum of squares of deviations of each value from t, t-xk (k=1, 2, ..., n) as y. That is, y=(t-x1)^2+(t-x2)^2+...+(t-xn)^2. Prove that y is minimized when t=𝑥¯ (the mean of x).'

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'Please explain what necessary and sufficient conditions are.'

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'Explain the similarities and differences between examples 3 to 8 and examples 9 to 12.'

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'Explain the concavity and points of inflection of a function graph.'

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'(2) $\\int t \\sqrt[4]{t^{3}} d t=\\int t^{\\frac{7}{4}} d t=\\frac{t^{\\frac{7}{4}+1}}{\\frac{7}{4}+1}+C=\\frac{4}{11} t^{\\frac{11}{4}}+C=\\frac{4}{11} t^{2} \\sqrt[4]{t^{3}}+C$'

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'(2)\\\\n\\\\[\\\egin{array}{l}\\\\n0 \\leqq|\\cos x| \\leqq 1, e^{-x}>0 \\text { so } \\quad e^{-x} \\geqq e^{-x}|\\cos x| \\\\\\\\n\\text { Therefore } \\quad a_{1}=\\int_{0}^{\\\\pi}\\left(e^{-x}-e^{-x}|\\cos x|\\right) d x \\\\\\\\n=\\left[-e^{-x}\\right]_{0}^{\\\\pi}-\\int_{0}^{\\\\frac{\\\\pi}{2}} e^{-x} \\cos x d x+\\int_{\\\\frac{\\\\pi}{2}}^{\\\\pi} e^{-x} \\cos x d x \\\\\\\\n=1-e^{-\\\\pi}-\\\\frac{1}{2}\\left[e^{-x}(\\\\sin x-\\\\cos x)\\right]_{0}^{\\\\frac{\\\\pi}{2}} \\\\\\\\n\\quad+\\\\frac{1}{2}\\left[e^{-x}(\\\\sin x-\\\\cos x)\\right]_{\\\\frac{\\\\pi}{2}}^{\\\\pi} \\\\\\\\n=\\\\frac{1}{2}\\left(1-2 e^{-\\\\frac{\\\\pi}{2}}-e^{-\\\\pi}\\right)\n\\end{array}\\\\]'

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'Volume of a rotation around the y-axis (3) Let a≤b. The volume V of a solid formed by rotating the portion of the graph of y=f(x) for a≤x≤b around the y-axis, bounded by the x-axis and the two lines x=a, x=b, is given by V=2π∫a to b xf(x)dx. Show for a<c<d<b, and for a function y=f(x) that decreases monotonically in the intervals [a, c], [d, b] and increases monotonically in the interval [c, d] as shown in the diagram on the right. Also, find the value V_0 of V when f(x)=x^3, a=0, b=2.'

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'Practice finding the following definite integral.'

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'In this chapter, we will learn about solving problems involving finding the area of specific shapes using integrals.'

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'Investigate the increasing and decreasing behavior of the differentiable function f(x) and find its extreme values.'

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

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'Let the function y=f(x) be continuous.\n(1) Let a be a real constant. For all real numbers x, if the inequality |f(x)-f(a)| ≤ 2/3|x-a| holds, prove using the intermediate value theorem that the curve y=f(x) intersects the line y=x.\n(2) Furthermore, if for all real numbers x1, x2, the inequality |f(x1)-f(x2)| ≤ 2/3|x1-x2| holds, prove that there is only one intersection point for (1).'

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'Learning to solve problems related to finding the area and volume of shapes, length of curves, and solving simple differential equations.'

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'Find the following definite integral.\\( \\int_{-1-\\sqrt{5}}^{-1+\\sqrt{5}}\\left(2 x^{2}+4 x-8\\right) d x \\)'

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

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'Area between the parabola and the x-axis: Find the area between the parabola y = x^2 and the x-axis from x = 0 to x = 1.'

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'Find the area of the figure enclosed by the curve y=x^{3}-5 x^{2}+2 x+6 and the tangent line at the point (3,-6) on the curve.'

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Join our Discord#### Q.63

'(2) Calculate the following definite integral:\n(a) \\( \\int_{2}^{3}(x-2)(x-3) d x \\)'

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"First, let's review the method of using differential calculus to determine the number of real solutions of an equation."

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'Practice finding the following definite integrals.'

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'Define the derivative of the function f(x) at x=a.'

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'Find the definite integral \ \\int_{1}^{n} x \\log x d x \.'

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'Find the value of the following definite integral.'

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'\nCalculation of area, volume, and length of curves\nArea between the curve \\( x=g(y) \\) and the \ y \ axis\nArea enclosed by the curve \\( x=g(y) \\) and the \ y \ axis and two lines \ y=c, y=d \ with \ c < d \ where \\( S=\\int_{c}^{d}|g(y)| dy \\)\n\nArea of a curve represented by \\( x=f(t), \\quad y=g(t) \\) is \\( S=\\int_{a}^{b} y dx=\\int_{\\alpha}^{\eta} g(t) f^{\\prime}(t) dt \\)\n\nWhere, always \\( y \\geqq 0, \\quad a=f(\\alpha), \\quad b=f(\eta) \\) Volume of a solid where the volume of the solid with \\( S(x) \\) is \\( V=\\int_{a}^{b} S(x) dx \\) when \ a < b \'

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'Let the area enclosed by curves C1, C2, and the line x=π/2 be denoted by T. Express the conditions for T=2S in terms of a and b.'

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'Let f(x) and g(x) be continuous functions on the interval [a, b]. If f(a)>g(a) and f(b)<g(b), then show that the equation f(x)=g(x) has at least one real solution in the range a<x<b.'

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'(1) Find the value of c that satisfies the conditions of the Mean Value Theorem for the function f(x) and intervals: (a) f(x)=\\log x [1, e] (b) f(x)=e^{-x} [0,1]'

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'Prove that for a function f(x) that is continuous on the interval a ≤ x ≤ b (a < b),\nint_{a}^{b} f(x) dx = (b-a) f(c), where a < c < b\naccording to the mean value theorem for integrals.'

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"When the function y of x is represented as t, θ as parameters, find the derivative dy/dx as a function of t, θ according to the following equation. Here, the 'a' in (2) is a positive constant. (1) {x=t^3+2, y=t^2-1}, (2) {x=a(θ- sin θ), y=a(1- cos θ)}"

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'Find the function f(x) that satisfies the equation (1).'

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'(1) Consider a point P moving on a number line with its velocity v as a function of time v=f(t). Also, let the coordinate of P at time a be k.\n[1] The coordinate x of P at time b is x=k+∫[a,b] f(t) dt\n[2] The change in position of P from time a to time b is s=∫[a,b] f(t) dt\n[3] The distance traveled by P from time a to time b is l=∫[a,b]|f(t)| dt'

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

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'(1) For a continuous function f(x), prove the equation \\( \\int_{0}^{\\frac{\\pi}{2}} f(\\sin x) d x = \\int_{0}^{\\frac{\\pi}{2}} f(\\cos x) d x \\). \n(2) Find the definite integral \ I=\\int_{0}^{\\frac{\\pi}{2}} \\frac{\\sin x}{\\sin x + \\cos x} d x \.'

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'Express the area S of the region enclosed by curves \ C_{1}, C_{2} \ and the y-axis in terms of a and b.'

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'Calculate the volume between the curve y=f(x) and the x-axis'

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"Using definite integration, we draw the outline of curve C in the form of 'y as a function of x' as per (1)."

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"For a quantity f(t) that changes over time (such as the volume of an expanding solid), the rate of change of that quantity at time t is represented by f'(t), similar to the speed of 1."

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'Practice: Find the following definite integrals.'

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'Prove that when the function $f(x)$ is continuous on the interval $a \\leqq x \\leqq b(a<b)$, we have $\\int_{a}^{b} f(x) d x=(b-a) f(c)$, where $a<c<b$, according to the Mean Value Theorem for Integrals.'

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Join our Discord#### Q.92

'Use the Mean Value Theorem to prove an inequality. For example, consider the application of the Mean Value Theorem for the function f(x) = x^3 - 3x + 2 on the interval [0,1].'

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'Definite integral and limit of sum (partition method)\\( f(x) \\) is continuous on interval \ [a, b] \, dividing this interval into \ n \ parts with endpoints and division points as \ a=x_{0}, x_{1}, x_{2}, … x_{n}=b \ and \ \\frac{b-a}{n}=\\Delta x \\n'

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'Find the following definite integrals. In (2), \ a \ is a constant.'

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'Prove that if \\( f(x) \\) is differentiable at \ x=a \, then it is also continuous. However, explain why the converse (continuous functions are not necessarily differentiable) does not hold.'

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'In the coordinate space, there is an origin O and points A(1,-2,3), B(2,0,4), C(3,-1,5). Find the minimum magnitude of the vector OA+x*AB+y*AC and the values of real numbers x and y at that time.'

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

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'Evaluate the definite integral \\( \\int_{1}^{3} \\frac{\\left(x^{2}-1\\right)^{2}}{x^{4}} dx \\).'

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'Contributed to the rigorous development of calculus through concepts like the continuity of real numbers.'

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'Find the following definite integrals:\n(1) \\( \\int_{1}^{3} \\frac{\\left(x^{2}-1\\right)^{2}}{x^{4}} d x \\)\n(2) \ \\int_{1}^{3} \\frac{d x}{x^{2}-4 x} \\n(3) \ \\int_{0}^{1} \\frac{x^{2}+2}{x+2} d x \\n(4) \\( \\int_{0}^{1}\\left(e^{2 x}-e^{-x}\\right)^{2} d x \\)\n(5) \ \\int_{0}^{2 \\pi} \\cos ^{4} x d x \\n(6) \ \\int_{\\frac{\\pi}{6}}^{\\frac{\\pi}{2}} \\sin x \\sin 3 x d x \'

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'Prove y^(n)=a^(n-1)(n+ax)e^(ax) using mathematical induction.'

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'Please solve a problem of calculating definite integrals.'

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'Determine the range of values for the constant a in order for the curve y=(x^2+ax+3)e^x to have inflection points. Also, how many inflection points can be created at that time.'

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'Let f(x) and g(x) be continuous functions on the interval [a, b]. If f(a) > g(a) and f(b) < g(b), prove that the equation f(x) = g(x) has at least one real number solution in the range a < x < b.'

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"Length of Curve\nThe length of a curve \\( x = f(t), y = g(t) (\\alpha \\leqq t \\leqq \eta) \\) is\n\\[\\int_{\\alpha}^{\eta} \\sqrt{\\left(\\frac{dx}{dt}\\right)^{2} + \\left(\\frac{dy}{dt}\\right)^{2}} dt = \\int_{\\alpha}^{\eta} \\sqrt{\\left\\{f'(t)\\right\\}^{2} + \\left\\{g'(t)\\right\\}^{2}} dt\\n\\]"

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'Limit of trigonometric functions\nThere is a point O at the center, and a point P moving on the circumference of a circle with diameter AB of length 2r. Let the area of △ABP be S1 and the area of sector OPB be S2. Answer the following questions.\n(1) When ∠PAB=θ (0<θ<π/2), find S1 and S2.\n(2) As P approaches B, find the limit of S1/S2.'

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'Find the value of c that satisfies the conditions of the mean value theorem for the following functions and intervals: (1) f(x)=2 x^{2}-3 [a, b] (2) f(x)=e^{-x} [0,1] (3) f(x)=\\frac{1}{x} [2,4] (4) f(x)=\\sin x [0,2 \\pi]'

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'Let a, b be constants, m, n be non-negative integers, and define \\( I(m, n)=\\int_{a}^{b}(x-a)^{m}(x-b)^{n} d x \\).'

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'Since 17 \ \\frac{d x}{d t}=1, \\frac{d y}{d t}=2 t-2 \, therefore\\n\\\frac{d \oldsymbol{y}}{d x}=\\frac{\\frac{d y}{d t}}{\\frac{d x}{d t}}=\\frac{2 t-2}{1}=2 t-2\'

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'Find the value of the following definite integrals. (1) \\int_{0}^{1} \\frac{x}{\\sqrt{2-x^{2}}} dx (2) \\int_{1}^{e} 5^{\\log x} dx (3) \\int_{0}^{\\frac{\\pi}{2}} \\frac{\\sin 2 x}{3+\\cos^2 x} dx (4) \\int_{0}^{\\frac{\\pi}{2}} \\sin^2 x \\cos^3 x dx'

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'In a space, there are 3 points A(1,-1,1), B(-1,2,2), C(2,-1,-1). Find the minimum value of the magnitude of vector 59r = OA + xAB + yAC.'

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'Find the following definite integral: \n\\\int_{0}^{\\frac{\\pi}{2}} x^{2} \\cos ^{2} x d x \'

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'Calculate the following definite integral:\n∫_0^1 sqrt(1 - x^2) dx'

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'Calculate the definite integral of \\( \\int_{0}^{2} (x^3 + 2x^2 + x + 1) \\,dx \\)'

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'Explain the difference between necessary and sufficient conditions.'

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'Find the number of real solutions of the function f(x) defined as follows. f(0)=-1/2, f(1/3)=1/2, f(1/2)=1/3, f(2/3)=3/4, f(3/4)=4/5, f(1)=5/6, when f(x) is continuous, how many real solutions does f(x)-x=0 have at least for 0 ≤ x ≤ 1.'

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'Calculate the area between the curve y = x^2 and x = 1.'

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"In the mean value theorem (1), since c lies between a and b, we have b-a=h, by defining (c-a)/(b-a)=θ, we get b=a+h, c=a+θh. Therefore, the mean value theorem (1) can also be expressed as follows. (3) Mean Value Theorem (2): If the function f(x) is continuous on the interval [a, a+h], and differentiable on the interval (a, a+h), then there exists a real number θ that satisfies 0<θ<1, such that f(a+h)=f(a)+hf'(a+θh)."

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'Find the area enclosed by the curve y = √(4 - x^2), the x-axis, and the y-axis. (Hint: Use critical points to find the area)'

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'Find the following definite integral. (3) $\\int_{-\\sqrt{2}}^{\\sqrt{2}}\\left(x^{5}-4 x^{3}+3 x^{2}-x+2\\right) d x$'

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'How to find the volume of a solid using cross-sectional area. When the cross-sectional area when cut by a plane perpendicular to the x-axis is represented by a function S(x) with respect to x, find the volume V. Consider the range from a to b.'

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'Find the area of the shape enclosed by curve C and the x-axis and y-axis.'