Analysis - Single Variable Calculus | AI tutor The No.1 Homework Finishing Free App

'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'

'Differentiation method'

'Calculate the definite integral of the following expression.'

'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 \\)'

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

'Find the following definite integrals.'

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

'Find the value of the following definite integrals.'

'Calculate the derivative by using limits'

'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 \\)'

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

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

'Evaluate the following definite integrals.'

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

'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.'

'Evaluate the following definite integrals.'

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

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

'Find the following definite integrals.'

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

'Definite Integral and Differentiation'

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

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

'Find the indefinite integral of ∫ 1/x dx.'

'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 \'

'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)'

"Please explain MacLaurin's theorem."

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

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

'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'

'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 \\)'

'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.'

'Calculate the following definite integrals.'

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

'Find the following definite integrals.'

"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}"

'Compute the following definite integrals using the formulas.'

'Calculate $\\int_{2}^{2}(x+1)(x-3)dx$.'

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

'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).'

'Please explain what necessary and sufficient conditions are.'

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

'Equation and curve'

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

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

'(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}\\\\]'

'169 (2) (ア) x y^{′}-2 y=0'

"State Rolle's Theorem and prove it"

'Hamilton-Cayley theorem'

'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.'

'What is the Mean Value Theorem?'

'Practice finding the following definite integral.'

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

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

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

'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).'

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

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

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

'Find the following definite integrals.'

'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.'

'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.'

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

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

'Find the following definite integrals.'

'Practice finding the following definite integrals.'

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

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

'Find the value of the following definite integral.'

'\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 \'

'Find the following definite integrals.'

'Calculate the area between two curves.'

'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.'

'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.'

'(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]'

'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.'

"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)}"

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

'Maximum and minimum of a function'

'Chapter 8: Applications of Integration'

'(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'

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

'(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 \.'

'Extreme Value Theorem'

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

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

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

'Find the following definite integrals.'

"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."

'Practice: Find the following definite integrals.'

'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.'

'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].'

'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'

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

'Tangent and Normal Basics'

'Find the length of the following curves.'

''

'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.'

'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.'

'Integration by Parts Formula'

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

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

'Evaluate the following definite integrals.'

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

'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 \'

'Evaluate the following definite integrals.'

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

'The equation of the sphere is given by'

'Proof of the Mean Value Theorem'

'Please solve a problem of calculating definite integrals.'

'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.'

'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.'

"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\\]"

'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.'

'Find the following definite integrals.'

'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]'

''

'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 \\).'

'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\'

'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'

'Find the following definite integrals.'

'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.'

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

'Maximum and Minimum Area'

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

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

'Explain the difference between necessary and sufficient conditions.'

'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.'

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

"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)."

'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)'

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

'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.'

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