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## Functions and Analysis

### Advanced Functions - Exponential and Logarithmic Functions

#### Q.01

"Mathematics II\n261\nTherefore\n\\[\n\egin{aligned}\n& x^{4}+2 x^{3}-3 x^{2}-m x-n \\n= & x^{4}-2(a+b) x^{3}+\\left\\{(a+b)^{2}+2 a b\\} x^{2}-2 a b(a+b) x+a^{2} b^{2}\n\\end{aligned}\n\\]\n\n\nComparing the coefficients on both sides\n\\[\n\egin{array}{l}\n2=-2(a+b) \\cdots \\cdots \\\\\n-m=-2 a b(a+b)\n\\end{array}\n\\]\n\\[-3=(a+b)^{2}+2 a b\\]\n(3), \ -n=a^{2} b^{2} \ \ \\qquad \\nFrom (1) we have \ a+b=-1 \\nFrom this and (2) we have \ a b=-2 \\n\nSubstitute these into (3), (4) we get \ m=4, \\quad n=-4 \\n\ a, b \ satisfy the equation of the second degree \ t^{2}+t-2=0 \ so\n\a \\neq b\\n\nTherefore, the equation of the line we seek is \ \\quad y=4 x-4 \\n(2) Since \ a<b \ we have \ \\quad a=-2, b=1 \\nIn the interval \ -2 \\leqq x \\leqq 1 \\n\x^{4}+2 x^{3}-3 x^{2} \\geqq 4 x-4\\n\nSo\n\\[\egin{aligned}\nS= & \\int_{-2}^{1}\\left\\{\\left(x^{4}+2 x^{3}-3 x^{2}\\right)-(4 x-4)\\right\\} d x \\\\\n= & {\\left[\\frac{x^{5}}{5}+\\frac{x^{4}}{2}-x^{3}-2 x^{2}+4 x\\right]_{-2}^{1} } \\\\\n= & \\left(\\frac{1}{5}+\\frac{1}{2}-1-2+4\\right) \\\\\n& -\\left(-\\frac{32}{5}+8+8-8-8\\right)=\\frac{81}{10}\n\\end{aligned}\\]\nIt touches at \ x=a, b \. \\( \\Leftrightarrow f(x)-(m x+n)=0 \\) has repeated roots at \ x=a, b \\n\nRemains to\nProve that \\( \\int_{\\alpha}^{\eta}(x-\\alpha)^{2}(x-\eta)^{2} d x=\\frac{1}{30}(\eta-\\alpha)^{5} \\) (from this book \ p .324 \ ) \\( \\unlhd f(x)=x^{2}(x-1)(x+3) \\) Thus, when \\( f(x)=0 \\), we get \ x=-3,0,1 \ therefore, \\( y=f(x) \\) intersects the graph at \ x=-3,0,1 \ and intersects at \ x=0 \. Reference\n\\[\egin{aligned}\nS & =\\int_{-2}^{1}(x+2)^{2}(x-1)^{2} d x \\\\\n& =\\frac{1}{30}\\{1-(-2)\\}^{5}=\\frac{81}{10}\n\\end{aligned}\\]\n\\[\egin{aligned}\n(x-\\alpha)^{2}(x-\eta)^{2} & =(x-\\alpha)^{2}(x-\\alpha+\\alpha-\eta)^{2} \\\\\n& =(x-\\alpha)^{2}\\left\\{(x-\\alpha)^{2}+2(x-\\alpha)(\\alpha-\eta)+(\\alpha-\eta)^{2}\\right\\} \\\\\n& =(x-\\alpha)^{4}+2(\\alpha-\eta)(x-\\alpha)^{3}+(\\alpha-\eta)^{2}(x-\\alpha)^{2}\n\\end{aligned}\\]\n\\[\\text { Therefore } \egin{aligned}\n\\int_{\\alpha}^{\eta}(x-\\alpha)^{2}(x-\eta)^{2} d x & =\\left[\\frac{(x-\\alpha)^{5}}{5}+2(\\alpha-\eta) \\cdot \\frac{(x-\\alpha)^{4}}{4}+(\\alpha-\eta)^{2} \\cdot \\frac{(x-\\alpha)^{3}}{3}\\right]_{\\alpha}^{\eta} \\\\\n& =\\frac{(\eta-\\alpha)^{5}}{5}+\\frac{1}{2}(\\alpha-\eta)(\eta-\\alpha)^{4}+\\frac{1}{3}(\\alpha-\eta)^{2}(\eta-\\alpha)^{3} \\\\\n& =\\left(\\frac{1}{5}-\\frac{1}{2}+\\frac{1}{3}\\right)(\eta-\\alpha)^{5}=\\frac{1}{30}(\eta-\\alpha)^{5}\n\\end{aligned}\\]\nChapter 7\nExercises\nEnd of Solution\nExercises\n(176 \\Rightarrow) from this book \ p .325 \\n(1) \ x^{2}=\\frac{y}{\\sqrt{2}} \\nLet's assume (1).\nSubstituting (1) into \ x^{2}+y^{2}=1 \ gives us\n\ \\frac{y}{\\sqrt{2}}+y^{2}=1 \ which simplifies to \ \\sqrt{2} y^{2}+y-\\sqrt{2}=0 \\nFactoring the left side gives\n\\( (y+\\sqrt{2})(\\sqrt{2} y-1)=0 \\)\nSince \ y=\\sqrt{2} x^{2} \\geqq 0 \ we have \ \\quad y=\\frac{1}{\\sqrt{2}} \\nEliminating \ x \ from the system. To eliminate \ y \, we get\n\\[x^{2}+\\left(\\sqrt{2} x^{2}\\right)^{2}=1\\]\n\nwhich results in \ 2 x^{4}+x^{2}-1=0 \\n\\( \\left(x^{2}+1\\right)\\left(2 x^{2}-1\\right)=0 \\)\nSince \ x^{2}+1>0 \ we have \ \\quad x^{2}=\\frac{1}{2} \\nTherefore \ x= \\pm \\frac{1}{\\sqrt{2}} \"

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'For a polynomial f(x), the identity f(f(x))={f(x)}^{2} holds true. Find all the f(x) that satisfy this condition, with f(x) always being non-zero.'

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'(3) Let the first term be $a$ and the common ratio be $r(r>0)$. According to the conditions\n\\n\egin{\overlineray}{l}\na+a r+a r^{2}=21 \\\\ \\cdots \\\\ \\cdots \\\\ a r^{3}+a r^{4}+a r^{5}+a r^{6}+a r^{7}+a r^{8}=1512\n\\end{\overlineray}\n\\nFrom (2) we have $\\left(a+a r+a r^{2}\\right)r^{3}+\\left(a+a r+a r^{2}\\right)r^{6}=1512$\nSubstituting (1) gives $21 r^{3}+21 r^{6}=1512$\n\nTherefore\n\\nr^{6}+r^{3}-72=0\n\\n\nFactoring gives $\\left(r^{3}+9\\right)\\left(r^{3}-8\\right)=0$\n\nHence $r^{3}+9=0, r^{3}-8=0$\n\nSince $r>0$, we have $r^{3}=8$, so $r=2$\nSubstitute $r=2$ into (1) gives $7 a=21$, hence $a=3$\nTherefore, the first term is 3, and the sum of the first 5 terms is $\\frac{3\\left(2^{5}-1\\right)}{2-1}=93$'

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'Identify the features of the graph of the logarithmic function y=log_a x.'

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'The equation of the tangent line at point (2, -2) is y - (-2) = -3(x - 2), which simplifies to y = -3x + 4. The x-coordinate of the intersection of lines (1) and (2) is found to be x = 1 from x = -3x + 4, hence the area to be determined is denoted as S, where S = ∫_{0}^{1}(x - (-x² + x)) dx + ∫_{1}^{2}((-3x + 4) - (-x² + x)) dx = ∫_{0}^{1} x³ / 3 + ∫_{1}^{2}(x - 2)³ / 3 = [x³/3]_{0}^{1} + [(x-2)³/3]_{1}^{2} = 1/3 + 1/3 = 2/3.'

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'The function takes the maximum value of 2a√a + b when x = -√a, and the function takes the minimum value of -2a√a + b when x = √a.'

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'Simplify the following expressions: (1) $\\left(\\log _{27} 4+\\log _{9} 4\\right)\\left(\\log _{2} 27-\\log _{4} 3\\right)$ [Shinshu University] (2) $\\log _{2} 25 \\cdot \\log _{3} 16 \\cdot \\log _{5} 27$'

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'Solve the following equations. (1) $\\log_{3} x + \\log_{3} (x-2) = 1$ (2) $\\log_{2}(x^2 - x - 18) - \\log_{2}(x - 1) = 3$ (3) $\\log_{2}(x + 1) - \\log_{4}(x + 4) = 1$'

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'The x-coordinate of the intersection point of 2 curves is the solution to x^3-3x^2+2x=ax(x-2). Since x^3-3x^2+2x=x(x-1)(x-2), we have x(x-1)(x-2)=ax(x-2). Therefore x(x-2)(x-1-a)=0. Therefore x= 0,2, a+1. Since a>1, the general shape of the two curves is as shown in the right figure, and the condition for the two areas S1, S2 to be equal is S1=S2, which means S1-S2=0. Therefore'

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'Find the minimum value of the function y=log _{3} x+3 log _{x} 3(x>1).'

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'Solve the logarithmic equation \ \\log_{a} x = b \. Here, \ a \ and \ b \ are constants, and \ x \ is the variable.'

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'Consider the 4th degree function of x, f(x)=x^{4}-a x^{2}+b x, where a and b are real numbers.'

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'Find the maximum and minimum values of the function y = (log_2(x/4))^2 - log_2(x^2) + 6 for 2 ≤ x ≤ 16, and the corresponding values of x.'

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'Example 152 Maximum and minimum of various functions (using differentiation 2) (1) Find the minimum value of the function f(x)=2^{3x}-3*2^{x} and the corresponding value of x. (2) Find the maximum value of the function f(x)=log_{2} x+2 log_{2}(6-x) and the corresponding value of x.'

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'Compound interest calculation\nAssuming an annual interest rate of r, using compound interest calculation each year, find the following:\n(1) The principal T yen to make the total amount after n years S yen\n(2) Save P yen at the beginning of each year, and the total principal after n years is Sn yen'

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'Example 31 | Recurrence relation involving product and powers (using logarithms)'

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'Plot the graphs of the following functions. Also, describe the relationship between the functions and \ y=\\log _{4} x \.'

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"Mathematics II -247 Exercise 80 ⇒ This book page 302 f'(x)=3x^2+2ax Therefore, the equation of the tangent ℓ_t at point P(t, f(t)) is y-(t^3+at^2+b)=(3t^2+2at)(x-t) When ℓ_t passes through the origin -(t^3+at^2+b)=(3t^2+2at)(-t) 2t^3+at^2-b=0 Simplifying we get 2t^3+at^2-b=0 In the graph of a cubic function, since different tangents correspond to different points of contact, equation (1) needs to have a single real solution. Now, let g(t)=2t^3+at^2-b. Find the conditions for a, b such that the curve y=g(t) has exactly one point in common with the t-axis. g'(t)=6t^2+2at=6t(t+a/3) Setting g'(t)=0 we get t=0,-a/3 [1] When a=0, g'(t)=6t^2 ≥ 0 Therefore, since g(t)=2t^3-b is monotonically increasing, regardless of the value of b, the curve y=g(t) has exactly one point in common with the t-axis. [2] When a≠0 Let the smaller of 0,-a/3 be α, and the larger be β, then the table of increases and decreases for g(t) is as follows. For the curve y=g(t) to have exactly one point in common with the t-axis, the conditions are that the maximum and minimum are both positive or both negative. That is, g(0)g(-a/3) > 0 Since g(0)=-b, g(-a/3)=a^3/27 - b Therefore, -b(a^3/27 - b) > 0, or equivalently b(a^3/27 - b) < 0 Therefore b < 0 and b < a^3/27 or b > 0 and b > a^3/27 Therefore the conditions we seek are When a=0, b is all real numbers When a≠0, b < 0 and b < a^3/27 or b > 0 and b > a^3/27 Therefore, the region where the point (a, b) exists is shown in the figure on the right as the slanted part. Please note that the boundary line includes only the origin and excludes others."

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"(1) f'(x)=x^2-s^2=(x+s)(x-s) f'(x)=0 when x=-s, s [1] s>0 the table of increases and decreases of f(x) is as follows on the right. (i) when 0<s<2 f'(x) = + 0 - 0 + f(x) = increasing maximum & decreasing minimum & increasing f(x) is the minimum value at x=s therefore g(s)=f(s)=s^3 / 3 - s^2 * s+2 s^2=-2 / 3 s^3+2 s^2 (ii) when s ≥ 2 f(x) is the minimum value at x=2 therefore g(s)=f(2)=2^3 / 3-s^2 * 2+2 s^2=8 / 3 [2] when s=0 f(x)=x^3 / 3, f'(x)=x^2 ≥ 0 therefore 0 ≤ x ≤ 2 f(x) is the minimum value at x=0 therefore g(0)=f(0)=0"

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'Express the relative size of each set of numbers using inequalities.'

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'Solve the equation (1), 2(log_{2}x)^{2}+3log_{2}4x=8'

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'Properties of Logarithmic Functions\nProperties and graph of the logarithmic function \ y=\\log _{a} x \ where \ a>0, a \\neq 1 \.\n(1) Domain is all positive numbers, range is all real numbers.\n(2) Passes through points \\( (1,0),(a, 1) \\), with the \ y \ axis as its asymptote.\n(3) When \ a>1 \, as \ x \ increases, so does \ y \.\n\\n0<p<q \\Longleftrightarrow \\log _{a} p<\\log _{a} q\n\\nWhen \ 0<a<1 \, as \ x \ increases, \ y \ decreases.\n\\n0<p<q \\Longleftrightarrow \\log _{a} p>\\log _{a} q\n\'

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'Simplify the following expressions:\n1. \ \\log_{4} 8 + \\log_{4} 2 \\n2. \ \\log_{5} 75 - \\log_{5} 15 \\n3. \ \\log_{8} 64^{3} \\n4. \ \\log_{3} \\sqrt[4]{3^{5}} \\n5. \ \\log_{\\sqrt{3}} 27 \\n6. \ \\log_{2} 8 + \\log_{3} \\frac{1}{81} \'

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'Find the maximum and minimum values of the function y = 9^x - 2 \\ cdot 3^{x+1} + 81 (-3≤x≤3).'

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'Exponential Function\nProperties and Graph of Exponential Function \ y=a^{x} \\nLet \ a>0, a \\neq 1 \.\n(1) The domain is all real numbers, the range is all positive numbers.\n(2) Passes through the points \\( (0,1),(1, a) \\) and the x-axis is its asymptote.\n(3) When \ a>1 \, as \ x \ increases, \ y \ also increases.\n\\np<q \\Longleftrightarrow a^{p}<a^{q}\n\\nWhen \ 0<a<1 \, as \ x \ increases, \ y \ decreases.\n\\np<q \\Longleftrightarrow a^{p}>a^{q}\n\'

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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. By rearranging the right-hand side, we get f(x)=1+2 x \\int\\_{0}\\^{1} t f(t) d t+2 \\int\\_{0}\\^{1} f(t) d t \\int\\_{0}\\^{1} t f(t) d t=a. Let \\int\\_{0}\\^{1} f(t) d t=b, where a and b are constants, so a=\\int\\_{0}\\^{1} t(x)=2 a x+2 b+1 =\\left[\\frac{2}{3} a t\\^{3}+\\frac{2 b+1}{2} t\\^{2}\\right]\\_{0}\\^{1}=\\frac{2}{3} a+\\frac{2 b+1}{2}. Therefore, a=\\int\\_{0}\\^{1} t(2 a t+2 b+1) d t=\\int\\_{0}\\^{1}\\left\\{2 a t\\^{2}+(2 b+1) t\\right\\} d t implies 2 a-6 b-3=0. On the other hand, b=\\int\\_{0}\\^{1}(2 a t+2 b+1) d t=\\left[a t\\^{2}+(2 b+1) t\\right]\\_{0}\\^{1} =a+2 b+1, so a+b+1=0 (1), solving the system of equations gives a=-\\frac{3}{8}, b=-\\frac{5}{8}, hence f(x)=2\\left(-\\frac{3}{8}\\right) x+2\\left(-\\frac{5}{8}\\right)+1=-\\frac{3}{4} x-\\frac{1}{4} x can be treated as a constant.'

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'Please describe the domain and range of the exponential function y=a^{x}.'

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'Find the maximum and minimum values of the following functions.'

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'Investigate the relationship between the basic graph of y=a^x and the graphs of y=3^x and y=3^{-x}.'

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'Logarithms and Their Properties\nDefinition of Logarithms\n\ a>0, \\quad a \\neq 1, \\quad M>0 \\text { \overlinee given values. } \\n\ M=a^{p} \\Longleftrightarrow \\log _{a} M=p \'

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'Consider the curves C1: y=a e^{x}, C2: y=e^{-x}. When the constant a varies in the range 1≤a≤4, let D1 be the region enclosed by C1, C2, and the y-axis, and let D2 be the region enclosed by C1, C2, and the line x=log 1/2. (1) Find the value of a when the area of D1 is 1. (2) Find the minimum value of the sum of the areas of D1 and D2 and the corresponding value of a.'

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'Prove the inequality (1) using differentiation (basic)'

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"For the function f(x)=A e^x cos x + B e^x sin x (where A, B are constants), answer the following questions: (1) Find f'(x). (2) Express f''(x) in terms of f(x) and f'(x). (3) Find ∫ f(x) dx."

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'Famous function and its associated limit\nThe limit of the function \ y=\\frac{\\log x}{x} \ is \ \\lim _{x \\rightarrow \\infty} \\frac{\\log x}{x}=0 \.'

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'Find the following indefinite integrals. (3) 136\n(1) \ \\int x^{2} \\cos x d x \\n(2) \ \\int x^{2} e^{-x} d x \\n(3) \ \\int x \\tan ^{2} x d x \'

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'Prove that the inequality b log (a/b) ≤ a - b ≤ a log (a/b) holds when a > 0, b > 0.'

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'Let \\( f(x)=-e^{x} \\) and let \ b \ be a real number. Find the number of tangents to the curve \\( y=f(x) \\) that pass through the point \\( (0, b) \\). You may use \ \\lim _{x \\rightarrow-\\infty} x e^{x}=0 \.'

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'Find the following definite integral. $$ \\int \\frac{1}{\\sqrt{x^{2}+1}} dx $$'

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'For real numbers a, b, c, let F(x) = x^4 + a x^3 + b x^2 + a x + 1, f(x) = x^2 + c x + 1. Also, let T be the set obtained by removing the points 1 and -1 from the unit circle in the complex plane.'

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'For a constant k, determine the number of real solutions to the equation log(sin x+2)-k=0 for 0<x<2π.'

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'How to draw a reflection graph The function expressions in the problem of drawing a graph using differential methods were in the following 3 patterns:'

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'Let n be any positive integer, and let two functions f(x), g(x) be functions that are both differentiable n times. [Quite large] (1) Find the 4th derivative of the product f(x)g(x) d^4/dx^4{f(x)g(x)}. (2) Infer the coefficient of f^(n-r)(x)g^(r)(x) in the nth derivative d^n/dx^n{f(x)g(x)} of the product f(x)g(x), and prove the inference is correct using mathematical induction. Here, r is a non-negative integer not greater than n, and f^(0)(x)=f(x), g^(0)(x)=g(x). (3) Find the nth derivative h^(n)(x) of the function h(x)=x^3e^x, where n≥4.'

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'Let a > 0, b > 0, and f(x) = log ((x + a) / (b - x). Prove that the curve y = f(x) is symmetric about its point of inflection.'

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'Find the inverse functions of the following functions. Also, plot their graphs.'

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'Find the following definite integrals: (1) Tokyo Institute of Technology, (2) Yokohama National University (1) $\\int_{1}^{2} \\frac{\\log x}{x^{2}} dx$ (2) $\\int_{0}^{2 \\pi} x^{2}|\\sin x| dx$'

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'Variation of function values, maximum and minimum, function graph'

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'Let n be an integer. Prove the following equations hold true: where, \ \\cos ^{0} x = 1 \, (4) \\( 138(\\log x)^{0} = 1 \\).\n (1) \\( \\int \\cos ^{n} x dx = \\frac{1}{n}\\{ \\sin x \\cos ^{n-1} x + (n-1) \\int \\cos ^{n-2} x dx \\} (n \\geqq 2) \\)\n (2) \\( \\int(\\log x)^{n} dx = x(\\log x)^{n} - n \\int(\\log x)^{n-1} dx \\) (n \\geqq 1)\n (3) \ \\int x^{n} \\sin x dx = -x^{n} \\cos x + n \\int x^{n-1} \\cos x dx \ (n \\geqq 1)'

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"How can we solve the embarrassing issue \ \\int \\frac{1}{\\sqrt{x^{2}+1}} d x \ in various ways? In the important example 141 (1), we solved it by assuming \ x+\\sqrt{x^{2}+1}=t \, but there are many other methods as well. First, let's take a look at the method of substituting \ x=\\tan \\theta \ as indicated in the previous page."

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'Solve the following problem. Let \\( f(x)=\\frac{1}{3} x^{3}+2 \\log |x| \\). For a real number \ a \, find the number of tangent lines to the curve \\( y=f(x) \\) with a slope equal to \ a \.'

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'Let \ a \ be a non-zero constant, and let \ A = \\int_{0}^{\\pi} e^{-a x} \\sin 2 x d x, B = \\int_{0}^{\\pi} e^{-a x} \\cos 2 x d x \. Find the values of \ A \ and \ B \.'

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'Given constants a, b, with ab ≠ 1. Find the condition for which the inverse function of y = (bx + 1) / (x + a) matches the original function.'

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'Basic 11: Condition for the inverse function to be equal to the original function'

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'Let n be a natural number greater than or equal to 2. Prove the following inequality:'

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'Find the indefinite integral of \\\int e^{2x+e^x} dx\.'

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'Prove that there exists a sequence of points Pₙ(xₙ, yₙ) satisfying P₁(1,1), xₙ₊₁=1/4 xₙ + 4/5 yₙ, yₙ₊₁=3/4 xₙ + 1/5 yₙ (n=1,2, ...) on a plane, and the sequence P₁, P₂, ... approaches a fixed point infinitely close.'

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'Prove the inequality \\int_{0}^{x} e^{-t^{2}} d t<x-\\frac{x^{3}}{3}+\\frac{x^{5}}{10} for x>0.'

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'Integration by substitution and integration by parts'

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'Find the following indefinite integrals. (1) $\\int x e^{-x} dx$ (2) $\\int x \\sin x dx$ (3) $\\int x^{2} \\log x dx$ (4) $\\int x \\cdot 3^{x} dx$ (5) $\\int \\frac{\\log(\\log x)}{x} dx$'

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'Please explain the characteristics of the square root function y = √(ax + b) graph (a ≠ 0).'

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'Find the range of constant a when a tangent line can be drawn from point (a, 0) on the x-axis to the graph of the function y=\\frac{x+3}{\\sqrt{x+1}}.'

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'Find the integral of the function f(x)=3 cos 2x+7 cos x over the interval [0, π] in the form of \\( \\int_{0}^{\\pi}|f(x)| dx \\).'

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'Find the equations of the lines and parabolas obtained by moving the following line and parabola parallel to the x-axis by -3 and to the y-axis by 1.'

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'The total cost of selling y chocolates, denoted as c(y), is given by c(y)=y^{2}. Find the values of the selling price p and the quantity y at which the profit of Company A (the difference between revenue and total cost) is maximized.'

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'63 (2) g(sqrt(2))=7-4 sqrt(2), g(-3)=33, g(1/2)=3/2, g(1-a)=2a^{2}+1'

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"In Example 30, we rationalize the denominators of each term before performing the calculations. However, in Example 31 (1), we proceed with the calculations without rationalizing the denominators. Let's think about the reason for this approach."

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'Let S(a) be the area enclosed by the line passing through the point (1,2) with slope a and the parabola y=x^2. Find the value of a that minimizes S(a) as a varies in the range 0 ≤ a ≤ 6.'

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'Draw the graphs of the following functions and describe their positional relationships with the function y=3^x.'

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'Draw the graphs of the following functions and describe their positional relationship with the function y=log_{2} x.'

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'Max and Min of Logarithmic Function (1): Find the maximum and minimum of the following logarithmic function.'

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'Find the maximum value, minimum value, and the corresponding values of x for the function y = log_2(x/2)log_2(x/8) (1/2 ≤ x ≤ 8).'

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'Given f(x) = a x^{2}(x-3) + b(a≠0) with a maximum value of 5 and a minimum value of -7 in the interval -1 ≤ x ≤ 1, determine the values of constants a and b.'

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'Calculate the common logarithm of 1.95 from data 1.'

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'Prove that the value of \\[P(m-kσ ≤ X ≤ m+kσ)\\] becomes a function of only k, regardless of the values of m and σ, when the random variable X follows the normal distribution N(m, σ^2).'

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'Describe the graph and properties of the following logarithmic functions.'

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'Consider the function y = -2(log₃(3x))³ + 3(log₃(x+1))² + 1, defined for 1550 1/3 ≤ x ≤ 3. Find the maximum and minimum values of the function y, and the corresponding x values.'

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'Describe the properties of the logarithmic function.'

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'For the graph of the function y = x ^ 2 (x > 0), use logarithmic scales for both the horizontal and vertical axes.'

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'Base conversion formula: Convert the bases of the following logarithms.'

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'Find the area enclosed by two parabolas denoted as S(a). Let the x-coordinates of the intersection points of the two parabolas be α and β (α < β), then from the figure on the right side:\n\nS(a) = ∫_{α}^{β} { -2(x - a)^2 + 3a - x^2 } dx\n\n= -3 ∫_{α}^{β} (x - α)(x - β) dx\n\n= -3・( -(1/6) ) (β - α)^3\n\n= (1/2)(β - α)^3\n\nThe solutions of the quadratic equation (1) are x = (2a ± √(-2a^2 + 9a))/3. Since α and β are the solutions of (1),\nβ - α = (2a + √(-2a^2 + 9a))/3 - (2a - √(-2a^2 + 9a))/3\n= (2/3) √(-2a^2 + 9a)\nTherefore, S(a) = (1/2)((2/3)√(-2a^2 + 9a))^3 = (4/27)(-2a^2 + 9a)^(3/2)\n\nSince -2a^2 + 9a = -2(a - (9/4))^2 + (81/8), in the range 0 < a < 9/2, -2a^2 + 9a is maximum at a = 9/4, and at this point S(a) is also maximum.\nThus, S(a) is maximum at a = 9/4\nS(9/4) = (4/27)((81/8))^(3/2) = (4/27) ・ (81/8) √(81/8) = 27√2/8.'

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'Answer the following questions on basic logarithm concepts. Compute the logarithm based on the given equations.'

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'Draw the graphs of the following functions and describe their relationship with the function y=log_{2} x.'

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'Conditions for the existence of solutions of logarithmic equations: Determine the conditions for the existence of solutions of the following logarithmic equation.'

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'The number of distinct real solutions n of the equation f(x)=0 equals the number of points of intersection between the curve y=f(x) and the x-axis. From (1), when a≤0, n=1, and from (2), when a>0, the minimum value -4√2a3/2+16 depends on the value of a and could be positive, 0, or negative, hence n=1,2,3. Therefore, summarizing (1) and (2), if n=1, then a<0, a=0, a>0 are all possible; if n=2, only a>0 is possible; if n=3, only a>0 is possible.'

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'What is the reason for the inclusion of a large number of problems in the blue chart?'

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'How can you further deepen your learning by utilizing digital content?'

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'Exercise answer 67 (2) \\frac{\\pi \\sqrt{1+\\pi^{2}}+\\log \\left(\\pi+\\sqrt{1+\\pi^{2}}\\right)}{2}'

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'Find the indefinite integrals. In (3), (4), where (a≠0, b≠0). (1) ∫e^{-x}cosxdx (2) ∫sin(logx)dx (3) ∫e^{ax}sinbxdx (4) ∫e^{ax}cosbxdx'

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'Translate the given text into multiple languages.'

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'Find the indefinite integral \ \\int e^{x} \\cos x dx \.'

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'Consider the function 47 f(x)=2 \\log(1+e^{x})-x-\\log 2. (1) Let f(x) denote the second derivative of f(x), show that the equation \\log f^{\\prime \\prime}(x)=-f(x) holds. (2) Find the definite integral \\int_{0}^{\\log 2}(x-\\log 2) e^{-f(x)} d x.'

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'Investigate the increasing and decreasing behavior of the function f(x)=x-1-log x, and prove the inequality log x ≤ x-1 for x>0.'

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"f'(x) = 1/ log x^3 (x^3 )' - 1/ log x^2 (x^2 )' = 1/(3 log x) * 3 x^2 - 1/(2 log x) * 2 x = ( x^2 - x ) / log x"

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"Using the above [4] [6] as formulas, let's try to find the following definite integral."

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'Find the following indefinite integrals. (1) \ \\int x \\cos 2 x \\, dx \ (2) \\( \\int(x+1)^{2} \\log x \\, dx \\) (3) \ \\int e^{\\sqrt{x}} \\, dx \'

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'(8) \\( y^{\\prime}=\\frac{(\\log x)^{\\prime} \\cdot x-\\log x \\cdot(x)^{\\prime}}{x^{2}}=\\frac{\\frac{1}{x} \\cdot x-\\log x \\cdot 1}{x^{2}} \\)\\n\\\n=\\frac{1-\\log x}{x^{2}}\\n\'

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'Important Example 165) Quantity and Integration Rotating a part of the curve y = e^x around the y-axis from 0 ≤ x ≤ 2 creates a container, into which water is poured at a rate of a (a positive constant) per unit time. Let V be the volume of water when the depth is h, and let S be the surface area of the water. (1) Find ∫(log y)^{2} dy. (2) Express V in terms of S. (3) Find the rate at which the water surface expands when S becomes π. [Shibaura Institute of Technology] Guidance (3) The rate at which the water surface expands is dS/dt, but it seems difficult to express S in terms of t. Therefore, using hint (2), use dV/dt = dV/dS * dS/dt to find the solution.'

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'Find the following indefinite integrals. (1) $\\int \\frac{dx}{\\sin x}$ (2) $\\int \\frac{\\cos x+\\sin 2x}{\\sin ^{2}x}dx$ (3) $\\int \\sin ^{2}x \\tan x dx$ (4) $\\int \\frac{\\sin x}{3+\\sin ^{2}x}dx$'

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'(1) Find the derivative of the function $F(x)=\\frac{1}{2}\\left\\{x \\sqrt{x^{2}+1}+\\log \\left(x+\\sqrt{x^{2}+1}\\right)\\right\\}, x>0$. (2) On the $x y$ plane, the point $P$ lies on the curve $C$ represented by the equation $x^{2}-y^{2}=1$ and is in the first quadrant. If the area enclosed by the line segment $OP$ connecting the origin $O$ and point $P$, the $x$-axis, and the curve $C$ is $\\frac{s}{2}$, express the coordinates of point $P$ in terms of $s$.'

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'(2)\n\\[\egin{aligned}\n\\frac{\\cos x+\\sin 2 x}{\\sin ^{2} x}= & \\frac{\\cos x+2 \\sin x \\cos x}{\\sin ^{2} x}=\\frac{1+2 \\sin x}{\\sin ^{2} x} \\cdot \\cos x \\\\\n\\sin x=t \\text { Let } & \\cos x d x=d t \\\\\n\\int \\frac{\\cos x+\\sin 2 x}{\\sin ^{2} x} d x & =\\int \\frac{1+2 \\sin x}{\\sin ^{2} x} \\cdot \\cos x d x=\\int \\frac{1+2 t}{t^{2}} d t \\\\\n& =\\int\\left(\\frac{1}{t^{2}}+\\frac{2}{t}\\right) d t=-\\frac{1}{t}+2 \\log |t|+C \\\\\n& =-\\frac{1}{\\sin x}+2 \\log |\\sin x|+C\n\\end{aligned}\\]'

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'Use equation (1) from exercise 152 on page 530, where (1) is V=2π∫₀πx{cosx-(-1)}dx.'

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'(2) \\\\ Let \ e^{x}+1=t \, then \ e^{x}=t-1, e^{x} dx = dt \\\\n\\[ \\int \\frac{e^{2x}}{(e^{x} + 1)^2} \\, dx = \\int \\frac{e^{x}}{(e^{x} + 1)^2} \\, e^{x} \\, dx= \\int \\frac{t-1}{t^2} \\, dt \\\\)\n\\ = \\int \\left( \\frac{1}{t} - \\frac{1}{t^2} \\right) \\, dt \\\\)\n\\ = \\log |t| + \\frac{1}{t} + C \\\\)\n\\ = \\log (e^{x}+1) + \\frac{1}{e^{x}+1} + C \\]'

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'Using the substitution integration formula (2), find the following integral.'

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'(2) Let the sequence \ \\left\\{I_{n}\\right\\} \ be defined by \\( I_{n}=\\int_{0}^{n} f_{n}(x) d x \\). Using the fact that \ 0 \\leqq x \\leqq 1 \ implies \\( \\log (1+x) \\leqq \\log 2 \\), prove that the sequence \ \\left\\{I_{n}\\right\\} \ converges and find its limit. You may use the fact that \ \\lim _{x \\rightarrow \\infty} \\frac{\\log x}{x}=0 \.'

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'Therefore \\[ \\int_{0}^{1} t f(t) d t = \\int_{0}^{1}(t \\sin \\pi t + a t) d t \\]\n\\[ = \\int_{0}^{1} t\\left(-\\frac{\\cos \\pi t}{\\pi}\\right)^{\\prime} d t + a \\int_{0}^{1} t d t \\]\n\ =\\left[-\\frac{t \\cos \\pi t}{\\pi}\\right]_{0}^{1} + \\int_{0}^{1} \\frac{\\cos \\pi t}{\\pi} d t + a\\left[\\frac{t^{2}}{2}\\right]_{0}^{1} \\]\n\\[ = \\frac{1}{\\pi} + \\left[\\frac{\\sin \\pi t}{\\pi^{2}}\\right]_{0}^{1} + \\frac{a}{2} = \\frac{1}{\\pi} + \\frac{a}{2} \\]\nTherefore \\[ \\frac{1}{\\pi} + \\frac{a}{2} = a \\] Solving this gives \\[ a = \\frac{2}{\\pi} \\nHence \\[ f(x) = \\sin \\pi x + \\frac{2}{\\pi} \\]'

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'Important question 115 Inverse Functions and Definite Integral\nLet the inverse function of the function y=e^{x}+e^{-x}, defined for x≥0, be y=g(x). Find ∫_{2}^{4} g(x) dx.'

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'Exercise 102 \\Rightarrow Page 453\n(1) \ x+\\sqrt{x^{2}+1}=t \ Letting \\( \\left(1+\\frac{x}{\\sqrt{x^{2}+1}}\\right) d x=d t \\)\nTherefore, \ \\quad \\frac{\\sqrt{x^{2}+1}+x}{\\sqrt{x^{2}+1}} d x = d t \\nHence, \ \\frac{1}{\\sqrt{x^{2}+1}} d x = \\frac{1}{t} d t \\nThus, \ \\int \\frac{1}{\\sqrt{x^{2}+1}} d x = \\int \\frac{1}{t} d t=\\log |t|+C \\n\\[ =\\log \\left( x+\\sqrt{x^{2}+1} \\right)+C \\]'

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'Since (2) (-x) e^((-x)^2) = -x e^(x^2), it follows that x e^(x^2) is an odd function.'

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'Translate the given text into multiple languages.'

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'Determine the range of real numbers for which the sequence {((x^2+2x-5)/(x^2-x+2))^n} converges. Also find the limit value at that point.'

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'Since this line passes through the point (0, Y(a)), Y(a) = (a^2 + 1)e^(-a^2/2)'

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'Example 82 | Number of Real Solutions of an Equation\nLet k be a real number. Find the number of real solutions of the equation x^{2}+3x+1=ke^{x}. You may use lim_{x \\rightarrow \\infty} x^{2}e^{-x}=0. [Similar to Yokohama National University]'

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'(1) Find the volume $V$ of the solid formed by rotating the region enclosed by the curves $y=x^{2}, y=\\sqrt{x}$ around the $y$-axis.\n(2) Let the curve $y=\\log 3 x$ be denoted by $C$. Find the volume $V$ of the solid formed by rotating the region enclosed by $C$, the tangent line passing through the origin $O$, and the $x$-axis around the $y$-axis.'

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'Mathematics II\n407\n[2] When \ p>2 \\n\\[\\frac{d S}{d p}=p \\log p+\\frac{p}{2}=\\frac{p}{2}(2 \\log p+1)>0\\]\nFrom [1], [2], the table showing the change in S is as follows.\nTherefore, \ S \ is minimum at \ p=\\frac{4}{3} \, and its minimum value is\n\egin{tabular}{c||c|c|c|c|c|c}\n\\hline\ p \ & 1 & \ \\cdots \ & \ \\frac{4}{3} \ & \ \\cdots \ & 2 & \ \\cdots \ \\\\\n\\hline\ \\frac{d S}{d p} \ & & - & 0 & + & & + \\\\\n\\hline\ S \ & & \ \\searrow \ & Local minimum & \ \\nearrow \ & 1 & \ \\nearrow \ \\\\\n\\hline\n\\end{tabular}\n\\egin{\overlineray}{l}\n\\text { p= } \\frac{4}{3} \\text { when } \\\\\na=\\frac{16}{9} \\log \\frac{4}{3}\n\\end{\overlineray}\\n\\[\egin{aligned}\n& \\frac{8}{3} \\log \\frac{4}{3}-\\frac{16}{3} \\log \\frac{4}{3}+\\frac{8}{3}+2 \\log 2-3 \\\\\n= & \\frac{1}{3}(8 \\log 3-10 \\log 2-1)\n\\end{aligned}\\]'

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'(5) Let \ \\log x=t \, then \ \\quad x=e^{t}, d x=e^{t} d t \'

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'Find the function that satisfies the following conditions.'

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'Using the mean value theorem, prove the following propositions:'

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'Translate the given text into multiple languages.'

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'Use formula (4) to evaluate the following integral.'

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'Find the extreme value of the function at x=1/√e. \n(1) At x=1/√e, the function has a minimum value of -1/(2e) \n(2) At x=-4/3, the function has a maximum value of 4√6/9, and at x=0, the function has a minimum value of 0'

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'Example 163 Moving Point on a Curve at Constant Speed\n547\nThere is a point P moving on the coordinate plane. Point P starts from (0,1) and moves along the curve y=(e^x+e^{-x})/2 (x≥0) at a speed of 1 unit per second. Let the coordinates of point P after t seconds be (f(t), g(t)). Find f(t), g(t).\n[Shinkei]\nConsider two ways to express the distance l from 0 seconds to t seconds.\n[1] Since moving at a speed of 1 unit per second, l=t\n[2] Since moving on the curve y=(e^x+e^{-x})/2 (x≥0), let the x-coordinate of point P after t seconds be p, then\nl=∫_{0}^{p}√(1+(dy/dx)^2)dx'

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"In the case of x>0, if f'(x)=0, then x+π/4=kπ, which means x=kπ-π/4 (k=1,2,3, ...). Since f''(x)=√2 e^(-x){sin(x+π/4)-cos(x+π/4)}"

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'Translate the given text into multiple languages.'

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'Find the derivative of the logarithmic function log_a x with respect to any base a.'

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'Important Example 87. Proof of inequality of 2 variables (2)\nIf 0 < a < b, show that the following inequality holds:\n\ \\sqrt{a b} < \\frac{b-a}{\\log b - \\log a} < \\frac{a+b}{2} \\n[Gifu University]'

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'Find the necessary and sufficient condition that q must satisfy for the line y = px + q to not have any points in common with the graph of the function y = log x.'

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'Find the inverse functions of the following functions.'

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'(2) \ \\int 3^{1-2 x} d x=-\\frac{1}{2} \\cdot \\frac{3^{1-2 x}}{\\log 3}+C = -\\frac{3^{1-2 x}}{2 \\log 3}+C \'

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'Exercise 97 \\Rightarrow Book p.447\n (3) \\(\\int \\log(x+3) d x\\)'

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'The integral \ \\int_{-\\infty}^{\\infty} e^{-x^{2}} d x \ is known as the Gaussian integral and is equal to \ \\sqrt{\\pi} \.'

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'Prove using mathematical induction the nth derivative and recurrence formula for f(x) = 1 / (1 + x^2).'

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'Practice (1) Find the x-coordinate of the intersection of two curves y=logx and y=a/x^{2} (a>0), denoted by p, express a in terms of 144p.'

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'(7)\\\\n\\\\[\\\\n\\\egin{aligned}\\\\n y^{\\\\prime} & =\\\\left(e^{x}\\\\right)^{\\\\prime} \\\\sin x+e^{x}(\\\\sin x)^{\\\\prime}=e^{x} \\\\sin x+e^{x} \\\\cos x \\\\n\\\\ & =e^{x}(\\\\sin x+\\\\cos x)\\\\n\\\\end{aligned}\\\\n\\\\]'

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'Given expression in mathematics \ \\mathbb{I} \\\n(4) \\( y^{\\prime}=\\frac{1-\\sin x}{1+\\sin x} \\cdot \\frac{\\cos x(1-\\sin x)-(1+\\sin x)(-\\cos x)}{(1-\\sin x)^{2}} \\)\\n\\[\\n=\\frac{2 \\cos x}{(1+\\sin x)(1-\\sin x)}=\\frac{2 \\cos x}{\\cos ^{2} x}=\\frac{2}{\\cos x}\\n\\]\\nAnother solution is \\( y=\\log (1+\\sin x)-\\log (1-\\sin x) \\), therefore\\n\\[\\n y^{\\prime}=\\frac{\\cos x}{1+\\sin x}-\\frac{-\\cos x}{1-\\sin x}=\\frac{2 \\cos x}{(1+\\sin x)(1-\\sin x)}=\\frac{2}{\\cos x}\\n\\]'

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'(3) Prove the inequality \\( \\sqrt{\\pi\\left(1-e^{-a^{2}}\\right)} \\leqq \\int_{-a}^{a} e^{-x^{2}} d x \\).'

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'Find the maximum and minimum values of the following functions:'

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'There are two points P and Q moving on the x-axis. At time t=0, the two points are at the origin O, and the speeds of P and Q at time t are respectively v_P(t)=a t (0 ≤ t) and v_Q(t)= {0 (0 ≤ t < 1), t log t (1 ≤ t). (1) Prove that Q will always overtake P. (2) Find the time when Q catches up to P and the maximum distance between P and Q within that time.'

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'Given text is translated into multiple languages.'

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'Let n be any positive integer, and let the two functions f(x) and g(x) be functions that are both n times differentiable.'

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'For any non-negative integers m and n, let Iₘ,ₙ = ∫₀^(π/2) sin^m x cos^n x dx.'

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'(1) $8\\log 2-1$ (2) $\\frac{1}{\\log 2}$ (3) $\\sqrt{3}-1-\\frac{\\pi}{12}$ (4) $\\frac{4\\sqrt{2}-2}{3}$ (5) $2\\sqrt{3}-\\frac{\\pi}{3}$'

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"Practice 70 => This Book p.399\n(1)\n$y'=(2 x+1) e^{-x}+(x^{2}+x-1) (-e^{-x})\n=-(x^{2}-x-2) e^{-x}\n=-(x+1)(x-2) e^{-x}\nx=-1,2"

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"(1) Find the derivative f'(x) of the function f(x) = log(x+√(1+x^2)) defined for x ≥ 0. (2) Find the length of the part of the curve defined by the polar equation r=θ(θ ≥ 0) for 0 ≤ θ ≤ π."

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'What are the characteristics of the exponential function graph?'

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'143 (1) x=0, π/2 for maximum value 1; x=π, 3π/2 for minimum value -1\n(2) x=log_{2} (fraction) sqrt{5} ± 1/2 for minimum value 1-10 sqrt{5}'

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'Illustrate the range of points $(x, y)$ that satisfy the inequality $\\log _{x} y+2 \\log _{y} x-3>0$.'

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"Find the values of constants a, b, and c such that the cubic function f(x)=2x^{3} + a x^{2} + b x + c satisfies the condition 6 f(x) = (2 x - 1) f'(x) + 6."

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'Find the maximum and minimum values of the following functions. (1) y=4^{x}-2^{x+2}(-1 \\leqq x \\leqq 3) (2) Let a>0, a \\neq 1. For the function y=a^{2 x}+a^{-2 x}-2\\left(a^{x}+a^{-x}\\right)+2, let a^{x}+a^{-x}=t. Express y in terms of t and find the minimum value of y. (3) y=\\left(\\frac{3}{4}\\right)^{x}(-1 \\leqq x \\leqq 2)'

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

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'Find the maximum value of the function y = log_4(x+2) + log_2(1-x) and the corresponding value of x.'

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'Given that F(x) takes a maximum value of 5 at x=1 and a minimum value of 4 at x=2, find the values of f(t) and α when α is a real constant and f(t) is a 2nd degree function.'

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'When 0 ≤ a < 1, the minimum value at x = -2 is -16a, and at x = 0 is 0; when a = 1, the minimum value at x = -2 is -16; when a > 1, the minimum value at x = -2 is -16a, and at x = a-1 is -(a-1)^3(a+3)'

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'(1) Find the minimum value of x^{2} + y^{2} when \\log _{2} x + \\log _{2} y = 3.\n(2) For positive real numbers x, y satisfying xy=100, find the minimum value of (\\log _{10} x)^{3} + (\\log _{10} y)^{3}, and the values of x and y at that minimum.\n(3) Let f(x) = (\\log _{2} \\frac{x}{a})(\\log _{2} \\frac{x}{b}) (where ab=8, a>b>0). If the minimum value of f(x) is -1, find the value of a^{2}. [Waseda University]'

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'Let a, b be constants. Prove the following inequality.'

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'Let a>0, a≠1, b>0. Illustrate all (a, b) points in the coordinate plane where the quadratic equation 4x²+4xlogₐb+1=0 has a unique solution in the range 0<x<1/2.'

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'Solve the following equations, simultaneous equations. In (3), assume 0<x<1, 0<y<1.'

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"From the condition of g(x), examine the sign of g(x) or f'(x) and create a table of the increase and decrease of f(x)."

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'Two differentiable functions, f(x) and g(x), defined on the entire set of real numbers satisfy the following conditions.'

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'Let n be a positive integer, and define I_{n} = \\int_{2}^{3} \\frac{(x-3)^{n}}{n x^{n}} dx. (1) Find I_{1}. (2) Find the range of values of \\left|\\frac{x-3}{x}\\right| for 2 \\leqq x \\leqq 3. Also, find \\lim _{n \\rightarrow \\infty} I_{n}. (3) Express I_{n+1} in terms of I_{n}. (4) Find \\sum_{n=1}^{\\infty} \\frac{1}{n(n+1)}\\left(-\\frac{1}{2}\\right)^{n}. 〔Kwansei Gakuin University〕'

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'Practice Let \ n \ be an integer. Prove the following equalities. Where, \ \\cos ^{0} x=1 \, \\( 203(\\log x)^{0}=1 \\).\n(1) \\( \\int \\cos ^{n} x d x=\\frac{1}{n}\\left\\{\\sin x \\cos ^{n-1} x+(n-1) \\int \\cos ^{n-2} x d x\\right\\}(n \\geqq 2) \\)\n(2) \\( \\int(\\log x)^{n} d x=x(\\log x)^{n}-n \\int(\\log x)^{n-1} d x \\quad(n \\geqq 1) \\)\n(3) \\( \\int x^{n} \\sin x d x=-x^{n} \\cos x+n \\int x^{n-1} \\cos x d x(n \\geqq 1) \\)'

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'Find the range of values for the constant a such that a tangent line can be drawn from the point (a, 0) to the curve y=e^{-x^{2}}.'

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'Find the volume of the solid obtained by rotating the region enclosed by the parabola y = 2 x - x^{2} and the x-axis around the y-axis once.'

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'Find the inverse functions of the following functions. Also, plot their graphs.'

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

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'Prove that \\( \\int x^{n} e^{-x} d x=-\\left(\\sum_{k=0}^{n} n \\mathrm{P}_{k} x^{n-k}\\right) e^{-x}+C(n is a natural number, C is an integration constant ) \\).'

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"When f(x) is a function that is twice differentiable, express \\frac{d^{2}}{d x^{2}} f(\\tan x) in terms of f'(\\tan x) and f''(\\tan x)."

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'Practice with the function f(x)=e^(kx)/(x^2+1) (k is a constant): (1) Find the value of k when f(x) has a local extremum at x=-2. (2) Determine the range of possible values of k for which f(x) has a local extremum.'

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

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'Translate the given text into multiple languages.'

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"Consider a differentiable function f(x) that satisfies the equation f(x) + ∫_{0}^{x} f(t) e^{x-t} dt = sin x. Find the derivative f'(x) of f(x). Furthermore, f'(x) = in the form of a square. Also, f(0) = "

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'Find the range of values for the constant a so that a tangent line can be drawn from point (a, 0) to the curve y=xe^x.'

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'Find the indefinite integral \ \\int e^{2 x+e^{x}} d x \.'

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'Practice for n being a natural number. Find the nth derivative of the following functions.'

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'Practice proving the following inequalities:\n(1) \\(\\sqrt{1+x} < 1 + \\frac{x}{2} (x>0)\\)\n(2) \\(e^{x} < 1 + x + \\frac{e}{2} x^{2} (0<x<1)\\)\n(3) \\(e^{x} > x^{2} (x>0)\\)\n(4) \\(\\sin x > x - \\frac{x^{3}}{6} \\quad(x>0)\\)'

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'Using natural number n, find the nth derivative y^{(n)} of the function y=(1-7x)^{-1}.'

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'Find the part of f(x) that is equal to 3/1x3 + 2log|x|.'

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'Let \n$n$ be a natural number greater than or equal to 2. Consider the functions $y=e^{x}$\n(1) and $y=e^{nx}-1$\n(2).\n(1) Show that the graphs of (1) and (2) have exactly one intersection point in the first quadrant.\n(2) Let the coordinates of the intersection point obtained in (1) be $(a_{n}, b_{n})$. Find $\\lim _{n \\rightarrow \\infty} a_{n}$ and $\\lim _{n \\rightarrow \\infty} na_{n}$.\n(3) Let the area enclosed by the graphs of (1) and (2) in the first quadrant and the $y$-axis be $S_{n}$. Find $\\lim _{n \\rightarrow \\infty} n S_{n}$.'

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'For the function f(x)=a e^{2 x} (where a is a constant), when the tangent line at the point (b, f(b)) on the curve y=f(x) is y=x and passes through (3, 15). Answer the following questions.'

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'(2) \\( \\frac{1}{4}(3 x+2) \\sqrt[3]{3 x+2}+C \\)'

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'For the function f(x)=e^(kx)/(x²+1) (where k is a constant), answer the following questions.'

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'Prove that the function f(x) = ax + xcosx - 2sinx has only one extremum between π/2 and π. Where -1 < a < 1.'

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'Let n be a natural number. Find the nth derivative of the function y=\\frac{1}{1-7x}.'

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'For the function f(x)=\\sqrt{x^{2}-1}, answer the following questions. Assume x>1. (1) Express c in terms of x which satisfies \\frac{f(x)-f(1)}{x-1}=f^{\\prime}(c), 1<c<x. (2) When (1) is true, find \\lim _{x \\rightarrow 1+0} \\frac{c-1}{x-1} and \\lim _{x \\rightarrow \\infty} \\frac{c-1}{x-1}. [Similar to Shinshu University]'

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"Consider a differentiable function $f(x)$ that satisfies the relation $f(x)+\\int_{0}^{x} f(t) e^{x-t} d t=\\sin x$. Find the derivative $f'(x)$ of $f(x)$, then $f'(x)= \\square$. Also, since $f(0)=1$, we have $f(x)= \\square$."

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'(1) There is a point P traveling on the number line starting from point 1 with a velocity of $v=t(t-1)(t-2)$ after t seconds. The position of P after 3 seconds from the start is A, and the distance traveled by P is B.\n\n(2) Let g be the acceleration due to gravity. A rocket with an acceleration of $\x0crac{1}{1+t}$ at t seconds after launch from the ground vertically with an initial velocity $v_{0}$. Find the speed and height of the rocket after t seconds.'

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'Sketch the general shape of the graph of the function y=(-x+1) e^{-x+1}. Given that lim _{x → ∞} x e^{-x}=0.'

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'Find the value of a and the coordinates of the point of tangency when the straight line y=x is a tangent to the curve y=a^x. Here, a>0 and a is not equal to 1.'

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'Prove the following inequalities, where n is a natural number. [Tohoku University]'

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'Derivatives of Exponential and Logarithmic Functions\nLet \a>0, a \\neq 1\.\n\\[ \egin{array}{l}\n\\cdot \\lim _{h \\rightarrow 0}(1+h)^{\\frac{1}{h}}=\\lim _{x \\rightarrow \\pm \\infty}\\left(1+\\frac{1}{x}\\right)^{x}=e \\quad(e=2.71828 \\cdots \\cdots) \\\\\n\\cdot\\left(e^{x}\\right)^{\\prime}=e^{x}, \\quad\\left(a^{x}\\right)^{\\prime}=a^{x} \\log a \\\\\n(\\log |x|)^{\\prime}=\\frac{1}{x}, \\quad\\left(\\log _{a}|x|\\right)^{\\prime}=\\frac{1}{x \\log a}\n\\end{array} \\]'

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'Find the following indefinite integrals: \n (1) $\\int x e^{2 x} d x$ \n (2) $\\int \\log (x+1) d x$ \n (3) $\\int x \\cos 2 x d x$'

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'177 (1) $\\frac{4}{15}(3 \\sqrt{x}-2)(1+\\sqrt{x}) \\sqrt{1+\\sqrt{x}}+C$ (2) $(\\log |\\sin x|)^{2}+C$ (3) $\\frac{1}{2}\\left(x^{2}-1\\right) e^{x^{2}}+C$'

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'Let n be a natural number greater than or equal to 2. Consider the functions y=e^x ... (1) and y=e^(nx)-1 ... (2).\n(1) Prove that the graphs of (1) and (2) have only one intersection point in the first quadrant.\n(2) Let the coordinates of the intersection point obtained in (1) be (a_n, b_n). Find lim n → ∞ a_n and lim n → ∞ n a_n.\n(3) Let the area enclosed by the graphs of (1) and (2) in the first quadrant and the y-axis be denoted as S_n. Find lim n → ∞ n S_n. [Tokyo Institute of Technology]'

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'Practice (2) For any natural number n, prove that (2nlogn)^{n}<e^{2nlogn} holds true.'

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'Find the value of the constant a when the curves y = x^2 - 2x and y = log x + a are tangent. Also, find the equation of the tangent line at the point of tangency.'

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'Find the following indefinite integrals. (1) $\\int x e^{-x} dx$ (2) $\\int x \\sin x dx$ (3) $\\int x^{2} \\log x dx$ (4) $\\int x \\cdot 3^{x} dx$ (5) $\\int \\frac{\\log(\\log x)}{x} dx$'

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'Transform example (1) by rationalizing the denominator and then integrate.'

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'(3) \x \\tan x+\\log|\\cos x|-\\frac{x^{2}}{2}+C\'

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'\ I_{n} = \\int_{0}^{\\frac{\\pi}{2}} \\sin ^{n} x d x, J_{n} = \\int_{0}^{\\frac{\\pi}{2}} \\cos ^{n} x d x \\left(n\\right. \ is an integer greater than or equal to 0). Prove that \\( I_{n}=J_{n} (n \\geqq 0) \\). Where \ \\sin ^{0} x = \\cos ^{0} x = 1 \.'

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"Given a function y=f(x) defined on all real numbers, which is twice differentiable and always satisfies f’’(x)=-2 f’(x)-2 f(x), answer the following questions: (1) Define a function F(x) as F(x)=e^x f(x), show that F’’(x)=-F(x). (2) Show that a function F(x) satisfying F’’(x)=-F(x) will lead to {F’(x)}^{2}+{F(x)}^{2} being a constant, and find lim_{x -> ∞} f(x). [Kochi Women's University]"

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'Prove the equation \\( \\left(\\cos \\frac{t}{2}\\right)\\left(\\cos \\frac{t}{4}\\right)\\left(\\cos \\frac{t}{8}\\right)=\\frac{\\sin t}{8 \\sin \\frac{t}{8}} \\).'

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'If each side of a cube with edge length a increases at a rate of b per second, then let V be the volume of the cube after t seconds, where V=(a+bt)^3. What is the rate of change of the volume of the cube t seconds after it starts increasing?'

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'Translate the given text into multiple languages.'

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'Therefore, the equation of the tangent at the point (t, log(t+2)) on the curve y=log(x+2) is y-log(t+2)=1/(t+2)(x-t) which simplifies to y=1/(t+2)x+log(t+2)-t/(t+2). The condition for two tangents (1) and (2) to be equal is e^s=1/(t+2) ...(3), -(s-1)e^s=log(t+2)-t/(t+2) <= the slopes and y-intercepts of (1) and (2) respectively are equal. From (3), t+2=1/e^s, therefore t=1/e^s-2. Substituting these into (4), -(s-1)e^s=-s-e^s*(1/e^s-2), thus (s+1)-(s+1)e^s=0, leading to (s+1)(1-e^s)=0, hence s=-1, e^s=1, so s=0,-1. Substituting these into (1), the required equations of the tangents are y=x+1 for s=0 and y=x/e+2/e for s=-1.'

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'Find the equation of the tangent line drawn from the given point P to the following curves, and determine the coordinates of the point of contact.'

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'Please compare the rate of increase of the functions \\( x^{q}(q>0) \\) and \ e^{x} \.'

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'Find the following indefinite integrals. Here, a ≠ 0, b ≠ 0. \n(1) ∫ e^{ax} sin bx dx\n(2) ∫ e^{ax} cos bx dx\n[Similar to Hiroshima City University]'

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'Find the extreme values of the function f(x) = x e^{-2x} and the coordinates of the inflection points of the curve y=f(x).'

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'Investigate the increase and decrease of the following function.'

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'Two differentiable functions $f(x), g(x)$ defined over the entire real numbers satisfy the following conditions:'

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'Find the following about the sequence a_n.\nAlso, considering \\( a_{n}=\\int_{(n-1) \\pi}^{n \\pi}\\left(e^{-x}-e^{-x}|\\cos x|\\right) d x \\) and letting x=t+(n-1) \\pi, we have dx=dt, then\n\\[\egin{aligned}a_{n} &=\\int_{0}^{\\pi}\\left[e^{-t-(n-1) \\pi}-e^{-t-(n-1) \\pi}|\\cos \\{t+(n-1) \\pi\\}|\\right. &=e^{-(n-1) \\pi} \\int_{0}^{\\pi}\\left(e^{-t}-e^{-t}|\\cos t|\\right) d t=e^{-(n-1) \\pi} a_{1} &=\\frac{1}{2} e^{-(n-1) \\pi}\\left(1-2 e^{-\\frac{\\pi}{2}}-e^{-\\pi}\\right)\\end{aligned}\\]'

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'Let a>0, b>0 and f(x)=log((x+a)/(b-x). Prove that the curve y=f(x) is symmetric about its inflection point.'

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'Find the increasing and decreasing intervals of the following functions: (1) y = x - 2√x (2) y = x³ / (x - 2) (3) y = 2x - log x'

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"Let F(x) be the primitive function of f(x), the following conditions [1], [2] hold. Find f'(x), and calculate 175f(x). It is given that x > 0. \n[1] F(x) = xf(x) - 1/x \n[2] F(1/sqrt{2}) = sqrt{2}"

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'Please compare the rate at which the functions \ \\log x \ and \\( x^{p}(p>0) \\) increase.'

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'Calculate f(x) based on the following conditions'

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'When real numbers a, b satisfy 0 < a < b < 1, compare the values of 2^a - 2a/(a-1) and 2^b - 2b/(b-1).'

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'Region D is the area shaded in red in the diagram on the right, so V1 = π∫1e a²(log x)² dx yields [detailed calculation omitted] π(e-2)a². Furthermore, from y= a log x, we have log x = y/a, hence x = e^(y/a), thus V2 = πe²a - π∫0a (e^(y/a))² dy = πe²a - π[(a/2)e^(2y/a)]0a = πe²a - π/2 a(e²-1) = π/2 a{2e²-(e²-1)} = π/2 (e²+1)a. Combining all calculations, we ultimately get π(e-2) a² = π(e²+1)/2 a, since a > 0, then 2(e-2)a = e²+1, hence a = (e²+1)/2(e-2)'

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'Find the area S enclosed by the following curve and line segments:'

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'Practice - Find the maximum and minimum values in the following functions: (1) \ y=\\frac{x^{2}-3 x}{x^{2}+3} \ [Similar to Kansai University] (2) \ y=e^{-x}+x-1 \ [Similar to Nagoya City University]'

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'For the curve C: x=\\frac{e^{t}+3 e^{-t}}{2}, y=e^{t}-2 e^{-t},\n(1) The equation of curve C is x^{2}+1 x y- y^{2}=25.\n(2) Express \\frac{d y}{d x} in terms of x and y.\n(3) At the point on curve C corresponding to t= , \\frac{d y}{d x}=-2.'

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'Prove that for any real number x, the inequality e^(-x^2) ≤ 1 / (1+x^2) holds.'

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'(1) Find the indefinite integral \ \\int e^{2 x+e^{x}} d x \.'

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'From g′(x)=d/dx g(x)=dy/dx = 1/(dx/dy) = 1/f′(y) f(1)=2, we get g(2)=1 from (1) and g′(2)=1/f′(1)=1/2'

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'Describe the characteristics of hyperbolas and their general shape.'

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'Find all linear functions g(x) that satisfy the condition g(f(x))=f(g(x)) for the cubic function f(x)=x³+bx+c.'

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'For constants $a>0, b$, consider the equation in terms of the real number $t$'

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'Using the mean value theorem, prove the following:\n\\nFor e^{-2}<a<b<1, \\quad a-b<b \\log b-a \\log a<b-a\n\'

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'Consider definite integral and recurrence relation 122'

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'Let e be a constant and let the curve 2x^{2}+y^{2}+8x+ey+6=0 be denoted by C. Which of the following statements about the curve C, when the value of e is varied, are correct?'

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'Find the following definite integrals. (1) $\\int_{0}^{\\frac{\\pi}{3}} x \\sin 2x dx$ (2) $\\int_{1}^{e} \\log x dx$ [Osaka Institute of Technology]'

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'Find the following indefinite integrals. (1) $\\int \\frac{\\log x}{x(\\log x+1)^{2}} dx$ (2) $\\int \\frac{e^{3 x}}{(e^{x}+1)^{2}} dx$'

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'Find the inverse functions of the following functions and plot their graphs.'

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'PRACTICE 100\n(1) Prove that x \\geqq 1, x \\log x \\geqq (x-1) \\log (x+1) holds.\n(2) For a natural number n, prove that (n!)^{2} \\geqq n^{n} holds.\n\n[Nagoya University]'

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'Investigate whether the function f(x) is continuous or discontinuous. Where [x] represents the greatest integer that does not exceed the real number x.'

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'Prove that the following equations hold true when PR n is an integer greater than or equal to 2. Where, \ \\cos ^{0} x=1, \\tan ^{0} x=1 \.'

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'Prove that when the function y=log x, the nth derivative of y is (-1)^(n-1) * (n-1)! / x^n.'

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'Question 99\n(1) x=e results in maximum value of e^{1/e}'

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'(2) \ \\log \\left|\\frac{x}{x+1}\\right| - \\frac{1}{x} + C \'

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'Translate the given text into multiple languages.'

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'When the continuous function f(x) satisfies the functional equation f(x)=e^{x} ∫_0^1 1/(e^t+1) dt+∫_0^1 f(t)/(e^t+1) dt, find f(x). [Kyoto Institute of Technology]'

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'Find the equation of the tangent line to the curve y=log(log x) at x=e^{2}.'

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"Using Taylor's theorem, show the third-order Taylor expansion of the function f(x) = e^x around x = 0."

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'\\( \\frac{x\\left(x^{2}+3 x+3\\right)}{3} \\log x - \\frac{x^{3}}{9} - \\frac{x^{2}}{2} - x + C \\)'

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'Find the area S enclosed by the curve y=(3-x)e^{x} and the x-axis, and the lines x=0, x=2.'

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'Find the following indefinite integral: \n\\( \\int_{e}^{e^e} \\frac{\\log (\\log x)}{x \\log x} dx \\)'

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'When real numbers a, b, c, and d satisfy ad-bc≠0, for the function f(x)=\\frac{a x+b}{c x+d}, answer the following questions. (1) Find the inverse function f^{-1}(x) of f(x). (2) Find the relationship between a, b, c, and d that satisfies f^{-1}(x)=f(x) and f(x)≠x.'

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'Find the volume V of the solid obtained by rotating the area enclosed by the following curve and line around the x-axis. (1) y=e^{x}, x-axis, x=0, x=1 (2) y=x^{2}-x, x-axis'

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"In the interval $0<x<2\\pi$, when $y'=0$, we have $x=\\frac{\\pi}{2}$ from $\\sin x-1=0$; and $x=\\frac{7}{6}\\pi, \\frac{11}{6}\\pi$ from $2\\sin x+1=0$. Therefore, the table of increase and decrease of $y$ is as follows."

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'Find the indefinite integral \ \\int \\log \\frac{1}{1+x} dx \.'

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'(4) Let \\( f(x)=e^{(x+1)^{2}}-e^{x^{2}+1} \\) 。 '

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'Find the inverse function of a given function and verify the conditions for the existence of the inverse function. For example, find the inverse of the function y=\x0crac{a x+b}{c x+d}. Verify the condition a d-b c \neq 0.'

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'Find the range of real numbers x for which the sequence {[(x^2-3x-1)/(x^2+x+1)]^n} converges. Also, find the limit value at that point.'

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'Prove that the graph of the function f(x)=log((x+a)/(3a-x)) (a>0) is symmetric about inflection points.'

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'Prove that the following equations hold when n is an integer greater than or equal to 2. Where cos^0x=1 and tan^0x=1.'

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'When the continuous function f(x) satisfies the relation f(x)=e^{x} \\int_{0}^{1} \\frac{1}{e^{t}+1} d t+\\int_{0}^{1} \\frac{f(t)}{e^{t}+1} d t, find f(x).'

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

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'Find the following definite integrals. (1) $\\int_{\\frac{1}{e}}^{e}|\\log x| dx$ (2) $\\int_{-2}^{3} \\sqrt{|x-2|} dx$'

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'96 \\( \\frac{1}{a^{2}+1} e^{a x}(\\sin x + a \\cos x) + C \\)'

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'Prove that the inequality a^b > b^a holds when e<a<b.'

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'Find the definite integral \ \\int_{0}^{\\pi}|\\sin x-\\sqrt{3} \\cos x| d x \.'

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'15\n(1) \\( y^{\\prime}=2(\\log x)^{\\prime}=\\frac{2}{x} \\)'

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'When -\\ frac {\\ pi} {2} \\ leqq \\ theta \\ leqq \\ frac {\\ pi} {3}, \\ cos \\ theta \\ geqq 0, so'

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'Find the extreme values of the function f(x)=x^{1/x}(x>0).'

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'For a natural number n, consider S_{n}(x)=x+x ⋅ (1-3x)/(1-2x) + x ⋅ ((1-3x)/(1-2x))^2 + … + x ⋅ ((1-3x)/(1-2x))^(n-1).'

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"(1) y' = 3^x * log3 + 1\nSince 3^x > 0 and log 3 > 0, y' is always greater than 0\nTherefore, it increases over the entire set of real numbers."

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'Therefore, (1) y=√[5]{(x+3)/(x+1)³} has y′=-{2(x+4)}/{5(x+1)(x+3)}=-{2(x+4)}/{5(x+1)√[5]{(x+1)³(x+3)⁴}} and (2) y=x^{x+1}(x>0) has y′=(log x + {1}/{x} + 1)x^{x+1}'

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'For a positive real number a, let the curve be y=e^{ax} and denote it as C. A line passing through the origin and tangent to the curve C at point P. Let D be the region bounded by C, the line, and the y-axis.'

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'Find the inverse functions of the following two functions. Also, plot their graphs.\n(1) y=-2x+3\n(2) y=log_{2}x\n(3) y=log_{\x0crac{1}{2}}x'

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'For the exponential function y=a^{x} and the logarithmic function y=\\log_{a} x, the limits can be understood from the graph as follows.'

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'Find the inverse function of the given functions.'

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"In today's consumer society, why are ready-made products chosen more often than customized products?"

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'Evaluate the indefinite integral of an irrational function (2) (special substitution integral)'

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'A point P moving on the coordinate plane with coordinates (x, y) given by x = 6e^{t}, y = e^{3t} + 3e^{-t}, where t is any real number.\n1. Eliminate t from the given equations and derive the equation y = f(x) that x and y satisfy.\n2. Illustrate the trajectory of point P.\n3. Find the velocity v of point P at time t.\n4. Determine the distance traveled by point P from t = 0 to t = 3.'

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'For the function f(x)=(ax+b)/(cx+d) (c≠0, ad-bc≠0), answer the following questions.'

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'Let N be a natural number, and define the function f(x) as f(x)=\\sum_{k=1}^{N} \\cos (2 k \\pi x). (1) For integers m, n, find \\int_{0}^{2 \\pi} \\cos (m x) \\cos (n x) dx. (2) Find \\int_{0}^{1} \\cos (4 \\pi x) f(x) dx.'

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'Find the range of real numbers x for which the given sequences converge. Also, determine the limiting value at that time.'

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'95 (3) \ -x - \\sin x - \\frac{1}{\\tan x} - \\frac{1}{\\sin x} + C \'

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'Find the following indefinite integrals:\n(1) \ \\int x \\cos 3 x d x \\n(2) \\( \\int \\log (x+2) d x \\)'

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'(2) The x-coordinates of the intersection points of two curves are the solutions to the equation xe^x = e^x. By rearranging the equation, the general shape of the graph is as shown in the diagram on the right. Therefore,'

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'Example 121 Indefinite integral by integration by parts (3) (same form occurs)'

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'Find the following definite integrals:\n1. $\\int_{2}^{4} \\sqrt{x} d x$\n2. $\\int_{-1}^{1} e^{x} d x$\n3. $\\int_{1}^{3} \\frac{d x}{x}$\n4. $\\int_{0}^{\\pi} \\cos t d t$\n5. $\\int_{0}^{2 \\pi} \\sin 2 x d x$'

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'Important Example 118 Minimum Value and Average of Functions, Median\n\nLet n be a natural number greater than 2, and let a_{1}, a_{2}, ..., a_{n} be real numbers satisfying a_{1} ≤ a_{2} ≤ ... ≤ a_{n}. Let m be the average, s be the standard deviation, and M be the median of n data a_{1}, a_{2}, ..., a_{n}.\n\n(1) Express the minimum value of the function f(x) = (x-a_{1})^2 + (x-a_{2})^2 + ... + (x-a_{n})^2, and the value of x when it is minimal, in terms of n, m, s, M.\n\n(2) Assume n is even. Show that the function g(x) = |x-a_{1}| + |x-a_{2}| + ... + |x-a_{n}| is minimized at x=M.\n\n(3) Assume n is even. Describe the necessary and sufficient conditions for the function g(x) in (2) to have only one x that minimizes it, using the necessary elements of a_{1}, a_{2}, ..., a_{n}.\n\n[Hiroshima Univ.]\n\nGuidance\nAlthough various characters appear, it is still a problem about functions, so think about it in connection with what you learned in Chapter 3. In particular, problems of maximum and minimum values are best solved using graphs. (1) Calculating f(x) will result in a quadratic function. Convert it to the standard form a(x-p)^2+q. (2) Divide the expression containing absolute values into cases—analyze accordingly. a_{1}, a_{2}, ..., a_{n} become the points of division. However, as it is difficult to understand with many absolute values, try examining the graphs for n=2,4 cases to predict at which values the minimum value occurs.'

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How can the parabola $y=x^{2}+4x+5$ be translated to coincide with the parabola $y=x^{2}-6x+8$?

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Find the equation of the parabola obtained by reflecting the parabola $y=-2 x^{2}+4 x-4$ over the $x$ axis, and then translating it 8 units in the $x$ direction and 4 units in the $y$ direction.