Calculus

Problem 101

Evaluate the integral of 9x2sinπx9 x^{2} \sin \pi x with respect to xx.

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Problem 102

Find f(m(k(14)))f(m(k(-14))) given f(x)=x22xf(x)=x^{2}-2x, m(x)=6x5m(x)=\frac{6}{x-5}, and k(x)=x5k(x)=\sqrt{-x-5}. A) 12 B) 15 C) 3 D) 0 E) None of these

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Problem 103

Find the area between the curve y=2lnxxy=\frac{2 \ln x}{x} and the xx-axis, 1xe1 \leq x \leq e.

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Problem 104

A growing circle's radius increases by 5 mm5 \mathrm{~mm} per second. Find the rate of area change when radius is 18 mm18 \mathrm{~mm}. (Round to nearest thousandth)

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Problem 105

Find the derivative of the function y=cscx4+x5y = \frac{\csc x}{4 + x^5}.

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Problem 106

Find false statement about f(x)f(x) with f(x)=x22x+1f'(x) = x^2 - 2x + 1. Options: a) f(x)f(x) increases always, b) f(x)f(x) has horizontal inflection at x=1x=1, c) f(x)f(x) is concave up always, d) f(x)f(x) is concave up for x>1x>1.

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Problem 107

Solve for xx in the equation arcsin(5xπ)=19\arcsin (5 x-\pi)=\frac{1}{9}.

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Problem 108

Find the instantaneous rate of change for f(x)=2x+1f(x)=\sqrt{2 x+1} at x=4x=4.

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Problem 109

Evaluate the definite integral 182x2+2xdx\int_{1}^{8} \frac{2 x^{2}+2}{\sqrt{x}} d x.

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Problem 110

Find the derivative dydx\frac{d y}{d x} of y=x(x9)(x3)y=\frac{x}{(x-9)(x-3)}.

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Problem 111

Find the average value of f(x)=2xf(x) = 2x on the interval [1,1][-1, 1].

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Problem 112

Find f(x)f(x) and the value of cc given 5x3+40=cxf(t)dt5x^3 + 40 = \int_c^x f(t) dt.

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Problem 113

Evaluate the indefinite integral of 1xlnx\frac{1}{x \ln x}.

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Problem 114

Approximate the decrease in fuel efficiency of a car when its speed changes from 65mph65 \mathrm{mph} to 75mph75 \mathrm{mph}, given that E(65)=0.25mpg/mphE'(65) = -0.25 \mathrm{mpg/mph}.

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Problem 115

Find the derivative of f(x)=9sin(sinx)f(x) = 9 \sin(\sin x).

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Problem 116

Find the derivative dydx\frac{dy}{dx} of the implicit function x+xy5x3=2x + xy - 5x^3 = -2.

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Problem 117

Solve 85=ln(6x)\frac{8}{5}=\ln (6 x) for xx.

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Problem 118

Find the derivative of y=5(723x3)10y=5\left(7-\frac{2}{3}x^{3}\right)^{10}.

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Problem 119

Trouve la valeur de aa pour que la tangente à f(x)=x2+axf(x)=x^{2}+a x en x=2x=2 ait pour équation y=8x4y=8x-4.

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Problem 120

Determine the exponential regression equation for the given data points (2,7),(3,10),(5,50),(8,415)(2,7),(3,10),(5,50),(8,415) and use it to estimate yy when x=7x=7.

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Problem 121

Find the length of the curve defined by x=y48+14y2x=\frac{y^{4}}{8}+\frac{1}{4 y^{2}} for 1y31 \leq y \leq 3.

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Problem 122

Find the approximate percent of adult height attained by girls at age 5 using the function f(x)=62+35log(x4)f(x)=62+35 \log (x-4), where xx represents the girl's age (from 5 to 15) and f(x)f(x) represents the percent of her adult height.

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Problem 123

Find the smallest kk for which f(k)(2)0f^{(k)}(2) \neq 0 and the value of this derivative, where f(x)=n=26nn2(7n+1)!(x2)7n+2f(x) = \sum_{n=2}^{\infty} \frac{6^{n} n^{2}}{(7 n+1) !}(x-2)^{7 n+2} is a Taylor series centered at x=2x=2.

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Problem 124

Find rr given the equation A=4πr2A=4 \pi r^{2}. Options: A. r=4πAr=\sqrt{\frac{4 \pi}{A}}, B. r=A4πr=\sqrt{\frac{A}{4 \pi}}, c. r=4πA2r=\frac{4 \pi A}{2}.

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Problem 125

Evaluate the integral 9dx(x+1)(x2+x)\int \frac{9 d x}{(x+1)(x^{2}+x)}. Consider the partial fraction decomposition 9(x+1)(x2+x)=9x9x(x+1)2\frac{9}{(x+1)(x^{2}+x)}=\frac{9}{x}-\frac{9 x}{(x+1)^{2}}. Identify any errors in the work shown.

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Problem 126

Find the value of h(1)h(1) where h(x)=9x6x11h(x) = \frac{9x}{6x-11}.

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Problem 127

Earthquake intensity is 10M=II010^{M}=\frac{I}{I_{0}}, where MM is magnitude, II is intensity, and I0I_{0} is reference intensity. Earthquake AA is 225 times as intense as BB, and BB has magnitude 3.8. Which equation determines magnitude mm of earthquake AA? m=log225+3.8m=\log 225+3.8

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Problem 128

Evaluate the absolute value of the integral of x24|x^2 - 4| from -2 to 2, and the absolute value of the integral of x24x^2 - 4 from -2 to 2. Are the results the same?

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Problem 129

Evaluate the definite integral 14dxx+1\int_{-1}^{4} \frac{d x}{\sqrt{|x+1|}} and determine if it converges or diverges.

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Problem 130

Encuentra la derivada de f(x)=ln5xf(x) = \ln 5x.

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Problem 131

Differentiate f(x)=(77125)xf(x)=\left(\frac{77}{125}\right)^{x} and rewrite the logarithms using only prime numbers.

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Problem 132

Determine the credit card balance after 6 months using the exponential function f(x)=800(1+0.122)xf(x)=800(1+0.122)^{x}, where xx represents the number of months. Round the answer to the nearest cent.

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Problem 133

Find the derivative of g(x)=ax3+ax2+ag(x)=a x^{3}+a x^{2}+a and show g(1+2)=g(1+2)a2g(1+\sqrt{2})=g'(1+\sqrt{2})-a \sqrt{2}.

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Problem 134

Find the range of y=2cos(x2π)3y=-2 \cos (-x-2 \pi)-3. The range is [5,1]\left[-5, -1\right].

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Problem 135

Find the value of cc such that the probability that ZZ (a standard normal random variable) exceeds cc is 0.9837, rounded to 2 decimal places.

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Problem 136

Find the derivative of the function g(x)=4x+3x2g(x)=\frac{4x+3}{x^2}.

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Problem 137

Find the limit of the expression (f(x)f(9))/(x3)(f(x) - f(9)) / (\sqrt{x} - 3) as xx approaches 9, given that ff is differentiable.

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Problem 138

Find the point on f(x)=x33x2f(x) = x^3 - 3x^2 where the tangent line slope equals the secant line slope through A(2,4)A(2,-4) and B(3,0)B(3,0), to 2 decimal places.

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Problem 139

Find the derivative yy' of the equation (6xy)4+2y3=14639(6x-y)^4 + 2y^3 = 14639 evaluated at the point (2,1)(-2,-1).

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Problem 140

Determine the function describing a toy rocket's height, its maximum height and time, the time interval it's above 268 feet, and when it hits the ground, given an initial velocity of 180180 feet/second from a 145145-foot building.

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Problem 141

Rewrite cosh6xsinh6x\cosh 6x - \sinh 6x in terms of exponentials and simplify.

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Problem 142

Determine the possible number of positive and negative real zeros for f(x)=4x4+3x39x2+6x9f(x) = -4x^4 + 3x^3 - 9x^2 + 6x - 9 using Descartes' Rule of Signs.

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Problem 143

Graph y=2cos2(x)y=2 \cos^{2}(x) and y=2ex+2x2y=2 e^{-x}+2 x-2 on [0,2π)[0, 2\pi). Find the intersection points and their coordinates to 4 decimal places. If no solution, enter "NO SOLUTION".

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Problem 144

Find the rate of change of height of a right circular cylinder with constant surface area 600π2600\pi^2 and radius increasing at 44 cm/sec.

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Problem 145

Find the approximate displacement of an object moving at v=2t+1v = 2t + 1 (m/s) for 0t80 \leq t \leq 8, using n=2n = 2 subintervals.

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Problem 146

Find the value of pp where the derivative H(p)H'(p) of the function H(p)=2p(1p)H(p) = 2p(1-p) is zero.

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Problem 147

Find the equation of the function passing through (3,2)(3,-2) with derivative dydx=3x4\frac{dy}{dx}=3x-4.

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Problem 148

Find the age xx where the percentage of Americans with coronary heart disease is 67%67\% using the logistic growth function P(x)=901+271e0.122xP(x)=\frac{90}{1+271 e^{-0.122 x}}.

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Problem 149

Find the approximation for f(24.85)f(24.85) using the tangent line of f(x)=4x1/2f(x) = -4x^{1/2} at x=25x = 25.

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Problem 150

Use implicit differentiation to find the derivative dy/dx. Find the slope of the curve at the point (9,1) for the equation 2xy+5x(3/2)y(1/2)=1532xy + 5x^(3/2)y^(-1/2) = 153.

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Problem 151

Solve ln(6x+2)=3\ln (6x + 2) = 3 for xx. The exact solution is x=(e32)/6x = (e^3 - 2)/6. Rounded to 4 decimal places, x=1.1709x = 1.1709.

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Problem 152

Find the concavity of the graph with x=t3+1x=t^{3}+1 and y=t4+ty=t^{4}+t at t=1t=1.

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Problem 153

Graph g(x)=1+log2(x+3)g(x) = 1 + \log_2(x + 3). Plot 2 points, draw asymptote, then graph. Provide domain and range in interval notation.

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Problem 154

Find the linear approximation to g(x)=x4g(x)=\sqrt[4]{x} at x=2x=2. Use it to approximate 34\sqrt[4]{3} and 104\sqrt[4]{10}. Compare to exact values. Analyze percent error and accuracy.

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Problem 155

Find the value of f(1)f(1) where f(x)=x34f(x) = \sqrt[3]{x} - 4.

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Problem 156

Find the correct statement about the exponential function k(x)=5(27)xk(x)=5\left(\frac{2}{7}\right)^{x}: A) Increasing, concave up B) Increasing, concave down C) Decreasing, concave up D) Decreasing, concave down.

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Problem 157

Find the first derivative of y=(29x214x3)5y = (2 - 9x^2 - 14x^3)^5 using the chain rule.

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Problem 158

Find (fg)(x)(f \circ g)(x) and (gf)(x)(g \circ f)(x) for f(x)=1xf(x)=\frac{1}{x} and g(x)=2sin(x)g(x)=2 \sin (x).

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Problem 159

Find the inverse function of the vertically dilated logarithmic function with base 10 and vertical asymptote at x=2x=2.

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Problem 160

(A) Find the exact cost of producing the 71st food processor, given C(x)=1900+40x0.2x2C(x)=1900+40x-0.2x^2. (B) Use the marginal cost to approximate the cost of producing the 71st food processor. (A) The exact cost of producing the 71st food processor is $2,662\$2,662. (B) Using the marginal cost, the approximate cost of producing the 71st food processor is $2,660\$2,660.

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Problem 161

Find the change in xx and yy when yy varies at a constant rate of 4 with respect to xx, for the given ranges of xx.
a. If xx varies from x=3.7x=3.7 to x=8.8x=8.8, then: i. The change in xx is Δx=5.1\Delta x=5.1 ii. The corresponding change in yy is Δy=20.4\Delta y=20.4
b. If xx varies from x=2x=2 to x=7x=-7, then: i. The change in xx is Δx=9\Delta x=-9 ii. The corresponding change in yy is Δy=36\Delta y=-36

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Problem 162

Sketch the function f(x)=x33x2f(x) = -x^3 - 3x^2 and select the correct graph.

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Problem 163

Find the derivative of y=ln[x(x52)10]y=\ln \left[x\left(x^{5}-2\right)^{10}\right] using logarithm properties.

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Problem 164

Find the derivative of y=(lnx)lnxy = (\ln x)^{\ln x}.

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Problem 165

Find and simplify the expressions for f(x+h)f(x+h), f(x+h)f(x)f(x+h) - f(x), and f(x+h)f(x)h\frac{f(x+h) - f(x)}{h} where f(x)=3x27x+6f(x) = 3x^{2} - 7x + 6.

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Problem 166

Find f(1)f'(1) if the normal line to ff at (1,2)(1,2) has equation x+5y=11x+5y=11.

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Problem 167

Find the derivative of f(x)=6x(24x)4f(x)=\frac{6 x}{(2-4 x)^{4}}.

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Problem 168

Find derivatives of s=Tx2+7xG+T2s=T x^{2}+7 x G+T^{2} with respect to xx, GG, and TT.

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Problem 169

The velocity needed to remove a foreign object from a 26-mm radius windpipe is V(r)=k(26r2r3)V(r) = k(26r^2 - r^3), where 0r260 \leq r \leq 26. The object that needs maximum velocity to remove has a radius of 1313 mm.

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Problem 170

Étudier la fonction g(x)=1+lnxxg(x)=1+\frac{\ln x}{x} définie sur ]0;+[\left]0;+\infty\right[. Résoudre g(x)=0g(x)=0 et déterminer le signe de gg. Étudier la famille de fonctions hn(x)=x2n+nlnxh_n(x)=x^2-n+n\ln x avec nNn\in\mathbb{N}^*, trouver leurs zéros et leur signe. Étudier la famille de fonctions fn(x)=xnnlnxxf_n(x)=x-n-\frac{n\ln x}{x}, leurs asymptotes, variations et minimum. Calculer une primitive de fnf_n.

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Problem 171

Find the derivative g(u)g'(u) of g(u)=3u2x+x2dxg(u) = \int_{3}^{u} \frac{-2}{x+x^{2}} dx using the Fundamental Theorem of Calculus.

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Problem 172

Rewrite f(x)=logaxf(x) = \log_a x to show f(x)=log1/ax-f(x) = \log_{1/a} x. Let y=f(x)y = f(x), then y=logaxy = \log_a x. Rewrite the left side of the equation using logarithm properties to show f(x)=log1/ax-f(x) = \log_{1/a} x.

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Problem 173

Find the Maclaurin series expansions up to 5th degree for y=sinxcosxy=\sin x \cos x, y=ln(x+1)y=\ln(x+1), and y=xcosxy=x \cos x.

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Problem 174

Solve the indefinite integral of cot1x\cot^{-1}x by selecting the correct antiderivative.

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Problem 175

Find the range of f(x)=1x24f(x) = \frac{1}{\sqrt{|x^2 - 4|}}.

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Problem 176

Find the derivative of the function y=x5x+6y=\frac{\sqrt{x}-5}{\sqrt{x}+6}.

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Problem 177

Find the value of f(2)f(-2) given f(x)=4x3+7xf(x)=4x^3+7x.

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Problem 178

Find the volume of a solid with a triangular base and semicircular cross-sections perpendicular to the xx-axis. The triangle has vertices at (0,0),(2,0),(0,2)(0,0), (2,0), (0,2).
Select one: V=02π8(2x)2dxV=\int_{0}^{2} \frac{\pi}{8}(2-x)^{2} d x V=02π4(x2)2dxV=\int_{0}^{2} \frac{\pi}{4}(x-2)^{2} d x

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Problem 179

Find the integral of 2f(x)+g(x)2f(x) + g(x) from 4 to 9 given 49f(x)dx=1\int_{4}^{9} f(x) dx=1 and 94g(x)dx=5\int_{9}^{4} g(x) dx=5.

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Problem 180

Solve the exponential equation e2x=18e^{-2x} = 18 for the value of xx.

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Problem 181

Find the value of xx if ln(x)ln(x)=12\ln(x) - \ln(\sqrt{x}) = \frac{1}{2}.

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Problem 182

Evaluate the integral π3π43cos(4x)dx\int_{-\frac{\pi}{3}}^{\frac{\pi}{4}} 3 \cos(4x) \, dx.

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Problem 183

Find the composite function f(r(x))f(r(x)) where f(x)=19x2f(x)=19x^2 and r(x)=18xr(x)=\sqrt{18x}.

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Problem 184

Differentiate the given functions: f(x)=x2+4xf(x)=x^{2}+4x, f(x)=5x32x4f(x)=5x^{3}-2x^{4}, f(x)=x57x4+3x3f(x)=x^{5}-7x^{4}+3x^{3}, g(x)=12x41xg(x)=\frac{1}{2}x^{4}-\frac{1}{x}, g(x)=(2x3)2g(x)=(2x-3)^{2}, g(x)=19x316x2g(x)=\frac{1}{9}x^{-3}-\frac{1}{6}x^{-2}, y=13x2xy=\frac{1}{3}\sqrt{x}-\frac{2}{\sqrt{x}}, y=6x43x3+10x5y=6\sqrt[4]{x}-3\sqrt[3]{x}+\frac{10}{\sqrt[5]{x}}, y=(x22x)3y=(x^{2}-2x)^{3}.

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Problem 185

Identify the domain and graph the function f(x)=x2f(x)=\sqrt{x-2}.

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Problem 186

Find the value of xx that satisfies the equation ln(3x)=2x5\ln(3x) = 2x - 5.

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Problem 187

Find the power series for the indefinite integral e8x17xdx\int \frac{e^{8x} - 1}{7x} dx, and give the first 5 nonzero terms.
f(x)=C+8x7+64x272!+512x373!+4096x474!+32768x575!+f(x) = C + \frac{8x}{7} + \frac{64x^2}{7 \cdot 2!} + \frac{512x^3}{7 \cdot 3!} + \frac{4096x^4}{7 \cdot 4!} + \frac{32768x^5}{7 \cdot 5!} + \cdots

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Problem 188

Solve for yy in terms of xx, given the equation ln(y9)ln4=x+lnx\ln(y-9) - \ln 4 = x + \ln x.

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Problem 189

Which limit definition represents the derivative of f(x)f(x)? Options: limh0f(x+h)f(x)h\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}, limxaf(x)f(a)xa\lim_{x\to a} \frac{f(x)-f(a)}{x-a}

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Problem 190

Find h(1)h'(-1) where h(x)=(x3+p(x))4h(x)=(x^{3}+p(x))^{4} and the given p(x)p(x) values.

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Problem 191

Evaluate the exponential function h(x)=1.5xh(x)=1.5^{x} for x=3x=\sqrt{3}, and round the result to the nearest hundredth.

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Problem 192

Find the marginal cost of the cost function C(x)=15,000+60x+1,000xC(x) = 15,000 + 60x + \frac{1,000}{x} at x=100x = 100. State the units.

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Problem 193

Find the derivative of y=x3y=\sqrt{x-3} using the definition of derivative.

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Problem 194

Solve for xx in the equation 5ex/78=05 e^{x / 7} - 8 = 0.

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Problem 195

Find the limit of 3an6bn3a_n - 6b_n if an6a_n \to 6 and bn9b_n \to 9.

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Problem 196

Find the derivative of y=3x2e1xy=3 x^{2} e^{\frac{1}{x}} and simplify.

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Problem 197

Find the limit of the sequence an=n/n3+8a_n = n/\sqrt{n^3 + 8} or state that it diverges.

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Problem 198

Identify local/absolute extrema and inflection points of y=(x3)3+4y=(x-3)^{3}+4. Find local max/min points. A. Local max point(s): (3,4)(3, 4) B. Local min point(s): (3,4)(3, 4)

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Problem 199

Find the expression to solve for xx in the equation 8x=118^{x}=11.

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Problem 200

Determine weekly sales decline after ad campaign using y=18,000(80.05x)y=18,000(8^{-0.05x}), where xx is weeks since end. Find sales at end and 2 weeks later, and whether sales reach $0\$ 0.

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