Calculus

Problem 101

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

See Solution

Problem 102

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

See Solution

Problem 103

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

See Solution

Problem 104

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

See Solution

Problem 105

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

See Solution

Problem 106

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

See Solution

Problem 107

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

See Solution

Problem 108

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

See Solution

Problem 109

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

See Solution

Problem 110

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

See Solution

Problem 111

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

See Solution

Problem 112

Find $f(x)$ and the value of $c$ given $5x^3 + 40 = \int_c^x f(t) dt$ .

See Solution

Problem 113

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

See Solution

Problem 114

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

See Solution

Problem 115

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

See Solution

Problem 116

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

See Solution

Problem 117

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

See Solution

Problem 118

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

See Solution

Problem 119

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

See Solution

Problem 120

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

See Solution

Problem 121

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

See Solution

Problem 122

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

See Solution

Problem 123

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

See Solution

Problem 124

Find $r$ given the equation $A=4 \pi r^{2}$ . Options: A. $r=\sqrt{\frac{4 \pi}{A}}$ , B. $r=\sqrt{\frac{A}{4 \pi}}$ , c. $r=\frac{4 \pi A}{2}$ .

See Solution

Problem 125

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

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Problem 126

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

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Problem 127

Earthquake intensity is $10^{M}=\frac{I}{I_{0}}$ , where $M$ is magnitude, $I$ is intensity, and $I_{0}$ is reference intensity. Earthquake $A$ is 225 times as intense as $B$ , and $B$ has magnitude 3.8. Which equation determines magnitude $m$ of earthquake $A$ ? $m=\log 225+3.8$

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Problem 128

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

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Problem 129

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

See Solution

Problem 130

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

See Solution

Problem 131

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

See Solution

Problem 132

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

See Solution

Problem 133

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

See Solution

Problem 134

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

See Solution

Problem 135

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

See Solution

Problem 136

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

See Solution

Problem 137

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

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Problem 138

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

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Problem 139

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

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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 $180$ feet/second from a $145$ -foot building.

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Problem 141

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

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Problem 142

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

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Problem 143

Graph $y=2 \cos^{2}(x)$ and $y=2 e^{-x}+2 x-2$ on $[0, 2\pi)$ . 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 $600\pi^2$ and radius increasing at $4$ cm/sec.

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Problem 145

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

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Problem 146

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

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Problem 147

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

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Problem 148

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

See Solution

Problem 149

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

See Solution

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 $2xy + 5x^(3/2)y^(-1/2) = 153$ .

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Problem 151

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

See Solution

Problem 152

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

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Problem 153

Graph $g(x) = 1 + \log_2(x + 3)$ . 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 $g(x)=\sqrt[4]{x}$ at $x=2$ . Use it to approximate $\sqrt[4]{3}$ and $\sqrt[4]{10}$ . Compare to exact values. Analyze percent error and accuracy.

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Problem 155

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

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Problem 156

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

See Solution

Problem 157

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

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Problem 158

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

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Problem 159

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

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Problem 160

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

See Solution

Problem 161

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

See Solution

Problem 162

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

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Problem 163

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

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Problem 164

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

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Problem 165

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

See Solution

Problem 166

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

See Solution

Problem 167

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

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Problem 168

Find derivatives of $s=T x^{2}+7 x G+T^{2}$ with respect to $x$ , $G$ , and $T$ .

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Problem 169

The velocity needed to remove a foreign object from a 26-mm radius windpipe is $V(r) = k(26r^2 - r^3)$ , where $0 \leq r \leq 26$ . The object that needs maximum velocity to remove has a radius of $13$ mm.

See Solution

Problem 170

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

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Problem 171

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

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Problem 172

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

See Solution

Problem 173

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

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Problem 174

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

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Problem 175

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

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Problem 176

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

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Problem 177

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

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Problem 178

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

See Solution

Problem 179

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

See Solution

Problem 180

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

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Problem 181

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

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Problem 182

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

See Solution

Problem 183

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

See Solution

Problem 184

Differentiate the given functions: $f(x)=x^{2}+4x$ , $f(x)=5x^{3}-2x^{4}$ , $f(x)=x^{5}-7x^{4}+3x^{3}$ , $g(x)=\frac{1}{2}x^{4}-\frac{1}{x}$ , $g(x)=(2x-3)^{2}$ , $g(x)=\frac{1}{9}x^{-3}-\frac{1}{6}x^{-2}$ , $y=\frac{1}{3}\sqrt{x}-\frac{2}{\sqrt{x}}$ , $y=6\sqrt[4]{x}-3\sqrt[3]{x}+\frac{10}{\sqrt[5]{x}}$ , $y=(x^{2}-2x)^{3}$ .

See Solution

Problem 185

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

See Solution

Problem 186

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

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Problem 187

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

See Solution

Problem 188

Solve for $y$ in terms of $x$ , given the equation $\ln(y-9) - \ln 4 = x + \ln x$ .

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Problem 189

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

See Solution

Problem 190

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

See Solution

Problem 191

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

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Problem 192

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

See Solution

Problem 193

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

See Solution

Problem 194

Solve for $x$ in the equation $5 e^{x / 7} - 8 = 0$ .

See Solution

Problem 195

Find the limit of $3a_n - 6b_n$ if $a_n \to 6$ and $b_n \to 9$ .

See Solution

Problem 196

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

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Problem 197

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

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Problem 198

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

See Solution

Problem 199

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

See Solution

Problem 200

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

See Solution

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Calculus

Problem 101

Evaluate the integral of 9x2sin⁡πx9 x^{2} \sin \pi x9x2sinπx with respect to xxx.

Problem 102

Find f(m(k(−14)))f(m(k(-14)))f(m(k(−14))) given f(x)=x2−2xf(x)=x^{2}-2xf(x)=x2−2x, m(x)=6x−5m(x)=\frac{6}{x-5}m(x)=x−56​, and k(x)=−x−5k(x)=\sqrt{-x-5}k(x)=−x−5​. A) 12 B) 15 C) 3 D) 0 E) None of these

Problem 103

Find the area between the curve y=2ln⁡xxy=\frac{2 \ln x}{x}y=x2lnx​ and the xxx-axis, 1≤x≤e1 \leq x \leq e1≤x≤e.

Problem 104

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

Problem 105

Find the derivative of the function y=csc⁡x4+x5y = \frac{\csc x}{4 + x^5}y=4+x5cscx​.

Problem 106

Find false statement about f(x)f(x)f(x) with f′(x)=x2−2x+1f'(x) = x^2 - 2x + 1f′(x)=x2−2x+1. Options: a) f(x)f(x)f(x) increases always, b) f(x)f(x)f(x) has horizontal inflection at x=1x=1x=1, c) f(x)f(x)f(x) is concave up always, d) f(x)f(x)f(x) is concave up for x>1x>1x>1.

Problem 107

Solve for xxx in the equation arcsin⁡(5x−π)=19\arcsin (5 x-\pi)=\frac{1}{9}arcsin(5x−π)=91​.

Problem 108

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

Problem 109

Evaluate the definite integral ∫182x2+2xdx\int_{1}^{8} \frac{2 x^{2}+2}{\sqrt{x}} d x∫18​x​2x2+2​dx.

Problem 110

Find the derivative dydx\frac{d y}{d x}dxdy​ of y=x(x−9)(x−3)y=\frac{x}{(x-9)(x-3)}y=(x−9)(x−3)x​.

Problem 111

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

Problem 112

Find f(x)f(x)f(x) and the value of ccc given 5x3+40=∫cxf(t)dt5x^3 + 40 = \int_c^x f(t) dt5x3+40=∫cx​f(t)dt.

Problem 113

Evaluate the indefinite integral of 1xln⁡x\frac{1}{x \ln x}xlnx1​.

Problem 114

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

Problem 115

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

Problem 116

Find the derivative dydx\frac{dy}{dx}dxdy​ of the implicit function x+xy−5x3=−2x + xy - 5x^3 = -2x+xy−5x3=−2.

Problem 117

Solve 85=ln⁡(6x)\frac{8}{5}=\ln (6 x)58​=ln(6x) for xxx.

Problem 118

Find the derivative of y=5(7−23x3)10y=5\left(7-\frac{2}{3}x^{3}\right)^{10}y=5(7−32​x3)10.

Problem 119

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

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)(2,7),(3,10),(5,50),(8,415) and use it to estimate yyy when x=7x=7x=7.

Problem 121

Find the length of the curve defined by x=y48+14y2x=\frac{y^{4}}{8}+\frac{1}{4 y^{2}}x=8y4​+4y21​ for 1≤y≤31 \leq y \leq 31≤y≤3.

Problem 122

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

Problem 123

Problem 124

Find rrr given the equation A=4πr2A=4 \pi r^{2}A=4πr2. Options: A. r=4πAr=\sqrt{\frac{4 \pi}{A}}r=A4π​​, B. r=A4πr=\sqrt{\frac{A}{4 \pi}}r=4πA​​, c. r=4πA2r=\frac{4 \pi A}{2}r=24πA​.

Problem 125

Evaluate the integral ∫9dx(x+1)(x2+x)\int \frac{9 d x}{(x+1)(x^{2}+x)}∫(x+1)(x2+x)9dx​. Consider the partial fraction decomposition 9(x+1)(x2+x)=9x−9x(x+1)2\frac{9}{(x+1)(x^{2}+x)}=\frac{9}{x}-\frac{9 x}{(x+1)^{2}}(x+1)(x2+x)9​=x9​−(x+1)29x​. Identify any errors in the work shown.

Problem 126

Find the value of h(1)h(1)h(1) where h(x)=9x6x−11h(x) = \frac{9x}{6x-11}h(x)=6x−119x​.

Problem 127

Problem 128

Evaluate the absolute value of the integral of ∣x2−4∣|x^2 - 4|∣x2−4∣ from -2 to 2, and the absolute value of the integral of x2−4x^2 - 4x2−4 from -2 to 2. Are the results the same?

Problem 129

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

Problem 130

Encuentra la derivada de f(x)=ln⁡5xf(x) = \ln 5xf(x)=ln5x.

Problem 131

Differentiate f(x)=(77125)xf(x)=\left(\frac{77}{125}\right)^{x}f(x)=(12577​)x and rewrite the logarithms using only prime numbers.

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}f(x)=800(1+0.122)x, where xxx represents the number of months. Round the answer to the nearest cent.

Problem 133

Find the derivative of g(x)=ax3+ax2+ag(x)=a x^{3}+a x^{2}+ag(x)=ax3+ax2+a and show g(1+2)=g′(1+2)−a2g(1+\sqrt{2})=g'(1+\sqrt{2})-a \sqrt{2}g(1+2​)=g′(1+2​)−a2​.

Problem 134

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

Problem 135

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

Problem 136

Find the derivative of the function g(x)=4x+3x2g(x)=\frac{4x+3}{x^2}g(x)=x24x+3​.

Problem 137

Find the limit of the expression (f(x)−f(9))/(x−3)(f(x) - f(9)) / (\sqrt{x} - 3)(f(x)−f(9))/(x​−3) as xxx approaches 9, given that fff is differentiable.

Problem 138

Find the point on f(x)=x3−3x2f(x) = x^3 - 3x^2f(x)=x3−3x2 where the tangent line slope equals the secant line slope through A(2,−4)A(2,-4)A(2,−4) and B(3,0)B(3,0)B(3,0), to 2 decimal places.

Problem 139

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

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 180180180 feet/second from a 145145145-foot building.

Problem 141

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

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

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

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

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

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

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

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

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

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

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

Find $f(x)$ and the value of $c$ given $5x^3 + 40 = \int_c^x f(t) dt$ .

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

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

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

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

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

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

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

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

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

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

Find $r$ given the equation $A=4 \pi r^{2}$ . Options: A. $r=\sqrt{\frac{4 \pi}{A}}$ , B. $r=\sqrt{\frac{A}{4 \pi}}$ , c. $r=\frac{4 \pi A}{2}$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 $180$ feet/second from a $145$ -foot building.

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

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

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

Find the rate of change of height of a right circular cylinder with constant surface area $600\pi^2$ and radius increasing at $4$ cm/sec.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Find derivatives of $s=T x^{2}+7 x G+T^{2}$ with respect to $x$ , $G$ , and $T$ .

The velocity needed to remove a foreign object from a 26-mm radius windpipe is $V(r) = k(26r^2 - r^3)$ , where $0 \leq r \leq 26$ . The object that needs maximum velocity to remove has a radius of $13$ mm.

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

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

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