Calculus

Problem 10101

Find the tangent line equation to $y=\sin(3x)+\cos(8x)$ at $\left(\frac{\pi}{6}, y\left(\frac{\pi}{6}\right)\right)$ .

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Problem 10102

A farmhouse is $a=1.5 \mathrm{~km}$ from a highway. An auto moves at $v=81 \mathrm{~km/h}$ . Find how fast distance $l$ increases when $s=3.7 \mathrm{~km}$ . Answer to three decimal places: $\mathrm{km/h}$ .

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Problem 10103

Find the derivative of $f(x)=x^{7}-2x^{5}+5x^{3}-7x$ .

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Problem 10104

Julian jogs on a circular track with radius $29 \mathrm{~m}$ . At $(21,20)$ , his $x$ -coordinate changes at $-5.25 \mathrm{~m/s}$ . Find $\frac{d y}{d t}$ . (Round to two decimal places.)

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Problem 10105

Differentiate $(f-g)^2$ and evaluate at $x=0$ using $f$ and $g$ values from the table.

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Problem 10106

Find the slope of the tangent line to the curve $5 \sin (x) + 2 \cos (y) - 6 \sin (x) \cos (y) + x = 6 \pi$ at $(6 \pi, 5 \pi / 2)$ .

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Problem 10107

Differentiate $\frac{1}{f-g}$ and evaluate at $x=2$ . Use values from the table. Leave the answer as a fraction.

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Problem 10108

Find the solution to the equation $2 \sqrt{y}=8-0.4 t$ with $y(0)=49$ and show $y$ is quadratic in $t$ .

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Problem 10109

Find $\frac{d P}{d t}$ for $b=1.6, P=10 \mathrm{kPa}, V=60 \mathrm{cm}^{3}$ , $\frac{d V}{d t}=80 \mathrm{cm}^{3}/\mathrm{min}$ .

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Problem 10110

Find the derivative of $f(x)=\frac{x^{3}}{3}+\frac{3}{x^{3}}$ .

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Problem 10111

Find the volume of the torus formed by revolving the circle $(x-17)^{2}+y^{2}=1$ around the $y$ -axis.

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Problem 10112

Brooke walks east at $5 \mathrm{~km/h}$ and Jamail west at $4 \mathrm{~km/h}$ . Find their distance apart after $11 \mathrm{~s}$ .

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Problem 10113

Given $y^{2}+x y-3 x=51$ and $\frac{d y}{d t}=-1$ at $x=-5$ , $y=-4$ , find $\frac{d x}{d t}$ .

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Problem 10114

A rocket moves up at $1300 \mathrm{~km} / \mathrm{hr}$ . Find the angle rate increase from an observer $18 \mathrm{~km}$ away after 3 min. $\frac{d \theta}{d t} \approx$

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Problem 10115

Differentiate $\frac{1}{f^{2}}$ and evaluate at $x=0$ . Use $f'(0)=4$ from the table. Answer in fraction form!

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Problem 10116

Helium fills a balloon at 4 ft³/s. Find the radius increase rate after 2 minutes using $V=\frac{4}{3}\pi r^{3}$ .

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Problem 10117

Differentiate $f(g(x))$ and evaluate at $x = -1$ using values from the table. What is $\frac{d y}{d x}$ ?

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Problem 10118

Water leaks from a conical tank at 6300 cm³/min. If the water rises at 28 cm/min when 2 m high, find the inflow rate.

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Problem 10119

Differentiate $(f g)^{\frac{1}{2}}$ and evaluate at $x=0$ . Leave your answer in fraction form!

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Problem 10120

Differentiate $(f g)^{-1}$ and evaluate at $x=-1$ . Leave the answer in fraction form!

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Problem 10121

Given $y=5 \sqrt{x}$ , find $\Delta y$ and $d y$ for $x=4$ and $\Delta x=d x=0.3$ .

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Problem 10122

Differentiate $\frac{g}{f}$ and evaluate at $x=1$ . Use the given values for $f(x)$ and $g(x)$ . Express as a fraction.

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Problem 10123

A baseball diamond is a square with sides of 90 ft. A player runs to first base at 29 ft/s. Find the rate of change of distance to second base halfway to first. (Round to two decimal places.)

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Problem 10124

Differentiate $f \cdot g$ and evaluate at $x=0$ . Use values from the table for $f(x)$ , $f'(x)$ , $g(x)$ , $g'(x)$ .

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Problem 10125

Differentiate $(f \cdot g)^3$ and evaluate at $x=-1$ using given values for $f, f', g, g'$ .

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Problem 10126

Differentiate $f-g$ and evaluate at $x=-1$ . Use values: $f'(-1)$ and $g'(-1)$ . Find $\frac{d}{dx}(f-g)$ .

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Problem 10127

Find $d y$ for $y=4 x^{2}+3 x+3$ at $x=5$ with $d x=0.2$ and $d x=0.4$ .

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Problem 10128

Differentiate $\sqrt{f+g}$ and evaluate at $x=1$ using the given values for $f$ and $g$ .

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Problem 10129

Find $\frac{d P}{d t}$ for $b=1.4$ , $P=7 \mathrm{kPa}$ , $V=110 \mathrm{~cm}^{3}$ , and $\frac{d V}{d t}=50 \mathrm{~cm}^{3}/\mathrm{min}$ .

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Problem 10130

Differentiate $(f g)^2$ and evaluate at $x=0$ . Use $f'(0)$ and $g'(0)$ from the provided values.

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Problem 10131

A car travels past a farmhouse 1.5 km away at 81 km/h. Find the speed of distance $l$ when the car is 3.7 km past the intersection. Answer to three decimal places.

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Problem 10132

Differentiate $f^2$ and evaluate at $x=2$ using given values for $f(x), f'(x), g(x), g'(x)$ .

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Problem 10133

Differentiate $g f$ and evaluate at $x=1$ . Use $f(1), f'(1), g(1), g'(1)$ from the table.

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Problem 10134

Find the percentage error in the volume and surface area of a cube with edge $30 \mathrm{~cm}$ and error $0.1 \mathrm{~cm}$ .

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Problem 10135

Water in a tank drains by Torricelli's Law: $\frac{d y}{d t}=-0.4 \sqrt{y}$ . Find depth after 0.5s if $y=36 \mathrm{~cm}$ .

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Problem 10136

Estimate the upper bound for error in $d(t)=16 t^{2}$ when $t=4$ seconds and $t=5$ seconds with $0.2$ sec accuracy.

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Problem 10137

Find the derivative of the function $f(x)=\tan^{2}(x^{4})$ . What is $f^{\prime}(x)$ ?

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Problem 10138

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{e^{x}-1}{\sin (3 x)}$ using L'Hospital's rule.

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Problem 10139

Find $y^{\prime}$ and $y^{\prime \prime}$ for $y=e^{8 e^{x}}$ .

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Problem 10140

Find the derivative of $y(x)=\cos(\tan(7x))$ . What is $d y / d x$ ? $d y / d x=$

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Problem 10141

Differentiate the function: $g(x)=(x+7 \sqrt{x}) e^{x}$ .

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Problem 10142

Evaluate the limit using L'Hopital's rule: $\lim _{x \rightarrow \frac{\pi}{2}} 7 \cos (3 x) \sec (-7 x)$

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Problem 10143

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{\sin (2 x)}{\sin (15 x)}$ using L'Hospital's rule if needed.

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Problem 10144

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{\sin (10 x)}{\tan (12 x)}$ using L'Hospital's rule.

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Problem 10145

Derive $E\left[e^{-X^{2} / 2}\right]$ for $X \sim \mathcal{N}(0,1)$ using $\int_{-\infty}^{\infty} \exp(-\frac{u^{2}}{2}) du=\sqrt{2 \pi}$ .

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Problem 10146

Evaluate the limit using L'Hôpital's rule: $\lim _{x \rightarrow 0} \frac{e^{x}+x-1}{4 x}$

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Problem 10147

Find the rate of volume increase of a cone when $r=5 \mathrm{~cm}$ , $h=20 \mathrm{~cm}$ , both growing at $7 \mathrm{~cm/s}$ .

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Problem 10148

Find the tangential and normal acceleration components for $\vec{r}(t)=\left\langle-3 t, 3 t^{4},-4 t^{5}\right\rangle$ at $t=1$ . $\vec{a}(1)=\square \vec{T}+\square \vec{N}$

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Problem 10149

Find the derivative of the function $f(x)=\cos^{2}\left(\frac{x}{x+2}\right)$ .

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Problem 10150

Find the limit as $x$ approaches 0 for $\frac{\tan x}{4 x}$ .

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Problem 10151

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{10 x^{3}}{e^{8 x}}$ using L'Hopital's rule.

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Problem 10152

A plane is at $6 \mathrm{~km}$ altitude and moves at $700 \mathrm{~km/h}$ . Find how fast the angle $\theta$ changes after $24 \mathrm{~min}$ . Give your answer to three decimal places: $\frac{d \theta}{d t} \approx$

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Problem 10153

Evaluate $\lim _{n \rightarrow \infty} \int_{0}^{\pi}\left|x^{2} \sin n x\right| d x$ using given steps.

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Problem 10154

Find the limit using L'Hospital's rule: $\lim _{x \rightarrow 0} \frac{e^{x}-x-1}{4 x^{2}}$

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Problem 10155

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{12^{x}-3^{x}}{x}$ using L'Hospital's rule.

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Problem 10156

Find the derivative of the function $f(x)=\sqrt{x} \tan ^{4}(\sqrt{x})$ . What is $f^{\prime}(x)$ ?

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Problem 10157

Brooke and Jamail walk on parallel paths. How far apart are they after 11 s and how fast is the distance changing?

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Problem 10158

Find $h^{\prime}(3)$ for $h(x)=\sqrt{7+6 f(x)}$ , given $f(3)=7$ and $f^{\prime}(3)=4$ .

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Problem 10159

Find the volume of the torus formed by revolving the circle $(x-17)^{2}+y^{2}=1$ around the $y$ -axis.

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Problem 10160

Find the derivative of the function $f(x)=\sqrt{3 x-\sin ^{2}(4 x)}$ . What is $f^{\prime}(x)$ ?

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Problem 10161

Find the derivative of the function $y=(9+\cos^{2} x)^{6}$ with respect to $x$ . What is $\frac{d y}{d x}=?$

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Problem 10162

Find the limit as $x$ approaches 0 for $\frac{\tan 8 x}{\sin 3 x}$ .

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Problem 10163

Find the limit as $x$ approaches 0 for $\frac{\sin 3x}{5x}$ .

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Problem 10164

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{e^{2 x}-1}$ .

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Problem 10165

Find the derivative of the function $f(x)=\frac{1}{(x^{3}-\sec(3x^{2}-5))^{3}}$ .

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Problem 10166

Find the derivative of $y=\tan(9 e^{x})$ with respect to $x$ : $d y / d x=$

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Problem 10167

Find the limit using L'Hospital's rule: $\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{1-\cos 9 x}.$

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Problem 10168

Find the limit: $\lim _{x \rightarrow \infty} x^{4} \cdot e^{-x}$ using L'Hospital's rule if needed.

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Problem 10169

Find the derivative of the function $f(x)=\frac{1+\csc \left(x^{9}\right)}{1-\cot \left(x^{9}\right)}$ .

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Problem 10170

Find the tangent line equation to $y(x)=\sin(2+x^{2})$ at $x=-4$ . The equation is $y=$

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Problem 10171

Differentiate $y=-2 \sin (\tan \sqrt{\sin x})$ . Find $y^{\prime}$ .

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Problem 10172

Find points where the tangent line of $f(x)=2 \sin x+\sin^{2} x$ is horizontal for $0 \leq x < 2\pi$ . List $x=$ .

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Problem 10173

Given $f(x)=\sin x$ , define $F(x)=f\left(x^{k}\right)$ and $G(x)=[f(x)]^{k}$ . Find (a) $F^{\prime}(x)$ and (b) $G^{\prime}(x)$ .

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Problem 10174

Find the derivative of the function $y=\left[x+\left(x+\sin ^{2}(x)\right)^{7}\right]^{3}$ . What is $y^{\prime}=?$

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Problem 10175

Find $\frac{d}{d t} \sin t$ when $t$ is in degrees and explain why radians are preferred in calculus.

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Problem 10176

Differentiate $y=-9 \cot^{2}(\sin t)$ . Find $y'$ .

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Problem 10177

Find functions $g(x)$ and $f(x)$ such that $y=(f \circ g)(x)$ for $y=(x+\sin (x))^{2}$ , then compute $(f \circ g)^{\prime}$ .

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Problem 10178

Find $\frac{d y}{d x}$ for $y=\sin(-2x^2+2x+1)$ using the chain rule. What is $\frac{d y}{d u}$ and $\frac{d u}{d x}$ ?

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Problem 10179

Find the derivative $f'(x)$ of $f(x)=\tan(4x)$ and calculate $f'(2)$ .

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Problem 10180

Find the second derivative of $y$ with respect to $x$ for the equation $x^{2}+4 y^{2}=4$ .

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Problem 10181

Find the absolute max and min of $f(x)=2-6 x^{2}$ for $-3<x \leq 1$ . Where do they occur?

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Problem 10182

Find the minimum surface area of a cylindrical container with volume $64 \pi$ in. $^{3}$ . Options: A) 4 in. $^{2}$ B) $48 \pi$ in. $^{2}$ C) $4 \pi$ in. $^{2}$ D) 48 in. $^{2}$

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Problem 10183

Find $c \in [0,2]$ for $\mathrm{f}(x)=\sin x^{2}$ that meets the Mean Value Theorem. Choices: A) 1.3099 B) 1.5261 C) -0.3784 D) 1.2533

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Problem 10184

Find the derivative $y^{\prime}$ for the function $y=\log x+e^{x}$ .

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Problem 10185

Find the derivative $y^{\prime}$ of the function $y=\cos (\pi x)$ .

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Problem 10186

Find the derivative of the function $f(x)=x^{2 x}$ .

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Problem 10187

Find all values of $c$ in $[0,3]$ for the function $f(x)=(1-x)^{3}$ that meet the Mean Value Theorem.

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Problem 10188

Find when the particle with position $S(t)=2\sqrt{t}(2-2t+t^2)$ is moving left for $t \geq 0$ .

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Problem 10189

Find the derivative of the function $f(x)=7 \sqrt{x} \cdot \cos \sqrt{x}$ .

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Problem 10190

Find points on the curve where the tangent line is horizontal for $y=2 x^{3}+3 x^{2}-12 x+2$ .

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Problem 10191

Find the derivative of $f(x)=\sqrt[3]{x^{2}+4}$ using the chain rule.

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Problem 10192

Find the derivative of $f(x)=x^{2} \cdot \sqrt{3x-4}$ .

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Problem 10193

Find the derivative of $f(x)=7 \ln (x^{2} e^{x})$ .

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Problem 10194

Evaluate the following using functions $f(x), g(x), h(x)$ and their derivatives at given points: a. $(g \cdot h)^{\prime}(10)$ b. $\left(\frac{f}{g}\right)^{\prime}(13)$ c. $(f \circ g)^{\prime}(10)$ d. $\left(f^{-1}\right)^{\prime}(11)$

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Problem 10195

Find the derivative of $f(x)=\frac{x}{\sqrt{x^{2}+2}}$ .

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Problem 10196

Gjeni $\frac{\mathrm{d} y}{\mathrm{~d} x}$ për funksionet: a) $x^{2}+2 x-3$ , b) $1-2 x-5 x^{2}$ , c) $x^{3}+2 x^{2}-3 x+1$ , d) $x^{2}-x^{4}+\pi$ , e) $x-\frac{1}{x}$ , f) $3+x+\frac{2}{x^{2}}$ , g) $x^{3}-\frac{1}{x^{3}}+\frac{3}{x}+5$ , h) $\frac{10}{x}-1-\frac{x}{10}$ , i) $\sqrt{x}+\frac{1}{\sqrt{x}}$ , j) $x^{5}-\frac{5}{\sqrt[3]{x}}$ , k) $3+\frac{2}{\sqrt{x}}-\frac{5}{\sqrt[3]{x^{2}}}$ , l) $\frac{3}{\pi}-\frac{2}{x^{2}}-x^{\frac{3}{2}}$ .

See Solution

Problem 10197

Find the limit as $n$ approaches 5 for the expression $\frac{7 n^{3}+8 n^{5}-n^{2}+1}{2-n+2 n^{2}-3 n^{3}+4 n^{4}-5 n^{5}}$ .

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Problem 10198

Find the first and second derivatives of the function $f(x)=a x^{3}+\frac{b}{x}+c(1+\ln x)$ .

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Problem 10199

Find the limit: $\lim _{\Delta x \rightarrow 0} \frac{(4 / \sqrt{x}+\Delta x)-4 / \sqrt{x}}{\Delta x}$ .

See Solution

Problem 10200

Invest \$4000 at 3% continuously compounded interest. (a) Find the account value after 9 years. (b) When will it reach \$48000?

See Solution

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Calculus

Problem 10101

Find the tangent line equation to y=sin⁡(3x)+cos⁡(8x)y=\sin(3x)+\cos(8x)y=sin(3x)+cos(8x) at (π6,y(π6))\left(\frac{\pi}{6}, y\left(\frac{\pi}{6}\right)\right)(6π​,y(6π​)).

Problem 10102

A farmhouse is a=1.5 kma=1.5 \mathrm{~km}a=1.5 km from a highway. An auto moves at v=81 km/hv=81 \mathrm{~km/h}v=81 km/h. Find how fast distance lll increases when s=3.7 kms=3.7 \mathrm{~km}s=3.7 km. Answer to three decimal places: km/h\mathrm{km/h}km/h.

Problem 10103

Find the derivative of f(x)=x7−2x5+5x3−7xf(x)=x^{7}-2x^{5}+5x^{3}-7xf(x)=x7−2x5+5x3−7x.

Problem 10104

Julian jogs on a circular track with radius 29 m29 \mathrm{~m}29 m. At (21,20)(21,20)(21,20), his xxx-coordinate changes at −5.25 m/s-5.25 \mathrm{~m/s}−5.25 m/s. Find dydt\frac{d y}{d t}dtdy​. (Round to two decimal places.)

Problem 10105

Differentiate (f−g)2(f-g)^2(f−g)2 and evaluate at x=0x=0x=0 using fff and ggg values from the table.

Problem 10106

Find the slope of the tangent line to the curve 5sin⁡(x)+2cos⁡(y)−6sin⁡(x)cos⁡(y)+x=6π5 \sin (x) + 2 \cos (y) - 6 \sin (x) \cos (y) + x = 6 \pi5sin(x)+2cos(y)−6sin(x)cos(y)+x=6π at (6π,5π/2)(6 \pi, 5 \pi / 2)(6π,5π/2).

Problem 10107

Differentiate 1f−g\frac{1}{f-g}f−g1​ and evaluate at x=2x=2x=2. Use values from the table. Leave the answer as a fraction.

Problem 10108

Find the solution to the equation 2y=8−0.4t2 \sqrt{y}=8-0.4 t2y​=8−0.4t with y(0)=49y(0)=49y(0)=49 and show yyy is quadratic in ttt.

Problem 10109

Find dPdt\frac{d P}{d t}dtdP​ for b=1.6,P=10kPa,V=60cm3b=1.6, P=10 \mathrm{kPa}, V=60 \mathrm{cm}^{3}b=1.6,P=10kPa,V=60cm3, dVdt=80cm3/min\frac{d V}{d t}=80 \mathrm{cm}^{3}/\mathrm{min}dtdV​=80cm3/min.

Problem 10110

Find the derivative of f(x)=x33+3x3f(x)=\frac{x^{3}}{3}+\frac{3}{x^{3}}f(x)=3x3​+x33​.

Problem 10111

Find the volume of the torus formed by revolving the circle (x−17)2+y2=1(x-17)^{2}+y^{2}=1(x−17)2+y2=1 around the yyy-axis.

Problem 10112

Brooke walks east at 5 km/h5 \mathrm{~km/h}5 km/h and Jamail west at 4 km/h4 \mathrm{~km/h}4 km/h. Find their distance apart after 11 s11 \mathrm{~s}11 s.

Problem 10113

Given y2+xy−3x=51y^{2}+x y-3 x=51y2+xy−3x=51 and dydt=−1\frac{d y}{d t}=-1dtdy​=−1 at x=−5x=-5x=−5, y=−4y=-4y=−4, find dxdt\frac{d x}{d t}dtdx​.

Problem 10114

A rocket moves up at 1300 km/hr1300 \mathrm{~km} / \mathrm{hr}1300 km/hr. Find the angle rate increase from an observer 18 km18 \mathrm{~km}18 km away after 3 min. dθdt≈\frac{d \theta}{d t} \approxdtdθ​≈

Problem 10115

Differentiate 1f2\frac{1}{f^{2}}f21​ and evaluate at x=0x=0x=0. Use f′(0)=4f'(0)=4f′(0)=4 from the table. Answer in fraction form!

Problem 10116

Helium fills a balloon at 4 ft³/s. Find the radius increase rate after 2 minutes using V=43πr3V=\frac{4}{3}\pi r^{3}V=34​πr3.

Problem 10117

Differentiate f(g(x))f(g(x))f(g(x)) and evaluate at x=−1x = -1x=−1 using values from the table. What is dydx\frac{d y}{d x}dxdy​?

Problem 10118

Water leaks from a conical tank at 6300 cm³/min. If the water rises at 28 cm/min when 2 m high, find the inflow rate.

Problem 10119

Differentiate (fg)12(f g)^{\frac{1}{2}}(fg)21​ and evaluate at x=0x=0x=0. Leave your answer in fraction form!

Problem 10120

Differentiate (fg)−1(f g)^{-1}(fg)−1 and evaluate at x=−1x=-1x=−1. Leave the answer in fraction form!

Problem 10121

Given y=5xy=5 \sqrt{x}y=5x​, find Δy\Delta yΔy and dyd ydy for x=4x=4x=4 and Δx=dx=0.3\Delta x=d x=0.3Δx=dx=0.3.

Problem 10122

Differentiate gf\frac{g}{f}fg​ and evaluate at x=1x=1x=1. Use the given values for f(x)f(x)f(x) and g(x)g(x)g(x). Express as a fraction.

Problem 10123

A baseball diamond is a square with sides of 90 ft. A player runs to first base at 29 ft/s. Find the rate of change of distance to second base halfway to first. (Round to two decimal places.)

Problem 10124

Differentiate f⋅gf \cdot gf⋅g and evaluate at x=0x=0x=0. Use values from the table for f(x)f(x)f(x), f′(x)f'(x)f′(x), g(x)g(x)g(x), g′(x)g'(x)g′(x).

Problem 10125

Differentiate (f⋅g)3(f \cdot g)^3(f⋅g)3 and evaluate at x=−1x=-1x=−1 using given values for f,f′,g,g′f, f', g, g'f,f′,g,g′.

Problem 10126

Differentiate f−gf-gf−g and evaluate at x=−1x=-1x=−1. Use values: f′(−1)f'(-1)f′(−1) and g′(−1)g'(-1)g′(−1). Find ddx(f−g)\frac{d}{dx}(f-g)dxd​(f−g).

Problem 10127

Find dyd ydy for y=4x2+3x+3y=4 x^{2}+3 x+3y=4x2+3x+3 at x=5x=5x=5 with dx=0.2d x=0.2dx=0.2 and dx=0.4d x=0.4dx=0.4.

Problem 10128

Differentiate f+g\sqrt{f+g}f+g​ and evaluate at x=1x=1x=1 using the given values for fff and ggg.

Problem 10129

Find dPdt\frac{d P}{d t}dtdP​ for b=1.4b=1.4b=1.4, P=7kPaP=7 \mathrm{kPa}P=7kPa, V=110 cm3V=110 \mathrm{~cm}^{3}V=110 cm3, and dVdt=50 cm3/min\frac{d V}{d t}=50 \mathrm{~cm}^{3}/\mathrm{min}dtdV​=50 cm3/min.

Problem 10130

Differentiate (fg)2(f g)^2(fg)2 and evaluate at x=0x=0x=0. Use f′(0)f'(0)f′(0) and g′(0)g'(0)g′(0) from the provided values.

Problem 10131

A car travels past a farmhouse 1.5 km away at 81 km/h. Find the speed of distance lll when the car is 3.7 km past the intersection. Answer to three decimal places.

Problem 10132

Differentiate f2f^2f2 and evaluate at x=2x=2x=2 using given values for f(x),f′(x),g(x),g′(x)f(x), f'(x), g(x), g'(x)f(x),f′(x),g(x),g′(x).

Problem 10133

Differentiate gfg fgf and evaluate at x=1x=1x=1. Use f(1),f′(1),g(1),g′(1)f(1), f'(1), g(1), g'(1)f(1),f′(1),g(1),g′(1) from the table.

Problem 10134

Find the percentage error in the volume and surface area of a cube with edge 30 cm30 \mathrm{~cm}30 cm and error 0.1 cm0.1 \mathrm{~cm}0.1 cm.

Problem 10135

Water in a tank drains by Torricelli's Law: dydt=−0.4y\frac{d y}{d t}=-0.4 \sqrt{y}dtdy​=−0.4y​. Find depth after 0.5s if y=36 cmy=36 \mathrm{~cm}y=36 cm.

Problem 10136

Estimate the upper bound for error in d(t)=16t2d(t)=16 t^{2}d(t)=16t2 when t=4t=4t=4 seconds and t=5t=5t=5 seconds with 0.20.20.2 sec accuracy.

Problem 10137

Find the derivative of the function f(x)=tan⁡2(x4)f(x)=\tan^{2}(x^{4})f(x)=tan2(x4). What is f′(x)f^{\prime}(x)f′(x)?

Problem 10138

Evaluate the limit: lim⁡x→0ex−1sin⁡(3x)\lim _{x \rightarrow 0} \frac{e^{x}-1}{\sin (3 x)}limx→0​sin(3x)ex−1​ using L'Hospital's rule.

Problem 10139

Find y′y^{\prime}y′ and y′′y^{\prime \prime}y′′ for y=e8exy=e^{8 e^{x}}y=e8ex.

Problem 10140

Find the tangent line equation to $y=\sin(3x)+\cos(8x)$ at $\left(\frac{\pi}{6}, y\left(\frac{\pi}{6}\right)\right)$ .

A farmhouse is $a=1.5 \mathrm{~km}$ from a highway. An auto moves at $v=81 \mathrm{~km/h}$ . Find how fast distance $l$ increases when $s=3.7 \mathrm{~km}$ . Answer to three decimal places: $\mathrm{km/h}$ .

Find the derivative of $f(x)=x^{7}-2x^{5}+5x^{3}-7x$ .

Julian jogs on a circular track with radius $29 \mathrm{~m}$ . At $(21,20)$ , his $x$ -coordinate changes at $-5.25 \mathrm{~m/s}$ . Find $\frac{d y}{d t}$ . (Round to two decimal places.)

Differentiate $(f-g)^2$ and evaluate at $x=0$ using $f$ and $g$ values from the table.

Find the slope of the tangent line to the curve $5 \sin (x) + 2 \cos (y) - 6 \sin (x) \cos (y) + x = 6 \pi$ at $(6 \pi, 5 \pi / 2)$ .

Differentiate $\frac{1}{f-g}$ and evaluate at $x=2$ . Use values from the table. Leave the answer as a fraction.

Find the solution to the equation $2 \sqrt{y}=8-0.4 t$ with $y(0)=49$ and show $y$ is quadratic in $t$ .

Find $\frac{d P}{d t}$ for $b=1.6, P=10 \mathrm{kPa}, V=60 \mathrm{cm}^{3}$ , $\frac{d V}{d t}=80 \mathrm{cm}^{3}/\mathrm{min}$ .

Find the derivative of $f(x)=\frac{x^{3}}{3}+\frac{3}{x^{3}}$ .

Find the volume of the torus formed by revolving the circle $(x-17)^{2}+y^{2}=1$ around the $y$ -axis.

Brooke walks east at $5 \mathrm{~km/h}$ and Jamail west at $4 \mathrm{~km/h}$ . Find their distance apart after $11 \mathrm{~s}$ .

Given $y^{2}+x y-3 x=51$ and $\frac{d y}{d t}=-1$ at $x=-5$ , $y=-4$ , find $\frac{d x}{d t}$ .

A rocket moves up at $1300 \mathrm{~km} / \mathrm{hr}$ . Find the angle rate increase from an observer $18 \mathrm{~km}$ away after 3 min. $\frac{d \theta}{d t} \approx$

Differentiate $\frac{1}{f^{2}}$ and evaluate at $x=0$ . Use $f'(0)=4$ from the table. Answer in fraction form!

Helium fills a balloon at 4 ft³/s. Find the radius increase rate after 2 minutes using $V=\frac{4}{3}\pi r^{3}$ .

Differentiate $f(g(x))$ and evaluate at $x = -1$ using values from the table. What is $\frac{d y}{d x}$ ?

Differentiate $(f g)^{\frac{1}{2}}$ and evaluate at $x=0$ . Leave your answer in fraction form!

Differentiate $(f g)^{-1}$ and evaluate at $x=-1$ . Leave the answer in fraction form!

Given $y=5 \sqrt{x}$ , find $\Delta y$ and $d y$ for $x=4$ and $\Delta x=d x=0.3$ .

Differentiate $\frac{g}{f}$ and evaluate at $x=1$ . Use the given values for $f(x)$ and $g(x)$ . Express as a fraction.

Differentiate $f \cdot g$ and evaluate at $x=0$ . Use values from the table for $f(x)$ , $f'(x)$ , $g(x)$ , $g'(x)$ .

Differentiate $(f \cdot g)^3$ and evaluate at $x=-1$ using given values for $f, f', g, g'$ .

Differentiate $f-g$ and evaluate at $x=-1$ . Use values: $f'(-1)$ and $g'(-1)$ . Find $\frac{d}{dx}(f-g)$ .

Find $d y$ for $y=4 x^{2}+3 x+3$ at $x=5$ with $d x=0.2$ and $d x=0.4$ .

Differentiate $\sqrt{f+g}$ and evaluate at $x=1$ using the given values for $f$ and $g$ .

Find $\frac{d P}{d t}$ for $b=1.4$ , $P=7 \mathrm{kPa}$ , $V=110 \mathrm{~cm}^{3}$ , and $\frac{d V}{d t}=50 \mathrm{~cm}^{3}/\mathrm{min}$ .

Differentiate $(f g)^2$ and evaluate at $x=0$ . Use $f'(0)$ and $g'(0)$ from the provided values.

A car travels past a farmhouse 1.5 km away at 81 km/h. Find the speed of distance $l$ when the car is 3.7 km past the intersection. Answer to three decimal places.

Differentiate $f^2$ and evaluate at $x=2$ using given values for $f(x), f'(x), g(x), g'(x)$ .

Differentiate $g f$ and evaluate at $x=1$ . Use $f(1), f'(1), g(1), g'(1)$ from the table.

Find the percentage error in the volume and surface area of a cube with edge $30 \mathrm{~cm}$ and error $0.1 \mathrm{~cm}$ .

Water in a tank drains by Torricelli's Law: $\frac{d y}{d t}=-0.4 \sqrt{y}$ . Find depth after 0.5s if $y=36 \mathrm{~cm}$ .

Estimate the upper bound for error in $d(t)=16 t^{2}$ when $t=4$ seconds and $t=5$ seconds with $0.2$ sec accuracy.

Find the derivative of the function $f(x)=\tan^{2}(x^{4})$ . What is $f^{\prime}(x)$ ?

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{e^{x}-1}{\sin (3 x)}$ using L'Hospital's rule.

Find $y^{\prime}$ and $y^{\prime \prime}$ for $y=e^{8 e^{x}}$ .

Find the derivative of $y(x)=\cos(\tan(7x))$ . What is $d y / d x$ ? $d y / d x=$

Differentiate the function: $g(x)=(x+7 \sqrt{x}) e^{x}$ .

Evaluate the limit using L'Hopital's rule: $\lim _{x \rightarrow \frac{\pi}{2}} 7 \cos (3 x) \sec (-7 x)$

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{\sin (2 x)}{\sin (15 x)}$ using L'Hospital's rule if needed.

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{\sin (10 x)}{\tan (12 x)}$ using L'Hospital's rule.

Derive $E\left[e^{-X^{2} / 2}\right]$ for $X \sim \mathcal{N}(0,1)$ using $\int_{-\infty}^{\infty} \exp(-\frac{u^{2}}{2}) du=\sqrt{2 \pi}$ .

Evaluate the limit using L'Hôpital's rule: $\lim _{x \rightarrow 0} \frac{e^{x}+x-1}{4 x}$

Find the rate of volume increase of a cone when $r=5 \mathrm{~cm}$ , $h=20 \mathrm{~cm}$ , both growing at $7 \mathrm{~cm/s}$ .

Find the tangential and normal acceleration components for $\vec{r}(t)=\left\langle-3 t, 3 t^{4},-4 t^{5}\right\rangle$ at $t=1$ . $\vec{a}(1)=\square \vec{T}+\square \vec{N}$

Find the derivative of the function $f(x)=\cos^{2}\left(\frac{x}{x+2}\right)$ .

Find the limit as $x$ approaches 0 for $\frac{\tan x}{4 x}$ .

Evaluate the limit: $\lim _{x \rightarrow \infty} \frac{10 x^{3}}{e^{8 x}}$ using L'Hopital's rule.

A plane is at $6 \mathrm{~km}$ altitude and moves at $700 \mathrm{~km/h}$ . Find how fast the angle $\theta$ changes after $24 \mathrm{~min}$ . Give your answer to three decimal places: $\frac{d \theta}{d t} \approx$

Evaluate $\lim _{n \rightarrow \infty} \int_{0}^{\pi}\left|x^{2} \sin n x\right| d x$ using given steps.

Find the limit using L'Hospital's rule: $\lim _{x \rightarrow 0} \frac{e^{x}-x-1}{4 x^{2}}$

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{12^{x}-3^{x}}{x}$ using L'Hospital's rule.

Find the derivative of the function $f(x)=\sqrt{x} \tan ^{4}(\sqrt{x})$ . What is $f^{\prime}(x)$ ?

Find $h^{\prime}(3)$ for $h(x)=\sqrt{7+6 f(x)}$ , given $f(3)=7$ and $f^{\prime}(3)=4$ .

Find the volume of the torus formed by revolving the circle $(x-17)^{2}+y^{2}=1$ around the $y$ -axis.

Find the derivative of the function $f(x)=\sqrt{3 x-\sin ^{2}(4 x)}$ . What is $f^{\prime}(x)$ ?

Find the derivative of the function $y=(9+\cos^{2} x)^{6}$ with respect to $x$ . What is $\frac{d y}{d x}=?$

Find the limit as $x$ approaches 0 for $\frac{\tan 8 x}{\sin 3 x}$ .

Find the limit as $x$ approaches 0 for $\frac{\sin 3x}{5x}$ .

Find the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{e^{2 x}-1}$ .

Find the derivative of the function $f(x)=\frac{1}{(x^{3}-\sec(3x^{2}-5))^{3}}$ .

Find the derivative of $y=\tan(9 e^{x})$ with respect to $x$ : $d y / d x=$

Find the limit using L'Hospital's rule: $\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{1-\cos 9 x}.$

Find the limit: $\lim _{x \rightarrow \infty} x^{4} \cdot e^{-x}$ using L'Hospital's rule if needed.

Find the derivative of the function $f(x)=\frac{1+\csc \left(x^{9}\right)}{1-\cot \left(x^{9}\right)}$ .

Find the tangent line equation to $y(x)=\sin(2+x^{2})$ at $x=-4$ . The equation is $y=$

Differentiate $y=-2 \sin (\tan \sqrt{\sin x})$ . Find $y^{\prime}$ .

Find points where the tangent line of $f(x)=2 \sin x+\sin^{2} x$ is horizontal for $0 \leq x < 2\pi$ . List $x=$ .

Given $f(x)=\sin x$ , define $F(x)=f\left(x^{k}\right)$ and $G(x)=[f(x)]^{k}$ . Find (a) $F^{\prime}(x)$ and (b) $G^{\prime}(x)$ .

Find the derivative of the function $y=\left[x+\left(x+\sin ^{2}(x)\right)^{7}\right]^{3}$ . What is $y^{\prime}=?$