Calculus

Problem 22101

Find $g(x)=\frac{1}{(x+2)^{3}}$ . Complete the table for $x$ near -2 and determine limits as $x \to -2$ .

See Solution

Problem 22102

Differentiate the series for $f(x)=\tan^{-1}(x^4)$ . Identify the function and interval of convergence. Choose the correct series.

See Solution

Problem 22103

Find the slope of the tangent line to the curve $2y + 2xy^3 = 2f(x)^2$ at the point $(1,1)$ using implicit differentiation.

See Solution

Problem 22104

Find the positive $x$ where the curve $y=x^{3} e^{-x / 6}$ has a horizontal tangent line. Provide the answer as a decimal.

See Solution

Problem 22105

Find the positive value of $x$ where $y=x^{3} e^{-x / 6}$ has a horizontal tangent line. Provide the answer as a decimal.

See Solution

Problem 22106

Find the first and second derivatives of the curve defined by $x=5+\sec t$ and $y=-1-5 \tan t$ .

See Solution

Problem 22107

Find the power series solution for $y^{\prime}(t)-y=0$ with $y(0)=30$ and identify the function.

See Solution

Problem 22108

Find the slope of the tangent line to the curve $2y + 2xy^3 = 2f(x)^2$ at point $(1,1)$ given $f(1)=2$ and $f'(1)=3$ .

See Solution

Problem 22109

Solve the differential equation $y^{\prime}(t)-4 y(t)=8$ , $y(0)=3$ . Find the power series and identify the function.

See Solution

Problem 22110

Find $g^{\prime}(2)$ for $g(x)=2+x^{3}+\tan^{-1}(f(x))$ given $f(2)=3$ and $f^{\prime}(2)=5$ .

See Solution

Problem 22111

An object falls from rest in a vacuum. How long to reach a velocity of $147 \mathrm{~m/s}$ with $g \approx 9.81 \mathrm{~m/s^2}$ ?

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Problem 22112

Estimate the integral $\int_{0}^{0.3} 2 e^{-x^{2}} d x$ using a series with error less than $10^{-8}$ . Round final answer to five decimal places.

See Solution

Problem 22113

Find $(\Delta a)_{m}$ for $a_{m}=\ln m$ and use it to compute $\sum_{m=1}^{N} \ln \left(1+\frac{1}{m}\right)$ .

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Problem 22114

Find all $x$ where the tangent line to $y=12x^2-\ln(x)$ is parallel to $6x+3y=17$ . Enter the largest $x$ as a rational number or decimal.

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Problem 22115

Find the $x$ -coordinate where the tangent line of the curve $x=t^{2}+1$ , $y=t^{2}-t$ has slope 27. Answer exactly. $x=$

See Solution

Problem 22116

Find all $x$ where the tangent to $y=12x^2-\ln(x)$ is parallel to $6x+3y=17$ . Provide the largest $x$ and if a second exists.

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Problem 22117

Complete the tables for $g(x) = \frac{1}{(x+2)^3}$ at $x = 10, 100, 1000$ and $x = -10, -100, -1000$ . What are the limits as $x \to \infty$ and $x \to -\infty$ ? Find vertical and horizontal asymptotes.

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Problem 22118

Calculate the distance an object falls in 20 seconds using the formula for free fall: $d = \frac{1}{2} g t^2$ where $g$ is the acceleration due to gravity.

See Solution

Problem 22119

Evaluate the integral using symmetry: $\int_{-1}^{1}\left(8 x^{8}-8\right) d x$ .

See Solution

Problem 22120

Determine if the statement is true or false: $\frac{d}{d \theta}[\ln (\cos (\theta))]=-\tan (\theta)$ .

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Problem 22121

Given that $f$ is an even function and $\int_{-5}^{5} f(x) dx = 10$ , find:
a. $\int_{0}^{5} f(x) dx$
b. $\int_{-5}^{5} x f(x) dx$

See Solution

Problem 22122

Approximate the integral $\int_{0}^{0.2} \sqrt{1+x^{4}} dx$ using a Taylor series, ensuring error < $10^{-4}$ . Round to four decimals.

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Problem 22123

Approximate the integral $\int_{0}^{0.28} \ln(1+x^{2}) dx$ using a Taylor series with error < $10^{-4}$ . Choose the correct series.

See Solution

Problem 22124

Calculate the average value of $f(x)=6 x^{2}$ over the interval $[2,4]$ .

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Problem 22125

(a) Rewrite $\int \frac{e^{x}}{x^{2}} d x$ using integration by parts with $d v=\frac{1}{x^{2}} d x$ and $u=e^{x}$ . No evaluation needed. (b) Compare the summation by parts formula $\sum_{n=a}^{b} u_{n}(\Delta v)_{n}=\left.u_{n} v_{n}\right|_{n=a}^{b+1}-\sum_{n=a}^{b} v_{n+1}(\Delta u)_{n}$ to integration by parts. (c) Use summation by parts on $\sum_{n=5}^{12} \frac{2^{n}}{n(n+1)}$ with $u_{n}=2^{n}$ and $(\Delta v)_{n}=\frac{1}{n(n+1)}$ . No evaluation needed. (d) Identify analogues between parts (a) and (c).

See Solution

Problem 22126

Analyze the functions:
a. For $f(x)=\frac{9 x^{2}+6}{x-7}$ , find limits as $x \rightarrow \infty$ , $-\infty$ , $7^{+}$ , $7^{-}$ .
b. For $f(x)=\frac{x-9}{x+5}$ , find limits as $x \rightarrow \infty$ , $-\infty$ , $-5^{+}$ , $-5^{-}$ .

See Solution

Problem 22127

Approximate the integral $\int_{0}^{0.28} \ln(1+x^{2}) \, dx$ using Taylor series for error < $10^{-4}$ . Find the value.

See Solution

Problem 22128

Approximate the integral $\int_{0}^{0.52} \frac{d x}{\sqrt{1+x^{6}}}$ using a Taylor series. Choose the correct option.

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Problem 22129

Evaluate the series $\sum_{n=1}^{\infty} \cos (n \pi)$ : find $S_{k}$ , check convergence with $\lim _{k \rightarrow \infty} S_{k}$ , and apply a test.

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Problem 22130

Calculate the average value of $f(x)=\frac{1}{x}$ on the interval $[2,2e]$ and sketch its graph with the average value marked.

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Problem 22131

Is $\sum_{n=0}^{\infty} a_{n}$ convergent if $a_{n+1}=\frac{a_{n}}{3}$ and $a_{0}=10$ ? Find its sum if it converges.

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Problem 22132

Approximate the integral $\int_{0}^{0.52} \frac{dx}{\sqrt{1+x^{6}}}$ using Taylor series, ensuring error < $10^{-4}$ .

See Solution

Problem 22133

Find the weight $W$ of a person 5 feet 2 inches tall using $\frac{d W}{d h}=0.0012 h^{2}+0.01 h$ and $W(70)=141.7$ .

See Solution

Problem 22134

Find the area between the curves $f(x)=5x-7$ and $g(x)=x$ from $x=2$ to $x=5$ . Area =

See Solution

Problem 22135

Find the limits: (a) for $g(x)=-2 f(x+7)+5$ , and (b) for $h(x)=-f(-x)$ as $x \to -\infty$ and $x \to \infty$ .

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Problem 22136

Determine if these series converge or diverge: (a) $\sum_{n=1}^{\infty} \frac{\tan (1 / n)}{n}$ , (b) $\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{5^{n} n !}$ , (c) $\sum_{n=1}^{\infty}(\sqrt[n]{2}-1)^{n}$ .

See Solution

Problem 22137

Approximate the integral $\int_{0}^{0.99} \frac{dx}{\sqrt{1+x^6}}$ using Taylor series, ensuring error < $10^{-4}$ . Find $\int_{0}^{0.49} \frac{dx}{\sqrt{1+x^6}}=\square$ (round to 4 decimal places).

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Problem 22138

Determine convergence of these series using the ratio or root test: (a) $\sum_{n=1}^{\infty} \frac{10^{n}}{n !}$ (b) $\sum_{n=1}^{\infty}\left(\frac{n^{2}+1}{2 n^{2}+1}\right)^{n}$ (c) $\sum_{n=1}^{\infty} \frac{3^{n}}{5^{n}+n}$ (d) $\sum_{n=2}^{\infty}\left(\frac{n+1}{n}\right)^{n^{2}}$ (e) $\sum_{n=2}^{\infty} \frac{1}{(\ln n)^{n}}$

See Solution

Problem 22139

Find the area between the curves $y=x^{2}-1$ and $y=2x+7$ . Enter the exact value.

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Problem 22140

Find the area between the curves $y=x^{2}-4x$ and $y=0$ over the interval $[-2,2]$ . Area $=$

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Problem 22141

Approximate the integral $\int_{0}^{0.49} \frac{d x}{\sqrt{1+x^{6}}}$ using Taylor series for error < $10^{-4}$ .

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Problem 22142

Find $\frac{d x}{d t}$ when $x=4$ given $\sqrt{x}+y=6$ and $\frac{d y}{d t}=2$ .

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Problem 22143

Find $\frac{d x}{d t}$ when $x=4$ given $\sqrt{x}+y=6$ and $\frac{d y}{d t}=2$ .

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Problem 22144

Find the area between the curves $y=e^{4 x}$ and $y=\frac{1}{3 x}$ from $x=1$ to $x=6$ . Area =

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Problem 22145

Find the area between the curves $y=x^{2}$ and $y=3x$ from $x=0$ to $x=6$ . Area $=$

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Problem 22146

Find the Taylor polynomials $p_{3}$ and $p_{4}$ at $a=0$ for $f(x)=(2+x)^{-3}$ . What is $p_{3}(x)$ ?

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Problem 22147

Find the area between the curves $f(x)=\sqrt{x}$ and $g(x)=x / 2$ over the interval $[0,16]$ .

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Problem 22148

Find the area between the curves $y=x \sqrt{x^{2}+25}$ and $y=0$ from $x=-2$ to $x=1$ . $\text{Area} =$

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Problem 22149

Find the average rate of change for $f(x)=x^{3}-2 x^{2}+x+1$ on $[0,2]$ and $[2,4]$ . Compare the results.

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Problem 22150

Find the 4th and 5th Taylor polynomials for $f(x)=6 \cos (x)$ at $a=\frac{\pi}{6}$ .

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Problem 22151

Find the 4th and 5th Taylor polynomials, $p_{4}$ and $p_{5}$ , for $f(x)=2 \cos (x)$ at $a=\frac{\pi}{6}$ .

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Problem 22152

Find the area between the curve $y=(x-1)e^{4x^{2}-8x}$ and $y=0$ from $x=0$ to $x=2$ . $\text{Area} =$

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Problem 22153

Find the consumer surplus for the demand function $p=D(x)=\sqrt{25-2x}$ at a market price of \$3.

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Problem 22154

Evaluate the triple integral: $\int_{-1}^{2} \int_{-\sqrt{8-x^{2}}}^{\sqrt{8-x^{2}}} \int_{0}^{x+1} 2 z d z d y d x$ .

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Problem 22155

Calculate the integral $\int x \ln \left(x^{2}\right) d x$ .

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Problem 22156

Find the average distance from the parabola $y=8 x(22-x)$ to the $x$ -axis over the interval $[0,22]$ .

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Problem 22157

Find the partial derivatives of the function $f(x, y, z)=\frac{1+x^{3} y^{2}}{z^{2}}$ with respect to $x$ , $y$ , and $z$ .

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Problem 22158

Find the average rate of change of $y=3 \sin (2 x)$ on $[0, \frac{\pi}{4}]$ . Choices: A. $-\frac{12}{\pi}$ B. $-\frac{6}{\pi}$ C. $\frac{6}{\pi}$ D. $\frac{12}{\pi}$

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Problem 22159

Evaluate the integral from 0 to 4 of the function $7 x^{2}-6 x+8$ . What is the result?

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Problem 22160

Find when the speed of the particle, defined by $x(t)=2 t^{3}-18 t^{2}-96 t$ , is increasing for $t \geq 0$ .

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Problem 22161

Evaluate the integral: $\int_{0}^{\pi / 3} 8 \sec ^{2} x \, dx = \square$ (Type an exact answer, using radicals as needed.)

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Problem 22162

Find the 2nd degree Taylor polynomial $p_{2}$ at $a=8$ for $f(x)=\sqrt[3]{x}$ . Determine $p_{2}(x)=\square$ .

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Problem 22163

Find the derivative of the integral $\frac{d}{d t} \int_{0}^{t^{8}} \sqrt{u} d u$ using two methods.

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Problem 22164

Evaluate the integral: $\int_{0}^{\frac{\pi}{8}} \sin 2 x \, dx = \square$ (Exact answer with radicals).

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Problem 22165

Calculate the integral from -7 to 7 of the absolute value function: $\int_{-7}^{7}|x| d x=\square$

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Problem 22166

Evaluate the integral from $7 \pi / 2$ to $\pi$ : $\int_{7 \pi / 2}^{\pi} \frac{1-\cos 2 t}{2} d t = \square($ Type an exact answer, using $\pi$ as needed. $)$

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Problem 22167

Find $\frac{d}{d t} \int_{0}^{t^{8}} \sqrt{u} d u$ by evaluating the integral or differentiating directly.

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Problem 22168

Find $\frac{\partial f}{\partial y}$ for $f(x, y)=x y^{8}+6$ at $(x, y)=(-4,-1)$ and explain the meaning.

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Problem 22169

Find the derivative of $\int_{0}^{x^3} e^{-2t} dt$ using two methods: a) evaluate then differentiate, b) differentiate directly.

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Problem 22170

Find the present value of a \$30,000/year income for 25 years at a 5\% continuous reinvestment rate.

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Problem 22171

Find the derivative $\frac{d y}{d x}$ for the integral $y=\int_{0}^{\tan x} \frac{d t}{1+t^{2}}$ . Simplify your answer.

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Problem 22172

Find points $(x, y)$ where $f(x, y)=x^{3}-2xy+\frac{1}{3}y^{2}+8$ has relative extrema.

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Problem 22173

Find $\theta$ for horizontal and vertical tangents on the polar curve $r=\theta$ , where $0 \leq \theta \leq 2\pi$ .

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Problem 22174

Calculate consumer surplus and producer surplus using demand $D(x) = -3x + 19$ and supply $S(x) = 4x + 5$ .

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Problem 22175

Find the velocity $v(t)=53(1-e^{-0.25 t})$ of a sky diver after 1 second and 4 seconds. Round to the nearest whole number.

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Problem 22176

Evaluate the integral: $\int_{0}^{2 \pi} \frac{(2+2 \cos \theta)^{2}}{2} d \theta$

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Problem 22177

Evaluate the expression: $\int_{0}^{2 \pi} \frac{(3+3 \sin \theta)^{2}}{2} d \theta - \int_{0}^{2 \pi} \frac{(6 \sin \theta)^{2}}{2} d \theta$ .

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Problem 22178

An artist has 9 m of wire to cut into a square and a circle. Where to cut for the smallest total area? Let $x$ be for the square.

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Problem 22179

Find values of $\theta$ for horizontal and vertical tangent lines on the polar curve $r=\theta$ , $0 \leq \theta \leq 2\pi$ .

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Problem 22180

Evaluate the integral $\int_{0}^{2 \pi} \sqrt{\left(\theta^{3}\right)^{2}+\left(3 \theta^{2}\right)^{2}} d \theta$ .

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Problem 22181

Find the area between $y=(x-1) e^{4 x^{2}-8 x}$ and $y=0$ from $x=0$ to $x=2$ . Area = $\frac{1}{4 e^{4}}-\frac{1}{4}$ .

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Problem 22182

Find $\frac{d y}{d x}$ for $y=\int_{0}^{\arcsin x} \cos t \, dt$ . What is $\frac{d y}{d x}=\square$ ?

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Problem 22183

Calculate the area between the $x$ -axis and $y=-x^{2}-3x$ for $-5 \leq x \leq 4$ . Area = $\square$ .

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Problem 22184

Calculate the average rate of change of $f(x)=-1 x^{2}+5 x-4$ from $x=3$ to $x=5$ .

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Problem 22185

Find the solution to the initial value problem: $\frac{d y}{d x}=\sqrt{5+x^{3}}$ , $y(5)=-6$ .

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Problem 22186

Find the integral solution for $y'=\cot x$ with $y(7)=6$ . Choose the correct option below. A. $y=\int_{7}^{x} \cot t d t+6$ B. $y=\int_{6}^{x} \cot t d t+7$ C. $y=\int_{7}^{x}(\cot t+6) d t$ D. $y=\int_{6}^{x}(\cot t+7) d t$

See Solution

Problem 22187

Find the limit as a definite integral: $\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n} 2 c_{k}^{8} \Delta x_{k}$ for $P$ in $[-9,1]$ .

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Problem 22188

Express the limit as a definite integral: $\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}\left(c_{k}^{6}-4 c_{k}\right) \Delta x_{k}$ for $P$ in $[-5,3]$ .

See Solution

Problem 22189

Evaluate the integral $\int_{1.5}^{2.5} x d x$ using $\int_{a}^{b} x d x=\frac{b^{2}}{2}-\frac{a^{2}}{2}$ . The result is $\square$ .

See Solution

Problem 22190

Graph the function and find the area using the integral: $\int_{-1}^{5}|2 x| d x$ .

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Problem 22191

Evaluate the integral using areas: $\int_{a}^{6 b} 2 s \, ds = \square$ where $0 < a < b$ .

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Problem 22192

Graph $y=-\frac{x^{2}-12}{x^{2}-4}$ : find domain, symmetries, derivatives, critical points, concavity, asymptotes, and extremes. Domain: $\square$ .

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Problem 22193

Calculate the integral from -3 to 3 of $5 + \sqrt{9 - x^2}$ . What is the exact answer, using $\pi$ if necessary?

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Problem 22194

Express the limit as a definite integral:
$\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}\left(\sec c_{k}\right) \Delta x_{k} \text{ for } P \text{ on } \left[-\frac{\pi}{4}, \frac{\pi}{12}\right]$

See Solution

Problem 22195

Evaluate $\int_{0}^{\sqrt[3]{12}} x^{2} d x$ using $\int_{a}^{b} x^{2} d x=\frac{b^{3}}{3}-\frac{a^{3}}{3}$ . Answer: \square.

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Problem 22196

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{(x-h)^{3}-x^{3}}{h}$ . Does it exist?

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Problem 22197

Evaluate the integral $\int_{3}^{7} 24 u^{2} d u$ using $\int_{a}^{b} x^{2} d x=\frac{b^{3}}{3}-\frac{a^{3}}{3}$ . $\int_{3}^{7} 24 u^{2} d u=\square$

See Solution

Problem 22198

How long does it take for a bacteria population to double with a growth rate of $3.1\%$ per hour? Round to the nearest hundredth.

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Problem 22199

Find the area between the curve $y=11 x$ and the $x$ -axis from $x=0$ to $x=b$ using a definite integral. Area = $\square$ .

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Problem 22200

Evaluate the integral $\int_{2}^{0}(9 x^{2}+x-4) dx$ . Use $\int_{a}^{b} x dx=\frac{b^{2}}{2}-\frac{a^{2}}{2}$ and $\int_{a}^{b} x^{2} dx=\frac{b^{3}}{3}-\frac{a^{3}}{3}$ . Find the value.

See Solution

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Calculus

Problem 22101

Find g(x)=1(x+2)3g(x)=\frac{1}{(x+2)^{3}}g(x)=(x+2)31​. Complete the table for xxx near -2 and determine limits as x→−2x \to -2x→−2.

Problem 22102

Differentiate the series for f(x)=tan⁡−1(x4)f(x)=\tan^{-1}(x^4)f(x)=tan−1(x4). Identify the function and interval of convergence. Choose the correct series.

Problem 22103

Find the slope of the tangent line to the curve 2y+2xy3=2f(x)22y + 2xy^3 = 2f(x)^22y+2xy3=2f(x)2 at the point (1,1)(1,1)(1,1) using implicit differentiation.

Problem 22104

Find the positive xxx where the curve y=x3e−x/6y=x^{3} e^{-x / 6}y=x3e−x/6 has a horizontal tangent line. Provide the answer as a decimal.

Problem 22105

Find the positive value of xxx where y=x3e−x/6y=x^{3} e^{-x / 6}y=x3e−x/6 has a horizontal tangent line. Provide the answer as a decimal.

Problem 22106

Find the first and second derivatives of the curve defined by x=5+sec⁡tx=5+\sec tx=5+sect and y=−1−5tan⁡ty=-1-5 \tan ty=−1−5tant.

Problem 22107

Find the power series solution for y′(t)−y=0y^{\prime}(t)-y=0y′(t)−y=0 with y(0)=30y(0)=30y(0)=30 and identify the function.

Problem 22108

Find the slope of the tangent line to the curve 2y+2xy3=2f(x)22y + 2xy^3 = 2f(x)^22y+2xy3=2f(x)2 at point (1,1)(1,1)(1,1) given f(1)=2f(1)=2f(1)=2 and f′(1)=3f'(1)=3f′(1)=3.

Problem 22109

Solve the differential equation y′(t)−4y(t)=8y^{\prime}(t)-4 y(t)=8y′(t)−4y(t)=8, y(0)=3y(0)=3y(0)=3. Find the power series and identify the function.

Problem 22110

Find g′(2)g^{\prime}(2)g′(2) for g(x)=2+x3+tan⁡−1(f(x))g(x)=2+x^{3}+\tan^{-1}(f(x))g(x)=2+x3+tan−1(f(x)) given f(2)=3f(2)=3f(2)=3 and f′(2)=5f^{\prime}(2)=5f′(2)=5.

Problem 22111

An object falls from rest in a vacuum. How long to reach a velocity of 147 m/s147 \mathrm{~m/s}147 m/s with g≈9.81 m/s2g \approx 9.81 \mathrm{~m/s^2}g≈9.81 m/s2?

Problem 22112

Estimate the integral ∫00.32e−x2dx\int_{0}^{0.3} 2 e^{-x^{2}} d x∫00.3​2e−x2dx using a series with error less than 10−810^{-8}10−8. Round final answer to five decimal places.

Problem 22113

Find (Δa)m(\Delta a)_{m}(Δa)m​ for am=ln⁡ma_{m}=\ln mam​=lnm and use it to compute ∑m=1Nln⁡(1+1m)\sum_{m=1}^{N} \ln \left(1+\frac{1}{m}\right)∑m=1N​ln(1+m1​).

Problem 22114

Find all xxx where the tangent line to y=12x2−ln⁡(x)y=12x^2-\ln(x)y=12x2−ln(x) is parallel to 6x+3y=176x+3y=176x+3y=17. Enter the largest xxx as a rational number or decimal.

Problem 22115

Find the xxx-coordinate where the tangent line of the curve x=t2+1x=t^{2}+1x=t2+1, y=t2−ty=t^{2}-ty=t2−t has slope 27. Answer exactly. x=x=x=

Problem 22116

Find all xxx where the tangent to y=12x2−ln⁡(x)y=12x^2-\ln(x)y=12x2−ln(x) is parallel to 6x+3y=176x+3y=176x+3y=17. Provide the largest xxx and if a second exists.

Problem 22117

Problem 22118

Calculate the distance an object falls in 20 seconds using the formula for free fall: d=12gt2d = \frac{1}{2} g t^2d=21​gt2 where ggg is the acceleration due to gravity.

Problem 22119

Evaluate the integral using symmetry: ∫−11(8x8−8)dx\int_{-1}^{1}\left(8 x^{8}-8\right) d x∫−11​(8x8−8)dx.

Problem 22120

Determine if the statement is true or false: ddθ[ln⁡(cos⁡(θ))]=−tan⁡(θ)\frac{d}{d \theta}[\ln (\cos (\theta))]=-\tan (\theta)dθd​[ln(cos(θ))]=−tan(θ).

Problem 22121

Given that fff is an even function and ∫−55f(x)dx=10\int_{-5}^{5} f(x) dx = 10∫−55​f(x)dx=10, find:a. ∫05f(x)dx\int_{0}^{5} f(x) dx∫05​f(x)dx b. ∫−55xf(x)dx\int_{-5}^{5} x f(x) dx∫−55​xf(x)dx

Problem 22122

Approximate the integral ∫00.21+x4dx\int_{0}^{0.2} \sqrt{1+x^{4}} dx∫00.2​1+x4​dx using a Taylor series, ensuring error < 10−410^{-4}10−4. Round to four decimals.

Problem 22123

Approximate the integral ∫00.28ln⁡(1+x2)dx\int_{0}^{0.28} \ln(1+x^{2}) dx∫00.28​ln(1+x2)dx using a Taylor series with error < 10−410^{-4}10−4. Choose the correct series.

Problem 22124

Calculate the average value of f(x)=6x2f(x)=6 x^{2}f(x)=6x2 over the interval [2,4][2,4][2,4].

Problem 22125

Problem 22126

Problem 22127

Approximate the integral ∫00.28ln⁡(1+x2) dx\int_{0}^{0.28} \ln(1+x^{2}) \, dx∫00.28​ln(1+x2)dx using Taylor series for error < 10−410^{-4}10−4. Find the value.

Problem 22128

Approximate the integral ∫00.52dx1+x6\int_{0}^{0.52} \frac{d x}{\sqrt{1+x^{6}}}∫00.52​1+x6​dx​ using a Taylor series. Choose the correct option.

Problem 22129

Evaluate the series ∑n=1∞cos⁡(nπ)\sum_{n=1}^{\infty} \cos (n \pi)∑n=1∞​cos(nπ): find SkS_{k}Sk​, check convergence with lim⁡k→∞Sk\lim _{k \rightarrow \infty} S_{k}limk→∞​Sk​, and apply a test.

Problem 22130

Calculate the average value of f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ on the interval [2,2e][2,2e][2,2e] and sketch its graph with the average value marked.

Problem 22131

Is ∑n=0∞an\sum_{n=0}^{\infty} a_{n}∑n=0∞​an​ convergent if an+1=an3a_{n+1}=\frac{a_{n}}{3}an+1​=3an​​ and a0=10a_{0}=10a0​=10? Find its sum if it converges.

Problem 22132

Approximate the integral ∫00.52dx1+x6\int_{0}^{0.52} \frac{dx}{\sqrt{1+x^{6}}}∫00.52​1+x6​dx​ using Taylor series, ensuring error < 10−410^{-4}10−4.

Problem 22133

Find the weight WWW of a person 5 feet 2 inches tall using dWdh=0.0012h2+0.01h\frac{d W}{d h}=0.0012 h^{2}+0.01 hdhdW​=0.0012h2+0.01h and W(70)=141.7W(70)=141.7W(70)=141.7.

Problem 22134

Find the area between the curves f(x)=5x−7f(x)=5x-7f(x)=5x−7 and g(x)=xg(x)=xg(x)=x from x=2x=2x=2 to x=5x=5x=5. Area =

Problem 22135

Find the limits: (a) for g(x)=−2f(x+7)+5g(x)=-2 f(x+7)+5g(x)=−2f(x+7)+5, and (b) for h(x)=−f(−x)h(x)=-f(-x)h(x)=−f(−x) as x→−∞x \to -\inftyx→−∞ and x→∞x \to \inftyx→∞.

Problem 22136

Problem 22137

Approximate the integral ∫00.99dx1+x6\int_{0}^{0.99} \frac{dx}{\sqrt{1+x^6}}∫00.99​1+x6​dx​ using Taylor series, ensuring error < 10−410^{-4}10−4. Find ∫00.49dx1+x6=□\int_{0}^{0.49} \frac{dx}{\sqrt{1+x^6}}=\square∫00.49​1+x6​dx​=□ (round to 4 decimal places).

Problem 22138

Problem 22139

Find the area between the curves y=x2−1y=x^{2}-1y=x2−1 and y=2x+7y=2x+7y=2x+7. Enter the exact value.

Problem 22140

Find the area between the curves y=x2−4xy=x^{2}-4xy=x2−4x and y=0y=0y=0 over the interval [−2,2][-2,2][−2,2]. Area ===

Problem 22141

Approximate the integral ∫00.49dx1+x6\int_{0}^{0.49} \frac{d x}{\sqrt{1+x^{6}}}∫00.49​1+x6​dx​ using Taylor series for error < 10−410^{-4}10−4.

Problem 22142

Find dxdt\frac{d x}{d t}dtdx​ when x=4x=4x=4 given x+y=6\sqrt{x}+y=6x​+y=6 and dydt=2\frac{d y}{d t}=2dtdy​=2.

Find $g(x)=\frac{1}{(x+2)^{3}}$ . Complete the table for $x$ near -2 and determine limits as $x \to -2$ .

Differentiate the series for $f(x)=\tan^{-1}(x^4)$ . Identify the function and interval of convergence. Choose the correct series.

Find the slope of the tangent line to the curve $2y + 2xy^3 = 2f(x)^2$ at the point $(1,1)$ using implicit differentiation.

Find the positive $x$ where the curve $y=x^{3} e^{-x / 6}$ has a horizontal tangent line. Provide the answer as a decimal.

Find the positive value of $x$ where $y=x^{3} e^{-x / 6}$ has a horizontal tangent line. Provide the answer as a decimal.

Find the first and second derivatives of the curve defined by $x=5+\sec t$ and $y=-1-5 \tan t$ .

Find the power series solution for $y^{\prime}(t)-y=0$ with $y(0)=30$ and identify the function.

Find the slope of the tangent line to the curve $2y + 2xy^3 = 2f(x)^2$ at point $(1,1)$ given $f(1)=2$ and $f'(1)=3$ .

Solve the differential equation $y^{\prime}(t)-4 y(t)=8$ , $y(0)=3$ . Find the power series and identify the function.

Find $g^{\prime}(2)$ for $g(x)=2+x^{3}+\tan^{-1}(f(x))$ given $f(2)=3$ and $f^{\prime}(2)=5$ .

An object falls from rest in a vacuum. How long to reach a velocity of $147 \mathrm{~m/s}$ with $g \approx 9.81 \mathrm{~m/s^2}$ ?

Estimate the integral $\int_{0}^{0.3} 2 e^{-x^{2}} d x$ using a series with error less than $10^{-8}$ . Round final answer to five decimal places.

Find $(\Delta a)_{m}$ for $a_{m}=\ln m$ and use it to compute $\sum_{m=1}^{N} \ln \left(1+\frac{1}{m}\right)$ .

Find all $x$ where the tangent line to $y=12x^2-\ln(x)$ is parallel to $6x+3y=17$ . Enter the largest $x$ as a rational number or decimal.

Find the $x$ -coordinate where the tangent line of the curve $x=t^{2}+1$ , $y=t^{2}-t$ has slope 27. Answer exactly. $x=$

Find all $x$ where the tangent to $y=12x^2-\ln(x)$ is parallel to $6x+3y=17$ . Provide the largest $x$ and if a second exists.

Calculate the distance an object falls in 20 seconds using the formula for free fall: $d = \frac{1}{2} g t^2$ where $g$ is the acceleration due to gravity.

Evaluate the integral using symmetry: $\int_{-1}^{1}\left(8 x^{8}-8\right) d x$ .

Determine if the statement is true or false: $\frac{d}{d \theta}[\ln (\cos (\theta))]=-\tan (\theta)$ .

Given that $f$ is an even function and $\int_{-5}^{5} f(x) dx = 10$ , find:
a. $\int_{0}^{5} f(x) dx$
b. $\int_{-5}^{5} x f(x) dx$

Approximate the integral $\int_{0}^{0.2} \sqrt{1+x^{4}} dx$ using a Taylor series, ensuring error < $10^{-4}$ . Round to four decimals.

Approximate the integral $\int_{0}^{0.28} \ln(1+x^{2}) dx$ using a Taylor series with error < $10^{-4}$ . Choose the correct series.

Calculate the average value of $f(x)=6 x^{2}$ over the interval $[2,4]$ .

Approximate the integral $\int_{0}^{0.28} \ln(1+x^{2}) \, dx$ using Taylor series for error < $10^{-4}$ . Find the value.

Approximate the integral $\int_{0}^{0.52} \frac{d x}{\sqrt{1+x^{6}}}$ using a Taylor series. Choose the correct option.

Evaluate the series $\sum_{n=1}^{\infty} \cos (n \pi)$ : find $S_{k}$ , check convergence with $\lim _{k \rightarrow \infty} S_{k}$ , and apply a test.

Calculate the average value of $f(x)=\frac{1}{x}$ on the interval $[2,2e]$ and sketch its graph with the average value marked.

Is $\sum_{n=0}^{\infty} a_{n}$ convergent if $a_{n+1}=\frac{a_{n}}{3}$ and $a_{0}=10$ ? Find its sum if it converges.

Approximate the integral $\int_{0}^{0.52} \frac{dx}{\sqrt{1+x^{6}}}$ using Taylor series, ensuring error < $10^{-4}$ .

Find the weight $W$ of a person 5 feet 2 inches tall using $\frac{d W}{d h}=0.0012 h^{2}+0.01 h$ and $W(70)=141.7$ .

Find the area between the curves $f(x)=5x-7$ and $g(x)=x$ from $x=2$ to $x=5$ . Area =

Find the limits: (a) for $g(x)=-2 f(x+7)+5$ , and (b) for $h(x)=-f(-x)$ as $x \to -\infty$ and $x \to \infty$ .

Approximate the integral $\int_{0}^{0.99} \frac{dx}{\sqrt{1+x^6}}$ using Taylor series, ensuring error < $10^{-4}$ . Find $\int_{0}^{0.49} \frac{dx}{\sqrt{1+x^6}}=\square$ (round to 4 decimal places).

Find the area between the curves $y=x^{2}-1$ and $y=2x+7$ . Enter the exact value.

Find the area between the curves $y=x^{2}-4x$ and $y=0$ over the interval $[-2,2]$ . Area $=$

Approximate the integral $\int_{0}^{0.49} \frac{d x}{\sqrt{1+x^{6}}}$ using Taylor series for error < $10^{-4}$ .

Find $\frac{d x}{d t}$ when $x=4$ given $\sqrt{x}+y=6$ and $\frac{d y}{d t}=2$ .

Find $\frac{d x}{d t}$ when $x=4$ given $\sqrt{x}+y=6$ and $\frac{d y}{d t}=2$ .

Find the area between the curves $y=e^{4 x}$ and $y=\frac{1}{3 x}$ from $x=1$ to $x=6$ . Area =

Find the area between the curves $y=x^{2}$ and $y=3x$ from $x=0$ to $x=6$ . Area $=$

Find the Taylor polynomials $p_{3}$ and $p_{4}$ at $a=0$ for $f(x)=(2+x)^{-3}$ . What is $p_{3}(x)$ ?

Find the area between the curves $f(x)=\sqrt{x}$ and $g(x)=x / 2$ over the interval $[0,16]$ .

Find the area between the curves $y=x \sqrt{x^{2}+25}$ and $y=0$ from $x=-2$ to $x=1$ . $\text{Area} =$

Find the average rate of change for $f(x)=x^{3}-2 x^{2}+x+1$ on $[0,2]$ and $[2,4]$ . Compare the results.

Find the 4th and 5th Taylor polynomials for $f(x)=6 \cos (x)$ at $a=\frac{\pi}{6}$ .

Find the 4th and 5th Taylor polynomials, $p_{4}$ and $p_{5}$ , for $f(x)=2 \cos (x)$ at $a=\frac{\pi}{6}$ .

Find the area between the curve $y=(x-1)e^{4x^{2}-8x}$ and $y=0$ from $x=0$ to $x=2$ . $\text{Area} =$

Find the consumer surplus for the demand function $p=D(x)=\sqrt{25-2x}$ at a market price of \$3.

Evaluate the triple integral: $\int_{-1}^{2} \int_{-\sqrt{8-x^{2}}}^{\sqrt{8-x^{2}}} \int_{0}^{x+1} 2 z d z d y d x$ .

Calculate the integral $\int x \ln \left(x^{2}\right) d x$ .

Find the average distance from the parabola $y=8 x(22-x)$ to the $x$ -axis over the interval $[0,22]$ .

Find the partial derivatives of the function $f(x, y, z)=\frac{1+x^{3} y^{2}}{z^{2}}$ with respect to $x$ , $y$ , and $z$ .

Find the average rate of change of $y=3 \sin (2 x)$ on $[0, \frac{\pi}{4}]$ . Choices: A. $-\frac{12}{\pi}$ B. $-\frac{6}{\pi}$ C. $\frac{6}{\pi}$ D. $\frac{12}{\pi}$

Evaluate the integral from 0 to 4 of the function $7 x^{2}-6 x+8$ . What is the result?

Find when the speed of the particle, defined by $x(t)=2 t^{3}-18 t^{2}-96 t$ , is increasing for $t \geq 0$ .

Evaluate the integral: $\int_{0}^{\pi / 3} 8 \sec ^{2} x \, dx = \square$ (Type an exact answer, using radicals as needed.)

Find the 2nd degree Taylor polynomial $p_{2}$ at $a=8$ for $f(x)=\sqrt[3]{x}$ . Determine $p_{2}(x)=\square$ .

Find the derivative of the integral $\frac{d}{d t} \int_{0}^{t^{8}} \sqrt{u} d u$ using two methods.

Evaluate the integral: $\int_{0}^{\frac{\pi}{8}} \sin 2 x \, dx = \square$ (Exact answer with radicals).

Calculate the integral from -7 to 7 of the absolute value function: $\int_{-7}^{7}|x| d x=\square$

Evaluate the integral from $7 \pi / 2$ to $\pi$ : $\int_{7 \pi / 2}^{\pi} \frac{1-\cos 2 t}{2} d t = \square($ Type an exact answer, using $\pi$ as needed. $)$

Find $\frac{d}{d t} \int_{0}^{t^{8}} \sqrt{u} d u$ by evaluating the integral or differentiating directly.

Find $\frac{\partial f}{\partial y}$ for $f(x, y)=x y^{8}+6$ at $(x, y)=(-4,-1)$ and explain the meaning.

Find the derivative of $\int_{0}^{x^3} e^{-2t} dt$ using two methods: a) evaluate then differentiate, b) differentiate directly.

Find the derivative $\frac{d y}{d x}$ for the integral $y=\int_{0}^{\tan x} \frac{d t}{1+t^{2}}$ . Simplify your answer.

Find points $(x, y)$ where $f(x, y)=x^{3}-2xy+\frac{1}{3}y^{2}+8$ has relative extrema.

Find $\theta$ for horizontal and vertical tangents on the polar curve $r=\theta$ , where $0 \leq \theta \leq 2\pi$ .

Calculate consumer surplus and producer surplus using demand $D(x) = -3x + 19$ and supply $S(x) = 4x + 5$ .