Calculus

Problem 19101

Prove that $\int_{0}^{1} \sin \left(x^{2}\right) d x$ cannot equal 2. Choose the correct reasoning.

See Solution

Problem 19102

Use the Integral Test to check if the series $\sum_{k=1}^{\infty} \frac{e^{4 k}}{1+e^{8 k}}$ converges. Identify conditions satisfied by $f(x)=\frac{e^{4 x}}{1+e^{8 x}}$ . Select A-F.

See Solution

Problem 19103

Solve the IVP: $\frac{d y}{d t}=2-8 t$ , with $y(0)=-8$ .

See Solution

Problem 19104

Use the Integral Test to check if the series $\sum_{k=1}^{\infty} \frac{e^{4 k}}{1+e^{8 k}}$ converges.

See Solution

Problem 19105

Solve $\frac{d z}{d t}=t^{-3 / 2}$ with $z(64)=-4$ .

See Solution

Problem 19106

Use the Integral Test to check if the series $\sum_{k=1}^{\infty} k e^{-3 k^{2}}$ converges. Which conditions apply to $f(x)=x e^{-3 x^{2}}$ ? A. $a_{k}=f(k)$ B. $f(x)$ is decreasing for $x \geq 1$ C. $f(x)$ is continuous for $x \geq 1$ D. $f(x)$ is increasing for $x \geq 1$ E. $f(x)$ is negative for $x \geq 1$ F. $f(x)$ is positive for $x \geq 1$ .

See Solution

Problem 19107

Use the Integral Test to check if the series $\sum_{k=1}^{\infty} k e^{-3 k^{2}}$ converges. Select: A. Converges, B. Diverges, C. Test not applicable.

See Solution

Problem 19108

Use the Integral Test to check if the series converges: $\sum_{k=2}^{\infty} \frac{9}{k(\ln k)^{2}}$ . Which conditions apply to $f(x)=\frac{9}{x(\ln x)^{2}}$ ?

See Solution

Problem 19109

Prove that for the curve $(2 y+1)^{3}-24 x=-3$ , the derivative is $\frac{dy}{dx} = \frac{4}{(2y+1)^2}$ .

See Solution

Problem 19110

Use the Integral Test to check the convergence of $\sum_{k=2}^{\infty} \frac{2}{k(\ln k)^{2}}$ . Which conditions apply? A. Positive for $x \geq 2$ B. Increasing for $x \geq 2$ C. Negative for $x \geq 2$ D. $a_{k}=f(k)$ for $k=2,3,4$ E. Continuous for $x \geq 2$ F. Decreasing for $x \geq 2$ .

See Solution

Problem 19111

A penny is flipped upward at $2.85 \mathrm{~m/s}$ . Calculate the maximum height it reaches above your hand.

See Solution

Problem 19112

Find the derivative $f'(x)$ for the function $f(x)=\frac{e^{x}}{7 x^{2}+1}$ .

See Solution

Problem 19113

Calculate the derivative $f'(x)$ for the function $f(x)=\frac{8x^{2}}{6x-5}$ .

See Solution

Problem 19114

Find the derivative $f'(x)$ for the function $f(x)=\frac{e^{x}}{3 x^{2}+8}$ .

See Solution

Problem 19115

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{e^{x}}{8 x^{2}+1}$ .

See Solution

Problem 19116

Verify if the function $P(x)=x^{5}-3 x^{3}+1$ has a zero between -1.8 and -1.7 using the intermediate value theorem.

See Solution

Problem 19117

Show that $P(x)=4 x^{2}-2 x-5$ has a zero between 1 and 2 using the intermediate value theorem. Approximate the zero.

See Solution

Problem 19118

Verify if the function $P(x)=x^{5}-4 x^{3}+1$ has a real zero between -2.04 and -2.02 using the intermediate value theorem.

See Solution

Problem 19119

Verify if $P(x)=x^{5}-3 x^{3}+1$ has a zero between -1.8 and -1.7 using the intermediate value theorem. Approximate the zero to the nearest hundredth: $x=\square$ .

See Solution

Problem 19120

Verify if the function $P(x)=x^{5}-4x^{3}+1$ has a zero between -2.04 and -2.02 using the intermediate value theorem. Approximate the zero to the nearest hundredth: $x=\square$ .

See Solution

Problem 19121

Use the intermediate value theorem to show there's a zero for $P(x)=8x^4-12x^2+16x-20$ between 1 and 1.5. Evaluate $P(1)$ and $P(1.5)$ . Why do they indicate a zero exists? A. Opposite signs, B. Same sign, C. $P(1) \neq P(1.5)$ , D. $P(1)<P(1.5)$ .

See Solution

Problem 19122

Verify if $P(x) = x^{5} - 3x^{3} + 1$ has a zero between -1.8 and -1.7 using the intermediate value theorem.

See Solution

Problem 19123

Find the differential $d y$ for $y=\ln(1+x^{2})$ at $x=2.6$ and $d x=0.3$ . Round to three decimal places.

See Solution

Problem 19124

Find the speed of a penny just before it hits the ground, falling from $1.26 \mathrm{~m}$ . Use energy concepts.

See Solution

Problem 19125

Find the derivative $f'(x)$ for the function $f(x)=\frac{9 x^{2}}{6 x-7}$ . What is $f'(x)$ ?

See Solution

Problem 19126

Determine the vertical and horizontal asymptotes for the function $f(x)=\frac{1}{x+3}$ .

See Solution

Problem 19127

Determine if the series $\sum_{k=1}^{\infty} \frac{k}{k^{2}+6}$ converges or diverges. Choose A-F and fill in the box.

See Solution

Problem 19128

Evaluate the integral $I=\int_{-\infty}^{-3} \frac{d x}{|x+2|}$ .

See Solution

Problem 19129

Determine if the series $\sum_{k=2}^{\infty} \frac{k^{6}}{k^{2 \pi}}$ converges or diverges using relevant tests.

See Solution

Problem 19130

Determine if the series $\sum_{k=2}^{\infty} \frac{k^{\pi}}{k^{4}}$ converges using the Divergence, Integral, or p-series test.

See Solution

Problem 19131

Find $G(2)$ and $G^{\prime}(2)$ for $G(x)=\int_{2}^{x} \sqrt{t^{3}+1} d t$ .

See Solution

Problem 19132

Determine if the series $\sum_{k=1}^{\infty} \frac{\sqrt{4 k^{2}+11}}{k}$ converges or diverges.

See Solution

Problem 19133

Check if the series converges or diverges: $\sum_{n=0}^{\infty} \frac{4^{n}-2^{n}}{9^{n}}$ .

See Solution

Problem 19134

Determine if the series $\sum_{k=2}^{\infty} \frac{1}{(k-1)^{4}}$ converges or diverges. What is $p$ ?

See Solution

Problem 19135

Find the subinterval length $\Delta x$ for $[0,2]$ with $n=4$ . List grid points $\mathrm{x}_{0}$ to $\mathrm{x}_{4}$ . Identify points for left, right, and midpoint Riemann sums.

See Solution

Problem 19136

Determine if the series $\sum_{k=1}^{\infty} \frac{4^{k}+6^{k}}{11^{k}}$ converges or diverges.

See Solution

Problem 19137

Determine if the series $\sum_{k=1}^{\infty} \frac{4^{k}+7^{k}}{11^{k}}$ converges or diverges.

See Solution

Problem 19138

Find the differential $d y$ for $y=\ln(1+x^{2})$ at $x=2.2$ and $d x=0.6$ . Round to three decimal places.

See Solution

Problem 19139

Solve the differential equation: $5 \frac{d y}{d \theta}=\frac{\mathrm{e}^{y} \sin ^{2}(\theta)}{y \sec (\theta)}$ . Use $C$ as a constant.

See Solution

Problem 19140

Estimate $f(5.2)$ using the linear approximation given $f(5)=5.1$ and $f'(5)=3.2$ . Provide your answer to three decimal places.

See Solution

Problem 19141

Determine if the series converges:
$\sum_{k=0}^{\infty} \frac{8(-1)^{k}}{9 k+7}$
Evaluate $\lim_{k \rightarrow \infty} a_k$ where $a_k = \frac{8}{9k+7}$ .

See Solution

Problem 19142

Determine if the series $\sum_{k=0}^{\infty} \frac{9(-1)^{k}}{8 k+5}$ converges. Define $a_{k}$ . Choose A, B, or C.

See Solution

Problem 19143

Find the indefinite integral: $\int e^{x} \sqrt{1-e^{x}} \, dx$

See Solution

Problem 19144

Determine if the series $\sum_{k=0}^{\infty} \frac{8(-1)^{k}}{9 k+7}$ converges. Define $a_{k}$ and choose its behavior: A, B, or C.

See Solution

Problem 19145

Calculate the indefinite integral: $\int \ln \left(e^{6 x-9}\right) d x$ .

See Solution

Problem 19146

Calculate the integral from 0 to 1 for $x e^{-7 x^{2}}$ .

See Solution

Problem 19147

Calculate the left and right Riemann sums for $f(x)=\frac{2}{x}+2$ on $[1,5]$ with $n=4$ .

See Solution

Problem 19148

Find critical numbers of $f(x)$ , intervals of increase/decrease, and all relative extrema for $f(x)=\frac{x^{2}-3 x-4}{x-2}$ .

See Solution

Problem 19149

Evaluate the integral: $\int_{0}^{\pi / 2} e^{\sin 5 \pi x} \cos 5 \pi x \, dx$ .

See Solution

Problem 19150

Find the positive critical number of the function $f(x)=(x^{2}-9)^{3}$ . Round your answer to three decimal places.

See Solution

Problem 19151

Estimate $f(9.3)$ using linear approximation given $f(9)=1.7$ and $f'(9)=4$ . Round to three decimal places.

See Solution

Problem 19152

Find the integral of $\sin^{2} \theta \cos \theta$ with respect to $\theta$ .

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Problem 19153

Determine if the series $\sum_{k=0}^{\infty} \frac{9(-1)^{k}}{8 k+5}$ converges. Evaluate $\lim_{k \to \infty} a_{k}$ .

See Solution

Problem 19154

Calculate the left and right Riemann sums for $f(x)=\frac{2}{x}+5$ on $[1,5]$ with $n=4$ . Left sum: $\square$ .

See Solution

Problem 19155

Evaluate the integral from 0 to $\frac{\pi}{2}$ : $\int_{0}^{\pi / 2} e^{\sin 5 \pi x} \cos 5 \pi x \, dx$ . Use a graphing tool to check your answer.

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Problem 19156

Find $\frac{dy}{dx}$ from the equation $V=-4.7x-1.5\ln(x)+1.8y-1.3\ln(y)$ , then evaluate at $x=3$ , $y=8$ . Round to three decimal places.

See Solution

Problem 19157

Find the derivative of $f(x)=9+2.6 \tan ^{-1}(4.6 \ln (x))$ and evaluate it at $x=6$ . Round to three decimal places.

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Problem 19158

Find the positive value of $x$ where $f''(x)=0$ for $f(x)=x \exp(-x^{2}/5)$ . Round to three decimal places.

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Problem 19159

A bacteria culture starts with 740 and grows proportionally. After 3 hours, it has 2220. Find $P(t)$ , $P(5)$ , and time for 2250.

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Problem 19160

Find the positive value of $x$ where the second derivative $f''(x)=0$ for the function $f(x)=x \exp(-x^{2}/3)$ . Round to three decimal places.

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Problem 19161

Calculate the integral: $\int \frac{4 x}{\sqrt{16-x^{2}}} d x$

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Problem 19162

A bacteria culture starts with 1000 and grows to 5000 in 5 hours. Find $P(t)$ , $P(6)$ , and time to reach 1790.

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Problem 19163

A bacteria culture starts with 660 and grows proportionally. Find $P(t)$ , population after 7 hours, and time to reach 2770.

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Problem 19164

Find the tangent line equation for $y=x^{\sin (x)}$ at the point $\left(\frac{\pi}{2}, \frac{\pi}{2}\right)$ .

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Problem 19165

Find the tangent line equation to the function at the point $(7,1)$ for $y=9^{x-7}$ .

See Solution

Problem 19166

A bacteria culture starts with 1000 and grows proportionally. Find $P(t)$ after $t$ hours, $P(8)$ , and time to reach 1010.

See Solution

Problem 19167

Find $\frac{d y}{d x}$ from the equation $V=-3.7 x-2.2 \ln (x)+2 y-1.4 \ln (y)$ and evaluate at $x=4$ , $y=10$ . Round to three decimal places.

See Solution

Problem 19168

Find the derivative of $f(x)=3+3.4 \tan^{-1}(3.6 \ln(x))$ and calculate $f'(8)$ rounded to three decimal places.

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Problem 19169

Find $d y / d x$ using logarithmic differentiation for $y=(9+x)^{4/x}$ . $\frac{d y}{d x}=\square$

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Problem 19170

Solve the differential equation $\frac{d P}{d t}=5 \sqrt{P t}$ with the initial condition $P(1)=5$ .

See Solution

Problem 19171

Find the derivative of the integral: $\frac{d}{d x} \int_{0}^{\sin x} \sqrt{\frac{2}{t^{2}-3 t-5}} d t$

See Solution

Problem 19172

Determine if the series $\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k}{14 k+1}$ converges or diverges.

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Problem 19173

Determine if the series $\sum_{k=1}^{\infty} \frac{8(-1)^{k+1}}{k^{7}}$ converges. Identify $a_{k}$ .

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Problem 19174

Berechnen Sie die Fläche zwischen den Graphen der Funktionen und der $x$ -Achse im Intervall [1; 5]. Funktionen: a) $f(x)=-x+5$ b) $f(x)=0,2 x^{2}+2$ c) $f(x)=x^{3}+1$ d) $f(x)=-\frac{3}{7} x^{2}+27$ Skizzieren Sie die Graphen und erstellen Sie eine Wertetabelle mit drei Punkten.

See Solution

Problem 19175

Determine if the series $\sum_{k=1}^{\infty} \frac{7(-1)^{k+1}}{k^{4}}$ converges or diverges.

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Problem 19176

Ein Körper bewegt sich mit dem Weg $s(t)=4 t^{2}$ .
a) Finde die momentane Änderungsrate von $s$ bei $t_{0}=1$ und $t_{1}=5$ . b) Was bedeutet die momentane Änderungsrate von $s$ ?

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Problem 19177

Determine if the series $\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k}{4 k^{3}+1}$ converges using the Alternating Series Test. Are the terms nonincreasing?

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Problem 19178

Find the displacement and total distance traveled for $v(t)=5t-7$ from $t=0$ to $t=2$ . Show your work using integrals.

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Problem 19179

Determine if the series $\sum_{k=0}^{\infty}\left(-\frac{3}{4}\right)^{k}$ converges. Describe $a_{k}>0$ . Choose A, B, or C.

See Solution

Problem 19180

Does the series $\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k^{6}+4 k^{3}+1}{k\left(k^{5}+1\right)}$ converge? Define $a_{k}$ .

See Solution

Problem 19181

Find how many terms of the series $\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k^{4}}$ must be summed for the remainder to be < $10^{-5}$ . The answer is $\square$ .

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Problem 19182

Determine if the series converges: $\sum_{k=0}^{\infty}\left(-\frac{3}{5}\right)^{k}$ . Choose A, B, C, D, or E and fill in the blanks.

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Problem 19183

Find the number of terms needed for the series $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{2}}$ so that the remainder is < $10^{-3}$ .

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Problem 19184

Determine if the series converges: $\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k^{18}+4 k^{9}+1}{k(k^{17}+1)}$ . Find $a_{k}$ where $a_{k}>0$ . What is $a_{k}=\square$ ?

See Solution

Problem 19185

Determine if the series converges: $\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k^{6}+3 k^{3}+10}{k(k^{5}+10)}$ . Find $a_{k} = \square$ .

See Solution

Problem 19186

Determine if the series $\sum_{k=0}^{\infty}\left(-\frac{3}{7}\right)^{k}$ converges or diverges.

See Solution

Problem 19187

Determine if the series converges: $\sum_{k=1}^{\infty} \frac{\cos \pi k}{k^{2}}$ . Describe $a_{k} = \frac{|\cos \pi k|}{k^{2}}$ . Choose A, B, or C.

See Solution

Problem 19188

Determine if the series $\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k^{6}+3 k^{3}+10}{k(k^{5}+10)}$ converges. Define $a_{k}=\frac{k^{5}+3 k^{2}+\frac{10}{k}}{k^{5}+10}$ . Is $a_{k}$ increasing, nonincreasing, or neither for $k > N$ ? Choose A, B, or C.

See Solution

Problem 19189

Determine if the series converges using the Alternating Series Test:
$\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k}{8 k^{5}+1}$
Are the terms nonincreasing in magnitude? Choose A or B.

See Solution

Problem 19190

Bestimme die Ableitung $f^{\prime}(x)$ für folgende Funktionen: a) $f(x)=3 x^{2}$ , b) $f(x)=5 x^{3}$ , c) $f(x)=\frac{1}{2} x^{4}$ , d) $f(x)=-2 x^{7}$ , e) $f(x)=\frac{1}{2} x^{-2}$ , f) $f(x)=-5 x^{-3}$ , g) $f(x)=-0,2 x^{5}$ , h) $f(x)=4 x^{\frac{1}{2}}$ .

See Solution

Problem 19191

Leiten Sie die Funktionen $f$ ab: a) $f(x)=2 x^{3}+5 x^{2}$ , b) $f(x)=4 x^{5}-3 x$ , c) $f(x)=2 x^{7}-3 x^{4}$ , d) $f(x)=0,25 x^{4}-x^{3}$ , e) $f(t)=-5 t^{4}-4 t^{3}+t$ , f) $f(z)=-\frac{2}{3} z^{3}+2 z^{2}+3$ , g) $f(a)=a^{2}-16 a+5$ , h) $f(x)=\frac{1}{5} x^{5}-3 x^{3}-2$ , i) $f(t)=t^{4}-2 t^{3}+\frac{1}{4} t^{2}-t$ .

See Solution

Problem 19192

Berechnen Sie die Integrale von $f(x)=\frac{5 x^{3}-2 x}{x}$ , $g(x)=x^{2}+\sqrt{x}$ und $h(x)=\frac{2}{x^{5}}+3$ .

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Problem 19193

Bestimme $f^{\prime}(x)$ für die Funktionen: 1a) $f(x)=x^{3}+x^{2}$ , 1b) $f(x)=x^{4}+x^{2}$ , 1c) $f(x)=x^{5}+x^{4}$ , 1d) $f(x)=x^{4}+1$ , 1e) $f(x)=x^{7}+x^{4}+x^{2}$ , 1f) $f(x)=x^{8}+x^{3}+5$ , 1g) $f(x)=x^{5}+x+3$ , 1h) $f(x)=x^{11}+x^{7}+x^{3}$ , 1i) $f(x)=x^{5}+x^{4}+x^{3}+x^{2}$ . 2a) $f(x)=3 x^{2}$ , 2b) $f(x)=5 x^{3}$ , 2c) $f(x)=\frac{1}{2} x^{4}$ , 2d) $f(x)=-2 x^{7}$ , 2e) $f(x)=\frac{1}{2} x^{-2}$ , 2f) $f(x)=-5 x^{-3}$ , 2g) $f(x)=-0,2 x^{5}$ , 2h) $f(x)=4 x^{\frac{1}{2}}$ . 3a) $f(x)=2 x^{3}+5 x^{2}$ , 3b) $f(x)=4 x^{5}-3 x$ , 3c) $f(x)=2 x^{7}-3 x^{4}$ , 3d) $f(x)=0,25 x^{4}-x^{3}$ , 3e) $f(t)=-5 t^{4}-4 t^{3}+t$ , 3f) $f(z)=-\frac{2}{3} z^{3}+2 z^{2}+3$ , 3g) $f(a)=a^{2}-16 a+5$ , 3h) $f(x)=\frac{1}{5} x^{5}-3 x^{3}-2$ , 3i) $f(t)=t^{4}-2 t^{3}+\frac{1}{4} t^{2}-t$ .

See Solution

Problem 19194

Find the derivative of $y=\sin x \cos x$ . What is $y^{\prime}$ ? A. $\frac{1}{2} \sin 2 x$ B. $\frac{1}{4} \cos 2 x$ C. $\cos 2 x$ D. None

See Solution

Problem 19195

Solve the IVP: $\frac{d y}{d x}=(9 x+2)^{3}$ with $y(0)=5$ .

See Solution

Problem 19196

Determine the derivative $f'(x)$ of the function $f(x)=\frac{2}{3} \sin^{\frac{3}{2}} x - \frac{2}{7} \sin^{\frac{7}{2}} x$ . Choose the correct option: A, B, C, or D.

See Solution

Problem 19197

Untersuchen Sie die Medikamentenkonzentration $f(t)=0,015 t^{3}-0,6 t^{2}+6 t$ im Blut.
a) Bestimmen Sie $f(1)$ . b) Finden Sie $t$ , wenn $f(t)=0$ . c) Analysieren Sie das Verhalten von $f(t)$ für $t \rightarrow \infty$ .

See Solution

Problem 19198

Untersuche das Verhalten der Funktion $g(x)=\frac{6 x}{x^{2}-1}$ für große $|x|$ und finde die Asymptoten.

See Solution

Problem 19199

Bestimme eine Stammfunktion für $f$ : a) $f(x)=4 x^{3}+2 x^{2}-1$ , d) $f(x)=\frac{3}{4} x^{2}+\frac{8}{7} x^{3}$ .

See Solution

Problem 19200

1. Given the utility function $U(x_{1}, x_{2})=x_{1}^{3} x_{2}^{3}$ , with prices \$15 and \$30 and a budget of \$900, find:
a) The budget constraint equation. b) The optimal bundle where $MRS$ equals the price ratio. c) The expression $\ln[U(x_{1}, x_{2})]$ . d) The optimal bundle using Lagrange method. Is it the same as in (b)?

See Solution

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Calculus

Problem 19101

Prove that ∫01sin⁡(x2)dx\int_{0}^{1} \sin \left(x^{2}\right) d x∫01​sin(x2)dx cannot equal 2. Choose the correct reasoning.

Problem 19102

Use the Integral Test to check if the series ∑k=1∞e4k1+e8k\sum_{k=1}^{\infty} \frac{e^{4 k}}{1+e^{8 k}}k=1∑∞​1+e8ke4k​ converges. Identify conditions satisfied by f(x)=e4x1+e8xf(x)=\frac{e^{4 x}}{1+e^{8 x}}f(x)=1+e8xe4x​. Select A-F.

Problem 19103

Solve the IVP: dydt=2−8t\frac{d y}{d t}=2-8 tdtdy​=2−8t, with y(0)=−8y(0)=-8y(0)=−8.

Problem 19104

Use the Integral Test to check if the series ∑k=1∞e4k1+e8k\sum_{k=1}^{\infty} \frac{e^{4 k}}{1+e^{8 k}}∑k=1∞​1+e8ke4k​ converges.

Problem 19105

Solve dzdt=t−3/2\frac{d z}{d t}=t^{-3 / 2}dtdz​=t−3/2 with z(64)=−4z(64)=-4z(64)=−4.

Problem 19106

Problem 19107

Use the Integral Test to check if the series ∑k=1∞ke−3k2\sum_{k=1}^{\infty} k e^{-3 k^{2}}∑k=1∞​ke−3k2 converges. Select: A. Converges, B. Diverges, C. Test not applicable.

Problem 19108

Use the Integral Test to check if the series converges: ∑k=2∞9k(ln⁡k)2\sum_{k=2}^{\infty} \frac{9}{k(\ln k)^{2}}k=2∑∞​k(lnk)29​. Which conditions apply to f(x)=9x(ln⁡x)2f(x)=\frac{9}{x(\ln x)^{2}}f(x)=x(lnx)29​?

Problem 19109

Prove that for the curve (2y+1)3−24x=−3(2 y+1)^{3}-24 x=-3(2y+1)3−24x=−3, the derivative is dydx=4(2y+1)2\frac{dy}{dx} = \frac{4}{(2y+1)^2}dxdy​=(2y+1)24​.

Problem 19110

Problem 19111

A penny is flipped upward at 2.85 m/s2.85 \mathrm{~m/s}2.85 m/s. Calculate the maximum height it reaches above your hand.

Problem 19112

Find the derivative f′(x)f'(x)f′(x) for the function f(x)=ex7x2+1f(x)=\frac{e^{x}}{7 x^{2}+1}f(x)=7x2+1ex​.

Problem 19113

Calculate the derivative f′(x)f'(x)f′(x) for the function f(x)=8x26x−5f(x)=\frac{8x^{2}}{6x-5}f(x)=6x−58x2​.

Problem 19114

Find the derivative f′(x)f'(x)f′(x) for the function f(x)=ex3x2+8f(x)=\frac{e^{x}}{3 x^{2}+8}f(x)=3x2+8ex​.

Problem 19115

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=ex8x2+1f(x)=\frac{e^{x}}{8 x^{2}+1}f(x)=8x2+1ex​.

Problem 19116

Verify if the function P(x)=x5−3x3+1P(x)=x^{5}-3 x^{3}+1P(x)=x5−3x3+1 has a zero between -1.8 and -1.7 using the intermediate value theorem.

Problem 19117

Show that P(x)=4x2−2x−5P(x)=4 x^{2}-2 x-5P(x)=4x2−2x−5 has a zero between 1 and 2 using the intermediate value theorem. Approximate the zero.

Problem 19118

Verify if the function P(x)=x5−4x3+1P(x)=x^{5}-4 x^{3}+1P(x)=x5−4x3+1 has a real zero between -2.04 and -2.02 using the intermediate value theorem.

Problem 19119

Verify if P(x)=x5−3x3+1P(x)=x^{5}-3 x^{3}+1P(x)=x5−3x3+1 has a zero between -1.8 and -1.7 using the intermediate value theorem. Approximate the zero to the nearest hundredth: x=□x=\squarex=□.

Problem 19120

Verify if the function P(x)=x5−4x3+1P(x)=x^{5}-4x^{3}+1P(x)=x5−4x3+1 has a zero between -2.04 and -2.02 using the intermediate value theorem. Approximate the zero to the nearest hundredth: x=□x=\squarex=□.

Problem 19121

Problem 19122

Verify if P(x)=x5−3x3+1P(x) = x^{5} - 3x^{3} + 1P(x)=x5−3x3+1 has a zero between -1.8 and -1.7 using the intermediate value theorem.

Problem 19123

Find the differential dyd ydy for y=ln⁡(1+x2)y=\ln(1+x^{2})y=ln(1+x2) at x=2.6x=2.6x=2.6 and dx=0.3d x=0.3dx=0.3. Round to three decimal places.

Problem 19124

Find the speed of a penny just before it hits the ground, falling from 1.26 m1.26 \mathrm{~m}1.26 m. Use energy concepts.

Problem 19125

Find the derivative f′(x)f'(x)f′(x) for the function f(x)=9x26x−7f(x)=\frac{9 x^{2}}{6 x-7}f(x)=6x−79x2​. What is f′(x)f'(x)f′(x)?

Problem 19126

Determine the vertical and horizontal asymptotes for the function f(x)=1x+3f(x)=\frac{1}{x+3}f(x)=x+31​.

Problem 19127

Determine if the series ∑k=1∞kk2+6\sum_{k=1}^{\infty} \frac{k}{k^{2}+6}∑k=1∞​k2+6k​ converges or diverges. Choose A-F and fill in the box.

Problem 19128

Evaluate the integral I=∫−∞−3dx∣x+2∣I=\int_{-\infty}^{-3} \frac{d x}{|x+2|}I=∫−∞−3​∣x+2∣dx​.

Problem 19129

Determine if the series ∑k=2∞k6k2π\sum_{k=2}^{\infty} \frac{k^{6}}{k^{2 \pi}}∑k=2∞​k2πk6​ converges or diverges using relevant tests.

Problem 19130

Determine if the series ∑k=2∞kπk4\sum_{k=2}^{\infty} \frac{k^{\pi}}{k^{4}}∑k=2∞​k4kπ​ converges using the Divergence, Integral, or p-series test.

Problem 19131

Find G(2)G(2)G(2) and G′(2)G^{\prime}(2)G′(2) for G(x)=∫2xt3+1dtG(x)=\int_{2}^{x} \sqrt{t^{3}+1} d tG(x)=∫2x​t3+1​dt.

Problem 19132

Determine if the series ∑k=1∞4k2+11k\sum_{k=1}^{\infty} \frac{\sqrt{4 k^{2}+11}}{k}∑k=1∞​k4k2+11​​ converges or diverges.

Problem 19133

Check if the series converges or diverges: ∑n=0∞4n−2n9n\sum_{n=0}^{\infty} \frac{4^{n}-2^{n}}{9^{n}}n=0∑∞​9n4n−2n​.

Problem 19134

Determine if the series ∑k=2∞1(k−1)4\sum_{k=2}^{\infty} \frac{1}{(k-1)^{4}}∑k=2∞​(k−1)41​ converges or diverges. What is ppp?

Problem 19135

Find the subinterval length Δx\Delta xΔx for [0,2][0,2][0,2] with n=4n=4n=4. List grid points x0\mathrm{x}_{0}x0​ to x4\mathrm{x}_{4}x4​. Identify points for left, right, and midpoint Riemann sums.

Problem 19136

Determine if the series ∑k=1∞4k+6k11k\sum_{k=1}^{\infty} \frac{4^{k}+6^{k}}{11^{k}}∑k=1∞​11k4k+6k​ converges or diverges.

Problem 19137

Determine if the series ∑k=1∞4k+7k11k\sum_{k=1}^{\infty} \frac{4^{k}+7^{k}}{11^{k}}k=1∑∞​11k4k+7k​ converges or diverges.

Problem 19138

Find the differential dyd ydy for y=ln⁡(1+x2)y=\ln(1+x^{2})y=ln(1+x2) at x=2.2x=2.2x=2.2 and dx=0.6d x=0.6dx=0.6. Round to three decimal places.

Problem 19139

Solve the differential equation: 5dydθ=eysin⁡2(θ)ysec⁡(θ)5 \frac{d y}{d \theta}=\frac{\mathrm{e}^{y} \sin ^{2}(\theta)}{y \sec (\theta)}5dθdy​=ysec(θ)eysin2(θ)​. Use CCC as a constant.

Problem 19140

Estimate f(5.2)f(5.2)f(5.2) using the linear approximation given f(5)=5.1f(5)=5.1f(5)=5.1 and f′(5)=3.2f'(5)=3.2f′(5)=3.2. Provide your answer to three decimal places.

Problem 19141

Determine if the series converges: ∑k=0∞8(−1)k9k+7\sum_{k=0}^{\infty} \frac{8(-1)^{k}}{9 k+7}k=0∑∞​9k+78(−1)k​ Evaluate lim⁡k→∞ak\lim_{k \rightarrow \infty} a_klimk→∞​ak​ where ak=89k+7a_k = \frac{8}{9k+7}ak​=9k+78​.

Prove that $\int_{0}^{1} \sin \left(x^{2}\right) d x$ cannot equal 2. Choose the correct reasoning.

Use the Integral Test to check if the series $\sum_{k=1}^{\infty} \frac{e^{4 k}}{1+e^{8 k}}$ converges. Identify conditions satisfied by $f(x)=\frac{e^{4 x}}{1+e^{8 x}}$ . Select A-F.

Solve the IVP: $\frac{d y}{d t}=2-8 t$ , with $y(0)=-8$ .

Use the Integral Test to check if the series $\sum_{k=1}^{\infty} \frac{e^{4 k}}{1+e^{8 k}}$ converges.

Solve $\frac{d z}{d t}=t^{-3 / 2}$ with $z(64)=-4$ .

Use the Integral Test to check if the series $\sum_{k=1}^{\infty} k e^{-3 k^{2}}$ converges. Select: A. Converges, B. Diverges, C. Test not applicable.

Use the Integral Test to check if the series converges: $\sum_{k=2}^{\infty} \frac{9}{k(\ln k)^{2}}$ . Which conditions apply to $f(x)=\frac{9}{x(\ln x)^{2}}$ ?

Prove that for the curve $(2 y+1)^{3}-24 x=-3$ , the derivative is $\frac{dy}{dx} = \frac{4}{(2y+1)^2}$ .

A penny is flipped upward at $2.85 \mathrm{~m/s}$ . Calculate the maximum height it reaches above your hand.

Find the derivative $f'(x)$ for the function $f(x)=\frac{e^{x}}{7 x^{2}+1}$ .

Calculate the derivative $f'(x)$ for the function $f(x)=\frac{8x^{2}}{6x-5}$ .

Find the derivative $f'(x)$ for the function $f(x)=\frac{e^{x}}{3 x^{2}+8}$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{e^{x}}{8 x^{2}+1}$ .

Verify if the function $P(x)=x^{5}-3 x^{3}+1$ has a zero between -1.8 and -1.7 using the intermediate value theorem.

Show that $P(x)=4 x^{2}-2 x-5$ has a zero between 1 and 2 using the intermediate value theorem. Approximate the zero.

Verify if the function $P(x)=x^{5}-4 x^{3}+1$ has a real zero between -2.04 and -2.02 using the intermediate value theorem.

Verify if $P(x)=x^{5}-3 x^{3}+1$ has a zero between -1.8 and -1.7 using the intermediate value theorem. Approximate the zero to the nearest hundredth: $x=\square$ .

Verify if the function $P(x)=x^{5}-4x^{3}+1$ has a zero between -2.04 and -2.02 using the intermediate value theorem. Approximate the zero to the nearest hundredth: $x=\square$ .

Verify if $P(x) = x^{5} - 3x^{3} + 1$ has a zero between -1.8 and -1.7 using the intermediate value theorem.

Find the differential $d y$ for $y=\ln(1+x^{2})$ at $x=2.6$ and $d x=0.3$ . Round to three decimal places.

Find the speed of a penny just before it hits the ground, falling from $1.26 \mathrm{~m}$ . Use energy concepts.

Find the derivative $f'(x)$ for the function $f(x)=\frac{9 x^{2}}{6 x-7}$ . What is $f'(x)$ ?

Determine the vertical and horizontal asymptotes for the function $f(x)=\frac{1}{x+3}$ .

Determine if the series $\sum_{k=1}^{\infty} \frac{k}{k^{2}+6}$ converges or diverges. Choose A-F and fill in the box.

Evaluate the integral $I=\int_{-\infty}^{-3} \frac{d x}{|x+2|}$ .

Determine if the series $\sum_{k=2}^{\infty} \frac{k^{6}}{k^{2 \pi}}$ converges or diverges using relevant tests.

Determine if the series $\sum_{k=2}^{\infty} \frac{k^{\pi}}{k^{4}}$ converges using the Divergence, Integral, or p-series test.

Find $G(2)$ and $G^{\prime}(2)$ for $G(x)=\int_{2}^{x} \sqrt{t^{3}+1} d t$ .

Determine if the series $\sum_{k=1}^{\infty} \frac{\sqrt{4 k^{2}+11}}{k}$ converges or diverges.

Check if the series converges or diverges: $\sum_{n=0}^{\infty} \frac{4^{n}-2^{n}}{9^{n}}$ .

Determine if the series $\sum_{k=2}^{\infty} \frac{1}{(k-1)^{4}}$ converges or diverges. What is $p$ ?

Find the subinterval length $\Delta x$ for $[0,2]$ with $n=4$ . List grid points $\mathrm{x}_{0}$ to $\mathrm{x}_{4}$ . Identify points for left, right, and midpoint Riemann sums.

Determine if the series $\sum_{k=1}^{\infty} \frac{4^{k}+6^{k}}{11^{k}}$ converges or diverges.

Determine if the series $\sum_{k=1}^{\infty} \frac{4^{k}+7^{k}}{11^{k}}$ converges or diverges.

Find the differential $d y$ for $y=\ln(1+x^{2})$ at $x=2.2$ and $d x=0.6$ . Round to three decimal places.

Solve the differential equation: $5 \frac{d y}{d \theta}=\frac{\mathrm{e}^{y} \sin ^{2}(\theta)}{y \sec (\theta)}$ . Use $C$ as a constant.

Estimate $f(5.2)$ using the linear approximation given $f(5)=5.1$ and $f'(5)=3.2$ . Provide your answer to three decimal places.

Determine if the series converges:
$\sum_{k=0}^{\infty} \frac{8(-1)^{k}}{9 k+7}$
Evaluate $\lim_{k \rightarrow \infty} a_k$ where $a_k = \frac{8}{9k+7}$ .

Determine if the series $\sum_{k=0}^{\infty} \frac{9(-1)^{k}}{8 k+5}$ converges. Define $a_{k}$ . Choose A, B, or C.

Find the indefinite integral: $\int e^{x} \sqrt{1-e^{x}} \, dx$

Determine if the series $\sum_{k=0}^{\infty} \frac{8(-1)^{k}}{9 k+7}$ converges. Define $a_{k}$ and choose its behavior: A, B, or C.

Calculate the indefinite integral: $\int \ln \left(e^{6 x-9}\right) d x$ .

Calculate the integral from 0 to 1 for $x e^{-7 x^{2}}$ .

Calculate the left and right Riemann sums for $f(x)=\frac{2}{x}+2$ on $[1,5]$ with $n=4$ .

Find critical numbers of $f(x)$ , intervals of increase/decrease, and all relative extrema for $f(x)=\frac{x^{2}-3 x-4}{x-2}$ .

Evaluate the integral: $\int_{0}^{\pi / 2} e^{\sin 5 \pi x} \cos 5 \pi x \, dx$ .

Find the positive critical number of the function $f(x)=(x^{2}-9)^{3}$ . Round your answer to three decimal places.

Estimate $f(9.3)$ using linear approximation given $f(9)=1.7$ and $f'(9)=4$ . Round to three decimal places.

Find the integral of $\sin^{2} \theta \cos \theta$ with respect to $\theta$ .

Determine if the series $\sum_{k=0}^{\infty} \frac{9(-1)^{k}}{8 k+5}$ converges. Evaluate $\lim_{k \to \infty} a_{k}$ .

Calculate the left and right Riemann sums for $f(x)=\frac{2}{x}+5$ on $[1,5]$ with $n=4$ . Left sum: $\square$ .

Evaluate the integral from 0 to $\frac{\pi}{2}$ : $\int_{0}^{\pi / 2} e^{\sin 5 \pi x} \cos 5 \pi x \, dx$ . Use a graphing tool to check your answer.

Find $\frac{dy}{dx}$ from the equation $V=-4.7x-1.5\ln(x)+1.8y-1.3\ln(y)$ , then evaluate at $x=3$ , $y=8$ . Round to three decimal places.

Find the derivative of $f(x)=9+2.6 \tan ^{-1}(4.6 \ln (x))$ and evaluate it at $x=6$ . Round to three decimal places.

Find the positive value of $x$ where $f''(x)=0$ for $f(x)=x \exp(-x^{2}/5)$ . Round to three decimal places.

A bacteria culture starts with 740 and grows proportionally. After 3 hours, it has 2220. Find $P(t)$ , $P(5)$ , and time for 2250.

Find the positive value of $x$ where the second derivative $f''(x)=0$ for the function $f(x)=x \exp(-x^{2}/3)$ . Round to three decimal places.

Calculate the integral: $\int \frac{4 x}{\sqrt{16-x^{2}}} d x$

A bacteria culture starts with 1000 and grows to 5000 in 5 hours. Find $P(t)$ , $P(6)$ , and time to reach 1790.

A bacteria culture starts with 660 and grows proportionally. Find $P(t)$ , population after 7 hours, and time to reach 2770.

Find the tangent line equation for $y=x^{\sin (x)}$ at the point $\left(\frac{\pi}{2}, \frac{\pi}{2}\right)$ .

Find the tangent line equation to the function at the point $(7,1)$ for $y=9^{x-7}$ .

A bacteria culture starts with 1000 and grows proportionally. Find $P(t)$ after $t$ hours, $P(8)$ , and time to reach 1010.

Find $\frac{d y}{d x}$ from the equation $V=-3.7 x-2.2 \ln (x)+2 y-1.4 \ln (y)$ and evaluate at $x=4$ , $y=10$ . Round to three decimal places.

Find the derivative of $f(x)=3+3.4 \tan^{-1}(3.6 \ln(x))$ and calculate $f'(8)$ rounded to three decimal places.

Find $d y / d x$ using logarithmic differentiation for $y=(9+x)^{4/x}$ . $\frac{d y}{d x}=\square$

Solve the differential equation $\frac{d P}{d t}=5 \sqrt{P t}$ with the initial condition $P(1)=5$ .

Find the derivative of the integral: $\frac{d}{d x} \int_{0}^{\sin x} \sqrt{\frac{2}{t^{2}-3 t-5}} d t$

Determine if the series $\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k}{14 k+1}$ converges or diverges.

Determine if the series $\sum_{k=1}^{\infty} \frac{8(-1)^{k+1}}{k^{7}}$ converges. Identify $a_{k}$ .