Calculus

Problem 16001

Untersuche die Monotonie, Beschränktheit und Konvergenz der Folge $a_n=\frac{n^{2}+1}{n^{2}+2n+2}$ . Bestimme den Grenzwert.

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Problem 16002

Find the absolute max and min of $f(x)=\sqrt[3]{x}$ on $[-3, 8]$ using the Closed Interval Method. Show your work.

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Problem 16003

Find local maxima and minima of $f(x)=x^{3}+\frac{5}{2} x^{2}-12 x-1$ using the First Derivative Test. Show your work.

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Problem 16004

Find $Q$ given $\frac{d Q}{d t}=1.6 \mathrm{~g/s}$ and $Q(0) = 100 - \left(\frac{1000}{10} +0\right)$ .

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Problem 16005

Graph $g(x)=2 x^{3}+8 x^{2}-3$ . Find $x$ -intercepts, local max/min, and intervals of increase/decrease. Round as needed.

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Problem 16006

Solve for $z$ in the equation $\int_{z}^{10} 2 x \, dx = 19$ .

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Problem 16007

Find the function $Q(t)$ given $\frac{d Q}{d t}=1.6 \mathrm{~g/s}$ and $Q(0)=100 - \frac{1000}{10}$ .

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Problem 16008

Find $\lim _{t \rightarrow+\infty} Q(t)$ for $Q(t)= 100 - \frac{1000}{10+t}$ and explain the meaning of the result.

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Problem 16009

Find the critical numbers of the function $h(t)=t^{3/4}-6t^{1/4}$ . Enter answers as a comma-separated list or DNE.

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Problem 16010

Find the critical numbers of the function $f(x)=x^{5} e^{-6 x}$ . Enter answers as a comma-separated list or DNE.

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Problem 16011

Find the max and min of $y=2 x e^{-4 x}$ on $[0,3]$ by checking critical points and endpoints. If none, enter DNE.

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Problem 16012

A 17 ft ladder leans against a wall. If the top slips down at $4 \mathrm{ft} / \mathrm{s}$ , how fast is the foot moving when the top is 13 ft high? The foot moves at $\square \mathrm{ft} / \mathrm{s}$ .

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Problem 16013

Problème :
1. L'azote $(\mathrm{N})$ et l'hydrogène $(\mathrm{H})$ forment l'ammoniac avec $Q(t)=100-\frac{1000}{10+t}$ . a) Trouvez $Q(0)$ et $Q(20)$ . b) Calculez la variation de $Q$ sur $[10 s, 20 s]$ . c) Taux de variation moyen sur (i) $[10 s, 20 s]$ (ii) $[20 s, 30 s]$ .

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Problem 16014

Find critical points $c$ for local minima of the function $f(x)=(x^{2}-4)e^{-x}$ for $x>0$ .

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Problem 16015

Given the function $f(x)=x^{4}-32 x^{2}+2$ , find where $f$ is increasing, decreasing, local max/min, inflection points, and concavity.

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Problem 16016

Determine the inflection points of the function $f(x)=2 x^{4}-8 x+3$ .

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Problem 16017

Determine the concavity intervals for the function $f(x) = x^3 - 12x$ .

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Problem 16018

Calculate the area between the function $f(x)$ and the $x$ -axis on the interval $[-6,2]$ .
$f(x)=\left\{\begin{array}{ll} x^{2}+3 x+4 & x<1 \\ 4 x+4 & x \geq 1 \end{array}\right.$
The area is $\square$ .

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Problem 16019

Find the inflection points of the function $h(x)=(2x^{2}-5)^{2}$ .

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Problem 16020

Find $\frac{d y}{d x}$ for the curve $e^{y}=x$ and express your answer in terms of $x$ .

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Problem 16021

2. L'azote $(\mathrm{N})$ et l'hydrogène $(\mathrm{H})$ forment l'ammoniac avec $Q(t)=100-\frac{1000}{10+t}$ .
a) Trouver $Q(0)$ et $Q(20)$ . b) Calculer la variation de $Q$ sur $[10 \mathrm{~s}, 20 \mathrm{~s}]$ . c) Taux de variation moyen sur : (i) $[10 s, 20 s]$ (ii) $[20 s, 30 s]$ d) Évaluer $\lim _{h \rightarrow 0^{+}} \frac{Q(h+0)-Q(0)}{h}$ . e) Trouver la fonction de variation de $Q$ . f) Évaluer : (i) $\left.\frac{d Q}{d t}\right|_{t=10 s}$ (ii) $\left.\frac{d Q}{d t}\right|_{t=1 \min }$ g) Déterminer l'évolution de $Q$ et son taux de variation instantané. h) Trouver $\frac{d Q}{d t}$ lorsque $Q=70 \mathrm{~g}$ .

See Solution

Problem 16022

Find the derivative $y'(x)$ of $y(x)$ from the equation $\cos(x^2 y) - \sin y = x$ .

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Problem 16023

Find the power series for $f(x)=\ln(x^{2}+4)$ .

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Problem 16024

Evaluate the integral from $e$ to $e^{25}$ of $\frac{1}{x \sqrt{\ln(x)}} \, dx$ .

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Problem 16025

Evaluate the integral: $\int \frac{\ln (1-t)}{t} d t$

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Problem 16026

Find the absolute extrema of $g(x) = 3x^4 - 24x^2$ on the interval $[-1, 1]$ .

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Problem 16027

Calculate the integral $\int x^{3} \sqrt{x^{2}+12} \, dx$ .

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Problem 16028

Evaluate the integral: $\int x^{3} \sqrt{x^{2}+12} \, dx$

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Problem 16029

2. L'azote $(\mathrm{N})$ et l'hydrogène $(\mathrm{H})$ forment l'ammoniac avec $Q(t)=100-\frac{1000}{10+t}$ .
a) Trouvez $Q(0)$ et $Q(20)$ . b) Calculez la variation de $Q$ sur $[10 \mathrm{~s}, 20 \mathrm{~s}]$ . c) Trouvez le taux de variation moyen sur $[10 s, 20 s]$ et $[20 s, 30 s]$ . d) Évaluez $\lim _{h \rightarrow 0^{+}} \frac{Q(h+0)-Q(0)}{h}$ et interprétez. e) Déterminez la fonction du rythme de variation de $Q$ . f) Évaluez $\left.\frac{d Q}{d t}\right|_{t=10 s}$ et $\left.\frac{d Q}{d t}\right|_{t=1 \mathrm{~min}}$ . g) Analysez la tendance de $Q$ et de son taux de variation lorsque $t$ augmente. h) Trouvez $\frac{d Q}{d t}$ pour $Q=70 \mathrm{~g}$ . i) Déterminez $Q$ pour $\frac{d Q}{d t}=1,6 \mathrm{~g} / \mathrm{s}$ . j) Évaluez $\lim _{t \rightarrow+\infty} Q(t)$ et interprétez. k) Tracez les graphiques de $Q$ et $\frac{d Q}{d t}$ .

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Problem 16030

Calculate the area between the function $f(x)$ and the $x$ -axis over the interval $[-5, 4]$ .

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Problem 16031

Find the derivative $d y / d x$ for the function $y=\sec^{-1}(x) + \csc^{-1}(x)$ .

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Problem 16032

Find the second derivative of $f(x)=\frac{x^{2}}{9+x}$ and evaluate $f^{\prime \prime}(0)$ and $f^{\prime \prime}(5)$ .

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Problem 16033

Find the derivative $\frac{d y}{d x}$ for the function $y=\tan^{-1}(\sqrt{6 x^{2}-1})$ .

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Problem 16034

Find the second derivative $f^{\prime \prime}(x)$ of $f(x)=2 x^{3}-3 x^{2}+3 x+2$ . Then calculate $f^{\prime \prime}(0)$ and $f^{\prime \prime}(7)$ .

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Problem 16035

Evaluate the integral: $\int 5 x \cos (8 x) \, dx$ . Use the constant $C$ for the integration constant.

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Problem 16036

Find $f$ where $f^{\prime}(x)=\frac{2}{\sqrt{x}}$ and $f(16)=25$ .

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Problem 16037

Find the antiderivative of $C^{\prime}(x)=2 x^{2}-3 x$ with $C(0)=4,000$ .

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Problem 16038

Find the limit as $x$ approaches infinity:
$\lim _{x \rightarrow \infty} \frac{(3-x)^{5}\left(4 x^{2}+1\right)^{2}}{\left(8+x^{2}-x\right)^{2}(7+x)(3-2 x)^{2}}$

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Problem 16039

Find the point of diminishing returns $(x, y)$ for the revenue function $R(x)=11,000-x^{3}+42 x^{2}+800 x$ , where $0 \leq x \leq 20$ .

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Problem 16040

Find the curve equation through $(2,3)$ with slope $\frac{dy}{dx}=3x-7$ .

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Problem 16041

Sketch the graph of a function $f$ with these properties: $f(0)=0$ , $f'(-3)=f'(4)=0$ , $f'(<0)$ on $(-3,0)$ and $(0,4)$ , $f'(<0)$ on $(-\infty,-3)$ and $(4,\infty)$ , $f''(<0)$ on $(-\infty,0)$ , $f''(>0)$ on $(0,\infty)$ .

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Problem 16042

Find the limit: $\lim _{x \rightarrow-\infty}\left(\sqrt{9 x^{2}+8 x}+3 x\right)$ .

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Problem 16043

Find the point of diminishing returns $(x, y)$ for $R(x)=-0.6 x^{3}+2.7 x^{2}+7 x$ . Round the answer to two decimal places.

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Problem 16044

Find the initial velocity and velocity after 6 seconds for $v(t)=58(1-e^{-0.18 t})$ . Round to the nearest whole number.

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Problem 16045

Find the point of diminishing returns $(x, y)$ for the revenue function $R(x) = 11,000 - x^3 + 39x^2 + 800x$ , $0 \leq x \leq 20$ .

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Problem 16046

Find the net area and the area above the $x$ -axis for the region bounded by $y=4-x^{2}$ . Set up the integral(s) needed.

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Problem 16047

Find an antiderivative of $f(x)=\sqrt{x}+\frac{1}{x}$ . Choose from the given options.

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Problem 16048

Let $f(x)$ be a continuous function. Identify all true statements about its antiderivative $F(x)$ .

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Problem 16049

Find critical points of $f(x)=\sin^{-1}(x)-7x$ for $-1<x<1$ . List them as a comma-separated list or DNE if none exist.

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Problem 16050

Find the area function $A(x)=\int_{0}^{x} 3 \cos t \, dt$ , graph $f(x)$ and $A(x)$ , then evaluate $A(\frac{\pi}{2})$ and $A(\pi)$ .

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Problem 16051

Find the min and max of $y=\sqrt{x+x^{2}}-10\sqrt{x}$ on $[0,4]$ . minimum: maximum:

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Problem 16052

A particle starts at $s(0)=3$ , $v(0)=5$ , with $a(t)=10 \sin t+3 \cos t$ . Order steps to find $s(7)$ .

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Problem 16053

Bestimme die Ableitung von $f(x) = \frac{5}{x^{3}}$ .

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Problem 16054

Find critical points of $f(x)=\frac{x}{x^{2}+36}$ . List them as a comma-separated list or enter DNE if none exist.

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Problem 16055

Find the area function $A(x)=\int_{0}^{x} 3 \sin t \, dt$ , graph $f(x)$ and $A(x)$ , then evaluate $A(\frac{\pi}{2})$ and $A(\pi)$ .

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Problem 16056

Evaluate the improper integral $\int_{0}^{8} \frac{1}{2\left(x^{1 / 3}\right)} d x$ and determine if it converges or diverges.

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Problem 16057

Find the equilibria of the fish population model $N_{t+1}=\frac{2 N_{t}}{1+\frac{N_{t}}{100}}$ and their stability. Choose A, B, C, or D.

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Problem 16058

Which Type 1 Improper Integrals CONVERGE? Choose all that apply: $\int_{2}^{\infty} \frac{1}{x} dx$ , $\int_{5}^{\infty} \frac{1}{x^{9}} dx$ , $\int_{4}^{\infty} \frac{1}{x^{1/3}} dx$ , $\int_{1}^{\infty} x^{3} dx$ , $\int^{\infty} x \sin(x^{2}) dx$ .

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Problem 16059

Evaluate the integral: $\int 2 x e^{-9 x} d x$

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Problem 16060

Find the value of $\int_{-1}^{5} g(x) d x$ given that the area between $f(x)$ and $g(x)$ is 6 and $\int_{-1}^{5} f(x) d x=2$ .

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Problem 16061

Given the function $f(x)=(x+4)(x-1)^{2}$ , find critical values, intervals of increase/decrease, local max/min, concavity, and inflection points.

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Problem 16062

For what value(s) of constant $a$ is the integral $\int_{a}^{3} \frac{x}{x^{2}-3 x+2} d x$ improper?

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Problem 16063

Find the value(s) of $p$ for which the integral $\int_{1}^{5} \frac{1}{x-p} d x$ is improper.

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Problem 16064

Find the derivative of $f(N_{t})=\overline{1+\frac{N_{t}}{100}}$ .

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Problem 16065

Which statement is TRUE by the Comparison Test?
1. $\int_{e^{10}}^{\infty} \frac{2^{x}}{x} dx$ diverges.
2. $\int_{e^{10}}^{\infty} \frac{\ln x}{x^{2}} dx$ diverges.
3. $\int_{1}^{\infty} \frac{\cos x}{x} dx$ converges.

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Problem 16066

Find the derivative of the function $g(s)=\int_{9}^{s}(t-t^{9})^{5} dt$ . What is $g'(s)$ ?

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Problem 16067

Find the partial derivative $f_{y}$ of the function $f(x, y)=\int_{y}^{x} \cos(t^{6}) dt$ .

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Problem 16068

Find the derivative of the function $h(x)=\int_{1}^{e^{x}} 9 \ln (t) dt$ . What is $h^{\prime}(x)$ ?

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Problem 16069

Find the 100th derivative of $f(x)=x e^{-x}$ . What is $f^{(100)}(x)$ ?

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Problem 16070

Find the 100th derivative of $f(x)=x e^{-x}$ . What is $f^{(100)}(x)=$ ?

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Problem 16071

Find the 100th derivative of the function $f(x)=x e^{-x}$ . What is $f^{(100)}(x)$ ?

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Problem 16072

Find the derivative of $y=\int_{\cos (x)}^{\sin (x)} \ln (3+9 v) \, dv$ , denoted as $g'(x)=\square$ .

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Problem 16073

Find the derivative of the function $y=\int_{4}^{3x+7} \frac{t}{1+t^{3}} dt$ .

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Problem 16074

Given $a(t)=t+6$ , $v(0)=4$ , $0 \leq t \leq 11$ , find $v(t)$ and the distance traveled in this interval.

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Problem 16075

Evaluate the integral: $\int 7 x^{2} \sin \pi x \, dx$

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Problem 16076

Show that the vector field $\mathbf{F}=\frac{2 x}{y^{2}+1} \mathbf{i}-\frac{2 y\left(x^{2}+1\right)}{\left(y^{2}+1\right)^{2}} \mathbf{j}$ is conservative. Calculate $\int_{C} \mathbf{F} \cdot d \mathbf{r}$ for the curve $C: x=t^{3}-5, y=t^{6}-t, 0 \leq t \leq 1$ .

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Problem 16077

Evaluate the line integral using Green's Theorem: $\oint_{C} e^{6 x+5 y} dx + e^{-3 y} dy$ , where $C$ is the triangle with vertices $(0,0),(1,0),(1,1)$ .

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Problem 16078

Find the distance traveled by a particle with acceleration $a(t)=t+6$ and initial velocity $v(0)=4$ from $t=0$ to $t=11$ .

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Problem 16079

Evaluate the integral from $e$ to $e^{36}$ of $\frac{1}{x \sqrt{\ln(x)}} \, dx$ .

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Problem 16080

Evaluate the integral: $\int(x+2) \sqrt{4 x+x^{2}} \, dx$ (use $C$ for the constant).

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Problem 16081

Evaluate the line integral $\int_{C} \sqrt{5+x^{3}} d x+6 x y d y$ using Green's theorem for triangle $C$ with vertices $(0,0),(1,0),(1,4)$ .

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Problem 16082

Evaluate the integral using Cauchy's residue theorem: $\oint_{C} \frac{z+1}{z^{2}(z-6 i)} d z$ for contours (a) $|z|=1$ , (b) $|z-6 i|=1$ , and (c) $|z-6 i|=36$ .

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Problem 16083

Evaluate the integrals using Cauchy's residue theorem for the given contours:
17. $\oint_{C} \frac{1}{(z-1)(z+2)^{2}} d z$ for (a) $|z|=\frac{1}{2}$ , (b) $|z|=\frac{3}{2}$ , (c) $|z|=3$ .
18. $\oint_{C} \frac{z+1}{z^{2}(z-2 i)} d z$ for (a) $|z|=1$ , (b) $|z-2 i|=1$ , (c) $|z-2 i|=4$ .
19. $\oint_{C} z^{3} e^{-1 / z^{2}} d z$ for (a) $|z|=5$ , (b) $|z+i|=2$ , (c) $|z-3|=1$ .
20. $\oint_{C} \frac{1}{z \sin z} d z$ for (a) $|z-2 i|=1$ , (b) $|z-2 i|=3$ , (c) $|z|=5$ .

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Problem 16084

Evaluate the line integral $\int_{C} \sqrt{5+x^{3}} d x+6 x y d y$ over triangle $C$ with vertices $(0,0),(1,0)$ , and $(1,4)$ .

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Problem 16085

Calculate the curl of the vector field $\mathbf{F}(x, y, z)=x^{5} \mathbf{i}+y^{5} \mathbf{j}+z^{5} \mathbf{k}$ .

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Problem 16086

Estimate $\int_{0}^{12} \frac{1}{x+3} dx$ using a left-hand sum with $n=3$ . Round to three decimal places.

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Problem 16087

Analyze the curve $f(x)=\frac{x-4}{x^{2}}$ : find its domain, intercepts, symmetry, derivatives, asymptotes, and intervals of increase/decrease/concavity.

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Problem 16088

Estimate $\int_{0}^{40} f(x) d x$ using left- and right-hand sums for the increasing function $f(x)$ .

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Problem 16089

Approximate the volume change of a sphere as its radius goes from $r=70 \mathrm{ft}$ to $r=70.1 \mathrm{ft}$ using $V(r)=\frac{4}{3} \pi r^{3}$ . Find $\Delta V \approx \square \mathrm{ft}^{3}$ .

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Problem 16090

Rewrite $y$ for differentiation and find $y^{\prime}$ for: (a) $y=\frac{\ln \left(\frac{\sqrt[4]{x^{3}}}{2}\right)}{e}$ ; (b) $y=7^{x} \ln \left(e x^{2}\right)$ . Approximate $\ln(0.9)$ and $\ln(1.1)$ using tangent lines.

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Problem 16091

Find the value of $\int_{1}^{6} f(x) \, dx$ for the function $f(x)$ defined by the points (1, 1), (2, 2), (3, 1), (4, 2), (5, 2), (6, 2).

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Problem 16092

Estimate $\int_{0}^{0.5} e^{-x^{2}} d x$ using $n=5$ rectangles for (a) Left-hand sum and (b) Right-hand sum. Round to three decimal places.

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Problem 16093

Find the value of $\int_{1}^{6} f(x) \, dx$ for the function $f(x)$ with points: $(1,1)$ , $(2,2)$ , $(3,1)$ , $(4,2)$ , $(5,2)$ , $(6,2)$ .

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Problem 16094

Rewrite $y$ for differentiation and find $y^{\prime}$ without simplifying. Show your work. (a) $y=\frac{\ln \left(\frac{\sqrt[4]{x^{3}}}{2}\right)}{e}$ (b) $y=7^{x} \ln \left(e x^{2}\right)$ Approximate $\ln (0.9)$ and $\ln (1.1)$ using tangent line and sketch your work.

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Problem 16095

Find the position of a particle at time $t=2$ given $a(t)=\langle 2,3\rangle$ , $v(0)=\langle 3,1\rangle$ , $p(0)=\langle 1,5\rangle$ .

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Problem 16096

Evaluate the integral: $\int 5 \ln \sqrt[3]{x} \, dx$

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Problem 16097

Find the initial population $P(0)$ and the population $P(8)$ after 8 years for $P(t)=\frac{400}{1+2 e^{-0.2 t}}$ .

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Problem 16098

Find the $y$ -coordinate of a particle's position at time $t=2$ given $\frac{d x}{d t}=\cos t^{2}$ , $\frac{d y}{d t}=e^{t-2}$ , starting at $(1,2)$ at $t=3$ .

See Solution

Problem 16099

Find the derivative of $f(x) = 8x^2 + 6x \ln(x) + 1$ .

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Problem 16100

Find the domain of $f(x)=\ln (x)-x$ , classify critical points, and determine where $f$ is concave up or down.

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Calculus

Problem 16001

Untersuche die Monotonie, Beschränktheit und Konvergenz der Folge an=n2+1n2+2n+2a_n=\frac{n^{2}+1}{n^{2}+2n+2}an​=n2+2n+2n2+1​. Bestimme den Grenzwert.

Problem 16002

Find the absolute max and min of f(x)=x3f(x)=\sqrt[3]{x}f(x)=3x​ on [−3,8][-3, 8][−3,8] using the Closed Interval Method. Show your work.

Problem 16003

Find local maxima and minima of f(x)=x3+52x2−12x−1f(x)=x^{3}+\frac{5}{2} x^{2}-12 x-1f(x)=x3+25​x2−12x−1 using the First Derivative Test. Show your work.

Problem 16004

Find QQQ given dQdt=1.6 g/s\frac{d Q}{d t}=1.6 \mathrm{~g/s}dtdQ​=1.6 g/s and Q(0)=100−(100010+0)Q(0) = 100 - \left(\frac{1000}{10} +0\right)Q(0)=100−(101000​+0).

Problem 16005

Graph g(x)=2x3+8x2−3g(x)=2 x^{3}+8 x^{2}-3g(x)=2x3+8x2−3. Find xxx-intercepts, local max/min, and intervals of increase/decrease. Round as needed.

Problem 16006

Solve for zzz in the equation ∫z102x dx=19\int_{z}^{10} 2 x \, dx = 19∫z10​2xdx=19.

Problem 16007

Find the function Q(t)Q(t)Q(t) given dQdt=1.6 g/s\frac{d Q}{d t}=1.6 \mathrm{~g/s}dtdQ​=1.6 g/s and Q(0)=100−100010Q(0)=100 - \frac{1000}{10}Q(0)=100−101000​.

Problem 16008

Find lim⁡t→+∞Q(t)\lim _{t \rightarrow+\infty} Q(t)limt→+∞​Q(t) for Q(t)=100−100010+tQ(t)= 100 - \frac{1000}{10+t}Q(t)=100−10+t1000​ and explain the meaning of the result.

Problem 16009

Find the critical numbers of the function h(t)=t3/4−6t1/4h(t)=t^{3/4}-6t^{1/4}h(t)=t3/4−6t1/4. Enter answers as a comma-separated list or DNE.

Problem 16010

Find the critical numbers of the function f(x)=x5e−6xf(x)=x^{5} e^{-6 x}f(x)=x5e−6x. Enter answers as a comma-separated list or DNE.

Problem 16011

Find the max and min of y=2xe−4xy=2 x e^{-4 x}y=2xe−4x on [0,3][0,3][0,3] by checking critical points and endpoints. If none, enter DNE.

Problem 16012

A 17 ft ladder leans against a wall. If the top slips down at 4ft/s4 \mathrm{ft} / \mathrm{s}4ft/s, how fast is the foot moving when the top is 13 ft high? The foot moves at □ft/s\square \mathrm{ft} / \mathrm{s}□ft/s.

Problem 16013

Problem 16014

Find critical points ccc for local minima of the function f(x)=(x2−4)e−xf(x)=(x^{2}-4)e^{-x}f(x)=(x2−4)e−x for x>0x>0x>0.

Problem 16015

Given the function f(x)=x4−32x2+2f(x)=x^{4}-32 x^{2}+2f(x)=x4−32x2+2, find where fff is increasing, decreasing, local max/min, inflection points, and concavity.

Problem 16016

Determine the inflection points of the function f(x)=2x4−8x+3f(x)=2 x^{4}-8 x+3f(x)=2x4−8x+3.

Problem 16017

Determine the concavity intervals for the function f(x)=x3−12xf(x) = x^3 - 12xf(x)=x3−12x.

Problem 16018

Calculate the area between the function f(x)f(x)f(x) and the xxx-axis on the interval [−6,2][-6,2][−6,2]. f(x)={x2+3x+4x<14x+4x≥1 f(x)=\left\{\begin{array}{ll} x^{2}+3 x+4 & x<1 \\ 4 x+4 & x \geq 1 \end{array}\right. f(x)={x2+3x+44x+4​x<1x≥1​The area is □\square□.

Problem 16019

Find the inflection points of the function h(x)=(2x2−5)2h(x)=(2x^{2}-5)^{2}h(x)=(2x2−5)2.

Problem 16020

Find dydx\frac{d y}{d x}dxdy​ for the curve ey=xe^{y}=xey=x and express your answer in terms of xxx.

Problem 16021

Problem 16022

Find the derivative y′(x)y'(x)y′(x) of y(x)y(x)y(x) from the equation cos⁡(x2y)−sin⁡y=x\cos(x^2 y) - \sin y = xcos(x2y)−siny=x.

Problem 16023

Find the power series for f(x)=ln⁡(x2+4)f(x)=\ln(x^{2}+4)f(x)=ln(x2+4).

Problem 16024

Evaluate the integral from eee to e25e^{25}e25 of 1xln⁡(x) dx\frac{1}{x \sqrt{\ln(x)}} \, dxxln(x)​1​dx.

Problem 16025

Evaluate the integral: ∫ln⁡(1−t)tdt\int \frac{\ln (1-t)}{t} d t∫tln(1−t)​dt

Problem 16026

Find the absolute extrema of g(x)=3x4−24x2g(x) = 3x^4 - 24x^2g(x)=3x4−24x2 on the interval [−1,1][-1, 1][−1,1].

Problem 16027

Calculate the integral ∫x3x2+12 dx\int x^{3} \sqrt{x^{2}+12} \, dx∫x3x2+12​dx.

Problem 16028

Evaluate the integral: ∫x3x2+12 dx\int x^{3} \sqrt{x^{2}+12} \, dx∫x3x2+12​dx

Problem 16029

Problem 16030

Calculate the area between the function f(x)f(x)f(x) and the xxx-axis over the interval [−5,4][-5, 4][−5,4].

Problem 16031

Find the derivative dy/dxd y / d xdy/dx for the function y=sec⁡−1(x)+csc⁡−1(x)y=\sec^{-1}(x) + \csc^{-1}(x)y=sec−1(x)+csc−1(x).

Problem 16032

Find the second derivative of f(x)=x29+xf(x)=\frac{x^{2}}{9+x}f(x)=9+xx2​ and evaluate f′′(0)f^{\prime \prime}(0)f′′(0) and f′′(5)f^{\prime \prime}(5)f′′(5).

Problem 16033

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=tan⁡−1(6x2−1)y=\tan^{-1}(\sqrt{6 x^{2}-1})y=tan−1(6x2−1​).

Problem 16034

Find the second derivative f′′(x)f^{\prime \prime}(x)f′′(x) of f(x)=2x3−3x2+3x+2f(x)=2 x^{3}-3 x^{2}+3 x+2f(x)=2x3−3x2+3x+2. Then calculate f′′(0)f^{\prime \prime}(0)f′′(0) and f′′(7)f^{\prime \prime}(7)f′′(7).

Problem 16035

Evaluate the integral: ∫5xcos⁡(8x) dx\int 5 x \cos (8 x) \, dx∫5xcos(8x)dx. Use the constant C C C for the integration constant.

Problem 16036

Find fff where f′(x)=2xf^{\prime}(x)=\frac{2}{\sqrt{x}}f′(x)=x​2​ and f(16)=25f(16)=25f(16)=25.

Problem 16037

Find the antiderivative of C′(x)=2x2−3xC^{\prime}(x)=2 x^{2}-3 xC′(x)=2x2−3x with C(0)=4,000C(0)=4,000C(0)=4,000.

Problem 16038

Find the limit as x x x approaches infinity: lim⁡x→∞(3−x)5(4x2+1)2(8+x2−x)2(7+x)(3−2x)2 \lim _{x \rightarrow \infty} \frac{(3-x)^{5}\left(4 x^{2}+1\right)^{2}}{\left(8+x^{2}-x\right)^{2}(7+x)(3-2 x)^{2}} x→∞lim​(8+x2−x)2(7+x)(3−2x)2(3−x)5(4x2+1)2​

Problem 16039

Find the point of diminishing returns (x,y)(x, y)(x,y) for the revenue function R(x)=11,000−x3+42x2+800xR(x)=11,000-x^{3}+42 x^{2}+800 xR(x)=11,000−x3+42x2+800x, where 0≤x≤200 \leq x \leq 200≤x≤20.

Problem 16040

Find the curve equation through (2,3)(2,3)(2,3) with slope dydx=3x−7\frac{dy}{dx}=3x-7dxdy​=3x−7.

Problem 16041

Problem 16042

Untersuche die Monotonie, Beschränktheit und Konvergenz der Folge $a_n=\frac{n^{2}+1}{n^{2}+2n+2}$ . Bestimme den Grenzwert.

Find the absolute max and min of $f(x)=\sqrt[3]{x}$ on $[-3, 8]$ using the Closed Interval Method. Show your work.

Find local maxima and minima of $f(x)=x^{3}+\frac{5}{2} x^{2}-12 x-1$ using the First Derivative Test. Show your work.

Find $Q$ given $\frac{d Q}{d t}=1.6 \mathrm{~g/s}$ and $Q(0) = 100 - \left(\frac{1000}{10} +0\right)$ .

Graph $g(x)=2 x^{3}+8 x^{2}-3$ . Find $x$ -intercepts, local max/min, and intervals of increase/decrease. Round as needed.

Solve for $z$ in the equation $\int_{z}^{10} 2 x \, dx = 19$ .

Find the function $Q(t)$ given $\frac{d Q}{d t}=1.6 \mathrm{~g/s}$ and $Q(0)=100 - \frac{1000}{10}$ .

Find $\lim _{t \rightarrow+\infty} Q(t)$ for $Q(t)= 100 - \frac{1000}{10+t}$ and explain the meaning of the result.

Find the critical numbers of the function $h(t)=t^{3/4}-6t^{1/4}$ . Enter answers as a comma-separated list or DNE.

Find the critical numbers of the function $f(x)=x^{5} e^{-6 x}$ . Enter answers as a comma-separated list or DNE.

Find the max and min of $y=2 x e^{-4 x}$ on $[0,3]$ by checking critical points and endpoints. If none, enter DNE.

A 17 ft ladder leans against a wall. If the top slips down at $4 \mathrm{ft} / \mathrm{s}$ , how fast is the foot moving when the top is 13 ft high? The foot moves at $\square \mathrm{ft} / \mathrm{s}$ .

Find critical points $c$ for local minima of the function $f(x)=(x^{2}-4)e^{-x}$ for $x>0$ .

Given the function $f(x)=x^{4}-32 x^{2}+2$ , find where $f$ is increasing, decreasing, local max/min, inflection points, and concavity.

Determine the inflection points of the function $f(x)=2 x^{4}-8 x+3$ .

Determine the concavity intervals for the function $f(x) = x^3 - 12x$ .

Calculate the area between the function $f(x)$ and the $x$ -axis on the interval $[-6,2]$ .
$f(x)=\left\{\begin{array}{ll} x^{2}+3 x+4 & x<1 \\ 4 x+4 & x \geq 1 \end{array}\right.$
The area is $\square$ .

Find the inflection points of the function $h(x)=(2x^{2}-5)^{2}$ .

Find $\frac{d y}{d x}$ for the curve $e^{y}=x$ and express your answer in terms of $x$ .

Find the derivative $y'(x)$ of $y(x)$ from the equation $\cos(x^2 y) - \sin y = x$ .

Find the power series for $f(x)=\ln(x^{2}+4)$ .

Evaluate the integral from $e$ to $e^{25}$ of $\frac{1}{x \sqrt{\ln(x)}} \, dx$ .

Evaluate the integral: $\int \frac{\ln (1-t)}{t} d t$

Find the absolute extrema of $g(x) = 3x^4 - 24x^2$ on the interval $[-1, 1]$ .

Calculate the integral $\int x^{3} \sqrt{x^{2}+12} \, dx$ .

Evaluate the integral: $\int x^{3} \sqrt{x^{2}+12} \, dx$

Calculate the area between the function $f(x)$ and the $x$ -axis over the interval $[-5, 4]$ .

Find the derivative $d y / d x$ for the function $y=\sec^{-1}(x) + \csc^{-1}(x)$ .

Find the second derivative of $f(x)=\frac{x^{2}}{9+x}$ and evaluate $f^{\prime \prime}(0)$ and $f^{\prime \prime}(5)$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\tan^{-1}(\sqrt{6 x^{2}-1})$ .

Find the second derivative $f^{\prime \prime}(x)$ of $f(x)=2 x^{3}-3 x^{2}+3 x+2$ . Then calculate $f^{\prime \prime}(0)$ and $f^{\prime \prime}(7)$ .

Evaluate the integral: $\int 5 x \cos (8 x) \, dx$ . Use the constant $C$ for the integration constant.

Find $f$ where $f^{\prime}(x)=\frac{2}{\sqrt{x}}$ and $f(16)=25$ .

Find the antiderivative of $C^{\prime}(x)=2 x^{2}-3 x$ with $C(0)=4,000$ .

Find the limit as $x$ approaches infinity:
$\lim _{x \rightarrow \infty} \frac{(3-x)^{5}\left(4 x^{2}+1\right)^{2}}{\left(8+x^{2}-x\right)^{2}(7+x)(3-2 x)^{2}}$

Find the point of diminishing returns $(x, y)$ for the revenue function $R(x)=11,000-x^{3}+42 x^{2}+800 x$ , where $0 \leq x \leq 20$ .

Find the curve equation through $(2,3)$ with slope $\frac{dy}{dx}=3x-7$ .

Find the limit: $\lim _{x \rightarrow-\infty}\left(\sqrt{9 x^{2}+8 x}+3 x\right)$ .

Find the point of diminishing returns $(x, y)$ for $R(x)=-0.6 x^{3}+2.7 x^{2}+7 x$ . Round the answer to two decimal places.

Find the initial velocity and velocity after 6 seconds for $v(t)=58(1-e^{-0.18 t})$ . Round to the nearest whole number.

Find the point of diminishing returns $(x, y)$ for the revenue function $R(x) = 11,000 - x^3 + 39x^2 + 800x$ , $0 \leq x \leq 20$ .

Find the net area and the area above the $x$ -axis for the region bounded by $y=4-x^{2}$ . Set up the integral(s) needed.

Find an antiderivative of $f(x)=\sqrt{x}+\frac{1}{x}$ . Choose from the given options.

Let $f(x)$ be a continuous function. Identify all true statements about its antiderivative $F(x)$ .

Find critical points of $f(x)=\sin^{-1}(x)-7x$ for $-1<x<1$ . List them as a comma-separated list or DNE if none exist.

Find the area function $A(x)=\int_{0}^{x} 3 \cos t \, dt$ , graph $f(x)$ and $A(x)$ , then evaluate $A(\frac{\pi}{2})$ and $A(\pi)$ .

Find the min and max of $y=\sqrt{x+x^{2}}-10\sqrt{x}$ on $[0,4]$ . minimum: maximum:

A particle starts at $s(0)=3$ , $v(0)=5$ , with $a(t)=10 \sin t+3 \cos t$ . Order steps to find $s(7)$ .

Bestimme die Ableitung von $f(x) = \frac{5}{x^{3}}$ .

Find critical points of $f(x)=\frac{x}{x^{2}+36}$ . List them as a comma-separated list or enter DNE if none exist.

Find the area function $A(x)=\int_{0}^{x} 3 \sin t \, dt$ , graph $f(x)$ and $A(x)$ , then evaluate $A(\frac{\pi}{2})$ and $A(\pi)$ .

Evaluate the improper integral $\int_{0}^{8} \frac{1}{2\left(x^{1 / 3}\right)} d x$ and determine if it converges or diverges.

Find the equilibria of the fish population model $N_{t+1}=\frac{2 N_{t}}{1+\frac{N_{t}}{100}}$ and their stability. Choose A, B, C, or D.

Evaluate the integral: $\int 2 x e^{-9 x} d x$

Find the value of $\int_{-1}^{5} g(x) d x$ given that the area between $f(x)$ and $g(x)$ is 6 and $\int_{-1}^{5} f(x) d x=2$ .

Given the function $f(x)=(x+4)(x-1)^{2}$ , find critical values, intervals of increase/decrease, local max/min, concavity, and inflection points.

For what value(s) of constant $a$ is the integral $\int_{a}^{3} \frac{x}{x^{2}-3 x+2} d x$ improper?

Find the value(s) of $p$ for which the integral $\int_{1}^{5} \frac{1}{x-p} d x$ is improper.

Find the derivative of $f(N_{t})=\overline{1+\frac{N_{t}}{100}}$ .

Find the derivative of the function $g(s)=\int_{9}^{s}(t-t^{9})^{5} dt$ . What is $g'(s)$ ?

Find the partial derivative $f_{y}$ of the function $f(x, y)=\int_{y}^{x} \cos(t^{6}) dt$ .

Find the derivative of the function $h(x)=\int_{1}^{e^{x}} 9 \ln (t) dt$ . What is $h^{\prime}(x)$ ?

Find the 100th derivative of $f(x)=x e^{-x}$ . What is $f^{(100)}(x)$ ?

Find the 100th derivative of $f(x)=x e^{-x}$ . What is $f^{(100)}(x)=$ ?

Find the 100th derivative of the function $f(x)=x e^{-x}$ . What is $f^{(100)}(x)$ ?

Find the derivative of $y=\int_{\cos (x)}^{\sin (x)} \ln (3+9 v) \, dv$ , denoted as $g'(x)=\square$ .

Find the derivative of the function $y=\int_{4}^{3x+7} \frac{t}{1+t^{3}} dt$ .

Given $a(t)=t+6$ , $v(0)=4$ , $0 \leq t \leq 11$ , find $v(t)$ and the distance traveled in this interval.