Calculus

Problem 21101

Evaluate $\int 16 \tan 2 x \sec ^{2} 2 x d x$ using substitutions $u=\tan 2 x$ and $u=\sec 2 x$ . Compare results.

See Solution

Problem 21102

Find the radius of convergence, $R$ , and interval, $I$ , for the series $\sum_{n=1}^{\infty} n !(6 x-1)^{n}$ .

See Solution

Problem 21103

Find the displacement of a ball with velocity $v(t)=-32 t+73$ from $t=0$ to $t=2$ seconds. Displacement is $\square$ feet.

See Solution

Problem 21104

Determine if the sequence $a_{n}=1+\frac{9^{n}}{8^{n}}$ has a limit. If yes, find it; if not, state it's divergent.

See Solution

Problem 21105

Calculate the Riemann sum $R(f, P, C)$ for $f(x)=x$ , $P=\{1,1.3,1.7,2\}$ , and $C=\{1.2,1.5,1.8\}$ . Round to two decimal places.

See Solution

Problem 21106

Find the ball's displacement from $t=0$ to $t=2$ using $v(t)=-32t+73$ . Then, find its position at $t=2$ given $s(0)=12$ .

See Solution

Problem 21107

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{2} \sin x+2 x \cos x-2 \sin x$ .

See Solution

Problem 21108

Determine if the sequence $a_{n}=\frac{5 \sqrt{n}}{\sqrt{n}+4}$ converges or diverges. If it converges, find the limit.

See Solution

Problem 21109

Find the sky diver's velocity after 3 seconds and 5 seconds using $v(t)=56(1-e^{-0.15 t})$ . Round to whole numbers.

See Solution

Problem 21110

Find the absolute extreme values of $f(x)=2 x^{3}-33 x^{2}+168 x$ on $[3,8]$ . Use a graphing tool to verify.

See Solution

Problem 21111

Prove $\int \tan ^{n} x d x=\frac{\tan ^{n-1} x}{n-1}-\int \tan ^{n-2} x d x$ for $n \neq 1$ . Evaluate $\int_{0}^{\frac{\pi}{3}} 3 \tan ^{3} x d x$ . Start by factoring: $\int \tan ^{n} x d x=\int\left(\tan ^{2} x\right)(\square) d x$ .

See Solution

Problem 21112

Evaluate $\int 16 \tan 2 x \sec ^{2} 2 x d x$ using $u=\tan 2 x$ and $u=\sec 2 x$ . Reconcile results.

See Solution

Problem 21113

Approximate the area under $f(x)=\frac{1}{2} x^{2}$ from 1 to 3 using 10 left rectangles. Then find the exact area using integration.

See Solution

Problem 21114

Find the maximum value of $f(6)$ given $f(2)=5$ , $f$ is continuous on $[2,6]$ , and $f'(x) \leq 18$ .

See Solution

Problem 21115

Prove the formula for positive integers $n \neq 1$ :
$\int \sec^{n} x \, dx = \frac{\sec^{n-2} x \tan x}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2} x \, dx$
Use integration by parts with $u=\sec^{n-2} x$ , $dv=\sec^{2} x \, dx$ . Find $v$ and $du$ .

See Solution

Problem 21116

Prove $f(x)=x^{3}-4x^{2}+5$ meets Mean Value Theorem conditions on $[0, 2]$ and find $c$ . $c=\square \quad \square \quad \text { Done }$

See Solution

Problem 21117

Find where the function $f(x)=-5 x^{3}+15 x^{2}+120 x+3$ is increasing or decreasing.

See Solution

Problem 21118

Find the absolute max and min of $g(x)=-\sqrt{4-x^{2}}$ on $[-1, 2]$ and graph it.

See Solution

Problem 21119

Find the limit: $a_{n}=\sqrt[n]{6^{n}+7^{n}}$ , as $n$ approaches infinity. What is $\lim_{n \rightarrow \infty} a_{n}$ ?

See Solution

Problem 21120

Prove the integral formula for positive integers $n \neq 1$ :
$\int \sec^{n} x \, dx = \frac{\sec^{n-2} x \tan x}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2} x \, dx$
Find $v$ and $d u$ using $u=\sec^{n-2} x$ and $d v=\sec^{2} x \, dx$ :
$v=\tan x \text{ and } d u=\square \, dx$

See Solution

Problem 21121

Calculate the average value of $\sin x$ from $0$ to $\pi$ using Riemann sums. Then find the average speed of an object traveling at speed $\sin t$ from $0$ to $\pi$ . Compare both results.

See Solution

Problem 21122

Find the max and min of $g(x)=4 \csc x$ on $\left[\frac{\pi}{4}, \frac{3\pi}{4}\right]$ . Graph the function.

See Solution

Problem 21123

Show that $f(x)=x^{3}-4 x^{2}+5$ meets the Mean Value Theorem on $[0, 2]$ and find $c$ . $c=\square \text { Done }$

See Solution

Problem 21124

Use Rolle's Theorem on $f(x)=2 x^{3}-6 x^{2}-2 x+6$ to find $c$ in $(-1,3)$ where $f'(c)=0$ .

See Solution

Problem 21125

Find the average sales from day 0 to day 2 for the function $S(x) = 200x + 9x^2$ .

See Solution

Problem 21126

Approximate the area under $f(x)=2 x^{2}$ from $-1$ to $3$ using a Riemann sum with $n=4$ midpoints. Area $\approx \square$ .

See Solution

Problem 21127

Apply Rolle's Theorem to $f(x)=2 x^{3}-6 x^{2}-2 x+6$ and find $c$ in $(-1,3)$ where $f'(c)=0$ . Options: a) $c=1+\frac{1}{3} \sqrt{3}, 1-\frac{1}{3} \sqrt{3}$ b) $c=1-2 \sqrt{3}$ c) $c=1-\frac{2}{3} \sqrt{3}, 1+\frac{2}{3} \sqrt{3}$ d) Rolle's theorem does not apply.

See Solution

Problem 21128

Approximate the area under $f(x)=3 x^{3}$ from $x=4$ to $x=6$ using a Riemann sum with $n=5$ left endpoints. Area: $\square$ .

See Solution

Problem 21129

Let $g(x)=\int_{\frac{\pi}{2}}^{x} \sin t \, dt$ . Show $g\left(\frac{3 \pi}{2}\right)=0$ and find $x$ where $g(x)<0$ in $[0,2\pi]$ .

See Solution

Problem 21130

Evaluate the integral: $\int_{0}^{\pi / 6} 5 \cos ^{5} 3 x \sin ^{2} 3 x d x=\square$ (Type an exact answer.)

See Solution

Problem 21131

Approximate the area under $f(x)=2 x^{3}$ from $x=4$ to $x=6$ using a Riemann sum with $n=5$ left endpoints. Area: $\square$ .

See Solution

Problem 21132

Solve: $\int \left( \frac{1}{t} \sqrt{t} + \frac{1}{t^{3} \sqrt{t}} \right) dt =$

See Solution

Problem 21133

Find the number $c$ that satisfies the mean value theorem for $f(x)=1+3 x^{\frac{1}{3}}$ on $[-8,1]$ .

See Solution

Problem 21134

Find where the function $f(x)=x-4 x^{-2}$ is decreasing. Options: a) never decreasing b) all real numbers c) $0-2 \leq x<0$ d) $0-2<x<0$

See Solution

Problem 21135

Find the max and min of $g(x)=4 \csc x$ on $\left[\frac{\pi}{4}, \frac{3\pi}{4}\right]$ and graph it.

See Solution

Problem 21136

Find where the function $f(x)=x-4 x^{-2}$ is decreasing. Options: a) never b) all real c) $0-2 \leq x<0$ d) $0-2<x<0$

See Solution

Problem 21137

If $f$ is continuous on $[a, b]$ and $f^{\prime}(x) > 0$ for all $x$ in $(a, b)$ , what can we conclude about $f$ ? a) $f$ is increasing on $[a, b]$ b) $f$ is increasing on $(a, b)$ c) $f(x)$ is decreasing on $[a, b]$ d) $f(x)$ is decreasing on $(a, b)$

See Solution

Problem 21138

Calculate the area under the curve $y=2x$ from $x=2$ to $x=4$ . The area is $\square$ .

See Solution

Problem 21139

Calculate the sum of the series: $\sum_{n=1}^{\infty} \frac{4^{n+1}}{10^{n}}$

See Solution

Problem 21140

Calculate the area under the curve $y=\sqrt{x}+5$ from $x=0$ to $x=4$ . The area is $\square$ .

See Solution

Problem 21141

Find the critical points of $f(x)=2 x^{2}-512 \sqrt{x}$ . Choose A for points or B if none exist.

See Solution

Problem 21142

Calculate the area under the curve $y=2x$ from $x=3$ to $x=6$ and find the antiderivative $F(x)$ .

See Solution

Problem 21143

Calculate 10 partial sums of the series $\sum_{n=1}^{\infty} \frac{4^{n+1}}{10^{n}}$ and round to five decimal places.

See Solution

Problem 21144

Find $c$ such that $\frac{f(6)-f(2)}{6-2}=f^{\prime}(c)$ for $f(x)=\ln(x-1)$ on $[2,6]$ .
$c=\square$

See Solution

Problem 21145

Find $c$ such that $\frac{f(0)-f(-2)}{0-(-2)}=f^{\prime}(c)$ for $f(x)=5x^{2}-2x-3$ on $[-2,0]$ . Answer: $\square$ .

See Solution

Problem 21146

Find the critical points of the function $f(x)=2 x^{2}-512 \sqrt{x}$ . What is $f^{\prime}(x)=\square$ ?

See Solution

Problem 21147

Find the insect population after 4 days if it grows at $180 + 10t + 0.6t^{2}$ per day, starting with 80 insects.

See Solution

Problem 21148

Find the total bicycles produced from week 2 to week 3 given the rate $95 + 108 t^{2} - 6 t$ .

See Solution

Problem 21149

Encuentra la población de bacterias después de 5 horas si comienza con 85 y crece a un ritmo del $12\%$ por hora.

See Solution

Problem 21150

Calculate the displacement from $t=2$ to $t=7$ for $v(t)=0.01 t^{2}+t$ m/s. Round to two decimal places.

See Solution

Problem 21151

Find the average rate of change of $f(x)=\sqrt{5x-2}$ for $0.5 \leq x \leq 3$ .

See Solution

Problem 21152

Find the displacement and distance traveled of a particle with velocity $v(t)=24-12t$ (in m/s) over $[0,9]$ .

See Solution

Problem 21153

A substance grows at a rate of $14\%$ daily. If it starts at 228 grams, find its mass after 2 days, rounded to 1 decimal.

See Solution

Problem 21154

Evaluate the sequence $a_{n}=\frac{9 n}{3 n+1}$ for convergence and the series $\sum_{n=1}^{\infty} a_{n}$ for convergence.

See Solution

Problem 21155

Is the series $\sum_{n=1}^{\infty} \frac{3}{n^{2}+n^{3}}$ convergent or divergent?

See Solution

Problem 21156

Find the displacement and distance traveled by a particle with velocity $v(t)=3 t^{2}-30 t+27$ from $t=0$ to $t=10$ .

See Solution

Problem 21157

Is the series $\sum_{n=1}^{\infty} \frac{4}{\pi^{n}}$ convergent or divergent? If convergent, find the sum.

See Solution

Problem 21158

Find the average rate of change of $w(t) = 5t^{2} - 5t + 1$ over the interval $[-2,1]$ .

See Solution

Problem 21159

Graph the function $f(x)=-4x^2-5$ on $[0,1]$ and find its average value. Choose the correct graph: A, B, or C. Average value: $\square$ .

See Solution

Problem 21160

What is the relationship between an antiderivative $F$ of $f$ and the area function $A$ of $f$ ?
A. $A=f(x)$ B. $A$ is a derivative of $f, A(x)=f^{\prime}(x)$ C. $A$ is an antiderivative of $f; A^{\prime}(x)=f(x)$

See Solution

Problem 21161

Graph the function $f(x) = -4x^{2} - 6$ on $[0,2]$ and find its average value over this interval. Choose the correct graph.

See Solution

Problem 21162

What is the relationship between the antiderivative $F$ of $f$ and the area function $A$ of $f$ ?
A. $A(x)=F^{\prime}(x)$ B. $F(x)=A^{\prime}(x)$ C. $F(x)=A(x)$ D. Both $F$ and $A$ are antiderivatives of $f$ , differing by a constant $C$ .

See Solution

Problem 21163

Evaluate the integral $\int_{1}^{2} \frac{x^{3}+4 x}{x^{4}+8 x^{2}+6} d x$ .

See Solution

Problem 21164

Analyze the end behavior of the function $p(x)=-7x^{2}+1$ using limits.

See Solution

Problem 21165

Evaluate the integral from 0 to $\frac{\pi}{8}$ of $\sin(8t)$ dt.

See Solution

Problem 21166

Determine the upper and lower bounds for the integral $I=\int_{0}^{3} \frac{1}{\sqrt{x^{3}+6}} d x$ .

See Solution

Problem 21167

Evaluate $r(\theta)=\theta+\sin^{2}\left(\frac{\theta}{3}\right)-8$ at $-3\pi$ and $3\pi$ . Find $r(-3\pi)=\square$ .

See Solution

Problem 21168

Show that the function $r(\theta)=\theta+\sin ^{2}\left(\frac{\theta}{3}\right)-8$ has one zero in $(-\infty, \infty)$ . Find $r'(\theta)$ .

See Solution

Problem 21169

Find and graph the area function $A(x)=\int_{-10}^{x} (t+10) dt$ and verify $A'(x)=f(x)$ .

See Solution

Problem 21170

Evaluate the integral from 2 to 4 of the function (2x + 3): $\int_{2}^{4}(2 x+3) d x$ .

See Solution

Problem 21171

Find the derivative of $y=2 x e^{-x}-2 e^{x^{4}}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 21172

Evaluate the integral from -2 to 2:
$\int_{-2}^{2}\left(21 x^{3}-22 x^{2}\right) d x$
using
$\int_{0}^{b} x^{2} d x=\frac{1}{3} b^{3}$ and $\int_{0}^{b} x^{3} d x=\frac{1}{4} b^{4}$ .

See Solution

Problem 21173

Find the derivative of $y=2 x e^{-x}-e^{x^{2}}$ . What is $\frac{d y}{d x}$ in terms of $e$ ?

See Solution

Problem 21174

Calculate the integral from $\pi / 3$ to $\pi / 2$ : $\int_{\pi / 3}^{\pi / 2} \sin x \, dx$ .

See Solution

Problem 21175

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{0}^{\frac{1}{2}} \frac{8 d x}{\sqrt{1-x^{2}}}$

See Solution

Problem 21176

Find critical points of $f$ given $f^{\prime}(x)=(x+2)e^{-x}$ . Determine intervals of increase/decrease and local extrema.

See Solution

Problem 21177

Find critical points of $f$ given $f'(x)=\frac{x^{2}(x-5)}{x+8}$ , and determine intervals of increase/decrease.

See Solution

Problem 21178

Find the half-life of a radioactive substance with a decay rate of $5.9\%$ per day using the continuous decay model.

See Solution

Problem 21179

Calculate the elasticity of demand from the function $q=310-5p$ when the price $p=33$ .

See Solution

Problem 21180

Determine where the function $f(x)=\frac{1}{x^{2}+1}$ is increasing. Verify with a number line analysis.

See Solution

Problem 21181

Using Boyle's law ( $PV=C$ ), find the rate of pressure increase when $P=80 \mathrm{lb/in}^2$ , $V=90 \mathrm{in}^3$ , and $dV/dt=-14 \mathrm{in}^3/\mathrm{s}$ .

See Solution

Problem 21182

Find the marginal profit from the profit function $P(x)=-x^{2}+12x-32$ at $x=6$ .

See Solution

Problem 21183

A baby weighs 6 lbs at birth and 9 lbs at 3 months. How much will she weigh at 6 months if weight increases proportionally? (A) 11.9 (B) 12.8 (C) 13.5 (D) 14.6

See Solution

Problem 21184

Berechnen Sie die Fläche zwischen $p(x) = 0,5 x^{2} + 1$ und der $x$ -Achse im Intervall $I = [-1, 2]$ durch Integration.

See Solution

Problem 21185

A 16 ft fence is 4 ft from a building. Find the shortest ladder length over the fence to the wall.
1. Find $L(\theta)=$ .
2. Find $L'(\theta)=$ .
3. Find $L(\theta_{\min}) \approx \quad$ feet (5 decimal places).

See Solution

Problem 21186

Differentiate $f(x)=\ln (\sin x)$ .

See Solution

Problem 21187

Bestimmen Sie die Teilintervalle und die Gesamtfläche zwischen der Funktion $f(x)=x^{3}+x^{2}-2 x$ und der $x$ -Achse im Intervall $I=[-2 ; 2]$ .

See Solution

Problem 21188

Gegeben ist die Funktion $f(x)=\frac{1}{3} x^{3}-\frac{3}{2} x^{2}-10 x$ . Finde die Nullstellen (auf 2 Dezimalstellen) und die Extremstellen.

See Solution

Problem 21189

Gegeben ist die Funktion $f(x)=\frac{1}{3} x^{3}-\frac{3}{2} x^{2}-10 x$ .
a) Finde die Nullstellen von $f$ (2 Dezimalstellen). b) Bestimme die Extremstellen und deren Art. c) Zeige, dass die Tangente bei $B(0,0)$ $t(x)=-10 x$ ist. d) Finde den weiteren Schnittpunkt $C$ der Tangente und $f$ . e) Skizziere das Dreieck im 4. Quadranten und berechne dessen Fläche.

See Solution

Problem 21190

Graph $f(x)=-4x^{2}-6$ on $[0,2]$ and find its average value. Choose the correct graph and calculate the average value.

See Solution

Problem 21191

Graph $f(t)=(t-8)^{2}$ and calculate its average value on $[0,15]$ .

See Solution

Problem 21192

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

See Solution

Problem 21193

Bestimmen Sie die Ableitung von $f(x)=\sqrt[5]{x^{3}}$ .

See Solution

Problem 21194

Find the derivative of the integral $\int_{1}^{\sin x} 6 t^{5} dt$ . Use both evaluation and direct differentiation methods.

See Solution

Problem 21195

Find the integral: $\int \frac{e^{x}}{\cos ^{2}\left(e^{x}\right)} d x$

See Solution

Problem 21196

Evaluate the integral using the Fundamental Theorem of Calculus: $\int_{-9 \pi / 2}^{9 \pi / 2}(\cos x-3) d x$ .

See Solution

Problem 21197

Calculate the heat conduction rate in watts through a $3.00 \mathrm{~m}^{2}$ window, $0.00635 \mathrm{~m}$ thick, with temps $5.00^{\circ} \mathrm{C}$ and $-10.0^{\circ} \mathrm{C}$ . Thermal conductivity is $0.84 \mathrm{~W} /\left(\mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right)$ . Answer: $5952.8 \mathrm{~W}$ .

See Solution

Problem 21198

Use integration by parts to evaluate the integral $\int \frac{\ln (x)}{x^{4}} \mathrm{~d} x$ . Find $f$ , $g'$ , $f'$ , and $g$ .

See Solution

Problem 21199

Find $f$ and $g'$ for $\int x^{2} \sin (2 x) dx$ using integration by parts, then compute the integral.

See Solution

Problem 21200

Berechnen Sie die Fläche zwischen den Funktionen $f(x)=\frac{1}{3} x^{2}+3$ und $g(x)=-x^{2}+1$ im Intervall $I=[-2, 3]$ durch Integration.

See Solution

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Calculus

Problem 21101

Evaluate ∫16tan⁡2xsec⁡22xdx\int 16 \tan 2 x \sec ^{2} 2 x d x∫16tan2xsec22xdx using substitutions u=tan⁡2xu=\tan 2 xu=tan2x and u=sec⁡2xu=\sec 2 xu=sec2x. Compare results.

Problem 21102

Find the radius of convergence, RRR, and interval, III, for the series ∑n=1∞n!(6x−1)n\sum_{n=1}^{\infty} n !(6 x-1)^{n}∑n=1∞​n!(6x−1)n.

Problem 21103

Find the displacement of a ball with velocity v(t)=−32t+73v(t)=-32 t+73v(t)=−32t+73 from t=0t=0t=0 to t=2t=2t=2 seconds. Displacement is □\square□ feet.

Problem 21104

Determine if the sequence an=1+9n8na_{n}=1+\frac{9^{n}}{8^{n}}an​=1+8n9n​ has a limit. If yes, find it; if not, state it's divergent.

Problem 21105

Calculate the Riemann sum R(f,P,C)R(f, P, C)R(f,P,C) for f(x)=xf(x)=xf(x)=x, P={1,1.3,1.7,2}P=\{1,1.3,1.7,2\}P={1,1.3,1.7,2}, and C={1.2,1.5,1.8}C=\{1.2,1.5,1.8\}C={1.2,1.5,1.8}. Round to two decimal places.

Problem 21106

Find the ball's displacement from t=0t=0t=0 to t=2t=2t=2 using v(t)=−32t+73v(t)=-32t+73v(t)=−32t+73. Then, find its position at t=2t=2t=2 given s(0)=12s(0)=12s(0)=12.

Problem 21107

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=x2sin⁡x+2xcos⁡x−2sin⁡xy=x^{2} \sin x+2 x \cos x-2 \sin xy=x2sinx+2xcosx−2sinx.

Problem 21108

Determine if the sequence an=5nn+4a_{n}=\frac{5 \sqrt{n}}{\sqrt{n}+4}an​=n​+45n​​ converges or diverges. If it converges, find the limit.

Problem 21109

Find the sky diver's velocity after 3 seconds and 5 seconds using v(t)=56(1−e−0.15t)v(t)=56(1-e^{-0.15 t})v(t)=56(1−e−0.15t). Round to whole numbers.

Problem 21110

Find the absolute extreme values of f(x)=2x3−33x2+168xf(x)=2 x^{3}-33 x^{2}+168 xf(x)=2x3−33x2+168x on [3,8][3,8][3,8]. Use a graphing tool to verify.

Problem 21111

Problem 21112

Evaluate ∫16tan⁡2xsec⁡22xdx\int 16 \tan 2 x \sec ^{2} 2 x d x∫16tan2xsec22xdx using u=tan⁡2xu=\tan 2 xu=tan2x and u=sec⁡2xu=\sec 2 xu=sec2x. Reconcile results.

Problem 21113

Approximate the area under f(x)=12x2f(x)=\frac{1}{2} x^{2}f(x)=21​x2 from 1 to 3 using 10 left rectangles. Then find the exact area using integration.

Problem 21114

Find the maximum value of f(6)f(6)f(6) given f(2)=5f(2)=5f(2)=5, fff is continuous on [2,6][2,6][2,6], and f′(x)≤18f'(x) \leq 18f′(x)≤18.

Problem 21115

Problem 21116

Prove f(x)=x3−4x2+5f(x)=x^{3}-4x^{2}+5f(x)=x3−4x2+5 meets Mean Value Theorem conditions on [0,2][0, 2][0,2] and find ccc. c=□□ Done c=\square \quad \square \quad \text { Done }c=□□ Done

Problem 21117

Find where the function f(x)=−5x3+15x2+120x+3f(x)=-5 x^{3}+15 x^{2}+120 x+3f(x)=−5x3+15x2+120x+3 is increasing or decreasing.

Problem 21118

Find the absolute max and min of g(x)=−4−x2g(x)=-\sqrt{4-x^{2}}g(x)=−4−x2​ on [−1,2][-1, 2][−1,2] and graph it.

Problem 21119

Find the limit: an=6n+7nna_{n}=\sqrt[n]{6^{n}+7^{n}}an​=n6n+7n​, as nnn approaches infinity. What is lim⁡n→∞an\lim_{n \rightarrow \infty} a_{n}limn→∞​an​?

Problem 21120

Problem 21121

Calculate the average value of sin⁡x\sin xsinx from 000 to π\piπ using Riemann sums. Then find the average speed of an object traveling at speed sin⁡t\sin tsint from 000 to π\piπ. Compare both results.

Problem 21122

Find the max and min of g(x)=4csc⁡xg(x)=4 \csc xg(x)=4cscx on [π4,3π4]\left[\frac{\pi}{4}, \frac{3\pi}{4}\right][4π​,43π​]. Graph the function.

Problem 21123

Show that f(x)=x3−4x2+5f(x)=x^{3}-4 x^{2}+5f(x)=x3−4x2+5 meets the Mean Value Theorem on [0,2][0, 2][0,2] and find ccc. c=□ Done c=\square \text { Done }c=□ Done

Problem 21124

Use Rolle's Theorem on f(x)=2x3−6x2−2x+6f(x)=2 x^{3}-6 x^{2}-2 x+6f(x)=2x3−6x2−2x+6 to find ccc in (−1,3)(-1,3)(−1,3) where f′(c)=0f'(c)=0f′(c)=0.

Problem 21125

Find the average sales from day 0 to day 2 for the function S(x)=200x+9x2S(x) = 200x + 9x^2S(x)=200x+9x2.

Problem 21126

Approximate the area under f(x)=2x2f(x)=2 x^{2}f(x)=2x2 from −1-1−1 to 333 using a Riemann sum with n=4n=4n=4 midpoints. Area ≈□\approx \square≈□.

Problem 21127

Problem 21128

Approximate the area under f(x)=3x3f(x)=3 x^{3}f(x)=3x3 from x=4x=4x=4 to x=6x=6x=6 using a Riemann sum with n=5n=5n=5 left endpoints. Area: □\square□.

Problem 21129

Let g(x)=∫π2xsin⁡t dtg(x)=\int_{\frac{\pi}{2}}^{x} \sin t \, dtg(x)=∫2π​x​sintdt. Show g(3π2)=0g\left(\frac{3 \pi}{2}\right)=0g(23π​)=0 and find xxx where g(x)<0g(x)<0g(x)<0 in [0,2π][0,2\pi][0,2π].

Problem 21130

Evaluate the integral: ∫0π/65cos⁡53xsin⁡23xdx=□\int_{0}^{\pi / 6} 5 \cos ^{5} 3 x \sin ^{2} 3 x d x=\square∫0π/6​5cos53xsin23xdx=□ (Type an exact answer.)

Problem 21131

Approximate the area under f(x)=2x3f(x)=2 x^{3}f(x)=2x3 from x=4x=4x=4 to x=6x=6x=6 using a Riemann sum with n=5n=5n=5 left endpoints. Area: □\square□.

Problem 21132

Solve: ∫(1tt+1t3t)dt=\int \left( \frac{1}{t} \sqrt{t} + \frac{1}{t^{3} \sqrt{t}} \right) dt = ∫(t1​t​+t3t​1​)dt=

Problem 21133

Find the number ccc that satisfies the mean value theorem for f(x)=1+3x13f(x)=1+3 x^{\frac{1}{3}}f(x)=1+3x31​ on [−8,1][-8,1][−8,1].

Problem 21134

Find where the function f(x)=x−4x−2f(x)=x-4 x^{-2}f(x)=x−4x−2 is decreasing. Options: a) never decreasing b) all real numbers c) 0−2≤x<00-2 \leq x<00−2≤x<0 d) 0−2<x<00-2<x<00−2<x<0

Problem 21135

Find the max and min of g(x)=4csc⁡xg(x)=4 \csc xg(x)=4cscx on [π4,3π4]\left[\frac{\pi}{4}, \frac{3\pi}{4}\right][4π​,43π​] and graph it.

Problem 21136

Find where the function f(x)=x−4x−2f(x)=x-4 x^{-2}f(x)=x−4x−2 is decreasing. Options: a) never b) all real c) 0−2≤x<00-2 \leq x<00−2≤x<0 d) 0−2<x<00-2<x<00−2<x<0

Problem 21137

Problem 21138

Calculate the area under the curve y=2xy=2xy=2x from x=2x=2x=2 to x=4x=4x=4. The area is □\square□.

Problem 21139

Calculate the sum of the series: ∑n=1∞4n+110n\sum_{n=1}^{\infty} \frac{4^{n+1}}{10^{n}}∑n=1∞​10n4n+1​

Problem 21140

Calculate the area under the curve y=x+5y=\sqrt{x}+5y=x​+5 from x=0x=0x=0 to x=4x=4x=4. The area is □\square□.

Problem 21141

Find the critical points of f(x)=2x2−512xf(x)=2 x^{2}-512 \sqrt{x}f(x)=2x2−512x​. Choose A for points or B if none exist.

Problem 21142

Calculate the area under the curve y=2xy=2xy=2x from x=3x=3x=3 to x=6x=6x=6 and find the antiderivative F(x)F(x)F(x).

Evaluate $\int 16 \tan 2 x \sec ^{2} 2 x d x$ using substitutions $u=\tan 2 x$ and $u=\sec 2 x$ . Compare results.

Find the radius of convergence, $R$ , and interval, $I$ , for the series $\sum_{n=1}^{\infty} n !(6 x-1)^{n}$ .

Find the displacement of a ball with velocity $v(t)=-32 t+73$ from $t=0$ to $t=2$ seconds. Displacement is $\square$ feet.

Determine if the sequence $a_{n}=1+\frac{9^{n}}{8^{n}}$ has a limit. If yes, find it; if not, state it's divergent.

Calculate the Riemann sum $R(f, P, C)$ for $f(x)=x$ , $P=\{1,1.3,1.7,2\}$ , and $C=\{1.2,1.5,1.8\}$ . Round to two decimal places.

Find the ball's displacement from $t=0$ to $t=2$ using $v(t)=-32t+73$ . Then, find its position at $t=2$ given $s(0)=12$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=x^{2} \sin x+2 x \cos x-2 \sin x$ .

Determine if the sequence $a_{n}=\frac{5 \sqrt{n}}{\sqrt{n}+4}$ converges or diverges. If it converges, find the limit.

Find the sky diver's velocity after 3 seconds and 5 seconds using $v(t)=56(1-e^{-0.15 t})$ . Round to whole numbers.

Find the absolute extreme values of $f(x)=2 x^{3}-33 x^{2}+168 x$ on $[3,8]$ . Use a graphing tool to verify.

Evaluate $\int 16 \tan 2 x \sec ^{2} 2 x d x$ using $u=\tan 2 x$ and $u=\sec 2 x$ . Reconcile results.

Approximate the area under $f(x)=\frac{1}{2} x^{2}$ from 1 to 3 using 10 left rectangles. Then find the exact area using integration.

Find the maximum value of $f(6)$ given $f(2)=5$ , $f$ is continuous on $[2,6]$ , and $f'(x) \leq 18$ .

Prove $f(x)=x^{3}-4x^{2}+5$ meets Mean Value Theorem conditions on $[0, 2]$ and find $c$ . $c=\square \quad \square \quad \text { Done }$

Find where the function $f(x)=-5 x^{3}+15 x^{2}+120 x+3$ is increasing or decreasing.

Find the absolute max and min of $g(x)=-\sqrt{4-x^{2}}$ on $[-1, 2]$ and graph it.

Find the limit: $a_{n}=\sqrt[n]{6^{n}+7^{n}}$ , as $n$ approaches infinity. What is $\lim_{n \rightarrow \infty} a_{n}$ ?

Calculate the average value of $\sin x$ from $0$ to $\pi$ using Riemann sums. Then find the average speed of an object traveling at speed $\sin t$ from $0$ to $\pi$ . Compare both results.

Find the max and min of $g(x)=4 \csc x$ on $\left[\frac{\pi}{4}, \frac{3\pi}{4}\right]$ . Graph the function.

Show that $f(x)=x^{3}-4 x^{2}+5$ meets the Mean Value Theorem on $[0, 2]$ and find $c$ . $c=\square \text { Done }$

Use Rolle's Theorem on $f(x)=2 x^{3}-6 x^{2}-2 x+6$ to find $c$ in $(-1,3)$ where $f'(c)=0$ .

Find the average sales from day 0 to day 2 for the function $S(x) = 200x + 9x^2$ .

Approximate the area under $f(x)=2 x^{2}$ from $-1$ to $3$ using a Riemann sum with $n=4$ midpoints. Area $\approx \square$ .

Approximate the area under $f(x)=3 x^{3}$ from $x=4$ to $x=6$ using a Riemann sum with $n=5$ left endpoints. Area: $\square$ .

Let $g(x)=\int_{\frac{\pi}{2}}^{x} \sin t \, dt$ . Show $g\left(\frac{3 \pi}{2}\right)=0$ and find $x$ where $g(x)<0$ in $[0,2\pi]$ .

Evaluate the integral: $\int_{0}^{\pi / 6} 5 \cos ^{5} 3 x \sin ^{2} 3 x d x=\square$ (Type an exact answer.)

Approximate the area under $f(x)=2 x^{3}$ from $x=4$ to $x=6$ using a Riemann sum with $n=5$ left endpoints. Area: $\square$ .

Solve: $\int \left( \frac{1}{t} \sqrt{t} + \frac{1}{t^{3} \sqrt{t}} \right) dt =$

Find the number $c$ that satisfies the mean value theorem for $f(x)=1+3 x^{\frac{1}{3}}$ on $[-8,1]$ .

Find where the function $f(x)=x-4 x^{-2}$ is decreasing. Options: a) never decreasing b) all real numbers c) $0-2 \leq x<0$ d) $0-2<x<0$

Find the max and min of $g(x)=4 \csc x$ on $\left[\frac{\pi}{4}, \frac{3\pi}{4}\right]$ and graph it.

Find where the function $f(x)=x-4 x^{-2}$ is decreasing. Options: a) never b) all real c) $0-2 \leq x<0$ d) $0-2<x<0$

Calculate the area under the curve $y=2x$ from $x=2$ to $x=4$ . The area is $\square$ .

Calculate the sum of the series: $\sum_{n=1}^{\infty} \frac{4^{n+1}}{10^{n}}$

Calculate the area under the curve $y=\sqrt{x}+5$ from $x=0$ to $x=4$ . The area is $\square$ .

Find the critical points of $f(x)=2 x^{2}-512 \sqrt{x}$ . Choose A for points or B if none exist.

Calculate the area under the curve $y=2x$ from $x=3$ to $x=6$ and find the antiderivative $F(x)$ .

Calculate 10 partial sums of the series $\sum_{n=1}^{\infty} \frac{4^{n+1}}{10^{n}}$ and round to five decimal places.

Find $c$ such that $\frac{f(6)-f(2)}{6-2}=f^{\prime}(c)$ for $f(x)=\ln(x-1)$ on $[2,6]$ .
$c=\square$

Find $c$ such that $\frac{f(0)-f(-2)}{0-(-2)}=f^{\prime}(c)$ for $f(x)=5x^{2}-2x-3$ on $[-2,0]$ . Answer: $\square$ .

Find the critical points of the function $f(x)=2 x^{2}-512 \sqrt{x}$ . What is $f^{\prime}(x)=\square$ ?

Find the insect population after 4 days if it grows at $180 + 10t + 0.6t^{2}$ per day, starting with 80 insects.

Find the total bicycles produced from week 2 to week 3 given the rate $95 + 108 t^{2} - 6 t$ .

Encuentra la población de bacterias después de 5 horas si comienza con 85 y crece a un ritmo del $12\%$ por hora.

Calculate the displacement from $t=2$ to $t=7$ for $v(t)=0.01 t^{2}+t$ m/s. Round to two decimal places.

Find the average rate of change of $f(x)=\sqrt{5x-2}$ for $0.5 \leq x \leq 3$ .

Find the displacement and distance traveled of a particle with velocity $v(t)=24-12t$ (in m/s) over $[0,9]$ .

A substance grows at a rate of $14\%$ daily. If it starts at 228 grams, find its mass after 2 days, rounded to 1 decimal.

Evaluate the sequence $a_{n}=\frac{9 n}{3 n+1}$ for convergence and the series $\sum_{n=1}^{\infty} a_{n}$ for convergence.

Is the series $\sum_{n=1}^{\infty} \frac{3}{n^{2}+n^{3}}$ convergent or divergent?

Find the displacement and distance traveled by a particle with velocity $v(t)=3 t^{2}-30 t+27$ from $t=0$ to $t=10$ .

Is the series $\sum_{n=1}^{\infty} \frac{4}{\pi^{n}}$ convergent or divergent? If convergent, find the sum.

Find the average rate of change of $w(t) = 5t^{2} - 5t + 1$ over the interval $[-2,1]$ .

Graph the function $f(x)=-4x^2-5$ on $[0,1]$ and find its average value. Choose the correct graph: A, B, or C. Average value: $\square$ .

What is the relationship between an antiderivative $F$ of $f$ and the area function $A$ of $f$ ?
A. $A=f(x)$ B. $A$ is a derivative of $f, A(x)=f^{\prime}(x)$ C. $A$ is an antiderivative of $f; A^{\prime}(x)=f(x)$

Graph the function $f(x) = -4x^{2} - 6$ on $[0,2]$ and find its average value over this interval. Choose the correct graph.

What is the relationship between the antiderivative $F$ of $f$ and the area function $A$ of $f$ ?
A. $A(x)=F^{\prime}(x)$ B. $F(x)=A^{\prime}(x)$ C. $F(x)=A(x)$ D. Both $F$ and $A$ are antiderivatives of $f$ , differing by a constant $C$ .

Evaluate the integral $\int_{1}^{2} \frac{x^{3}+4 x}{x^{4}+8 x^{2}+6} d x$ .

Analyze the end behavior of the function $p(x)=-7x^{2}+1$ using limits.

Evaluate the integral from 0 to $\frac{\pi}{8}$ of $\sin(8t)$ dt.

Determine the upper and lower bounds for the integral $I=\int_{0}^{3} \frac{1}{\sqrt{x^{3}+6}} d x$ .

Evaluate $r(\theta)=\theta+\sin^{2}\left(\frac{\theta}{3}\right)-8$ at $-3\pi$ and $3\pi$ . Find $r(-3\pi)=\square$ .

Show that the function $r(\theta)=\theta+\sin ^{2}\left(\frac{\theta}{3}\right)-8$ has one zero in $(-\infty, \infty)$ . Find $r'(\theta)$ .

Find and graph the area function $A(x)=\int_{-10}^{x} (t+10) dt$ and verify $A'(x)=f(x)$ .

Evaluate the integral from 2 to 4 of the function (2x + 3): $\int_{2}^{4}(2 x+3) d x$ .

Find the derivative of $y=2 x e^{-x}-2 e^{x^{4}}$ . What is $\frac{d y}{d x}$ ?

Evaluate the integral from -2 to 2:
$\int_{-2}^{2}\left(21 x^{3}-22 x^{2}\right) d x$
using
$\int_{0}^{b} x^{2} d x=\frac{1}{3} b^{3}$ and $\int_{0}^{b} x^{3} d x=\frac{1}{4} b^{4}$ .

Find the derivative of $y=2 x e^{-x}-e^{x^{2}}$ . What is $\frac{d y}{d x}$ in terms of $e$ ?

Calculate the integral from $\pi / 3$ to $\pi / 2$ : $\int_{\pi / 3}^{\pi / 2} \sin x \, dx$ .