Calculus

Problem 21501

Find the turning points of the curve defined by $x=3t$ and $y=12t-t^3$ .

See Solution

Problem 21502

Find the absolute min and max of $f(t) = t \sqrt{25 - t^2}$ on the interval $[-1, 5]$ .

See Solution

Problem 21503

Find the absolute maximum of $f(x)=x^{3}-x^{2}-x+1$ on $[0,2]$ .

See Solution

Problem 21504

Rewrite $\int \frac{e^{x}}{x^{2}} d x$ using integration by parts with $d v=\frac{1}{x^{2}} d x$ , $u=e^{x}$ . Compare summation by parts to integration by parts. Apply it to $\sum_{n=5}^{12} \frac{2^{n}}{n(n+1)}$ with $u_{n}=2^{n}$ , $(\Delta v)_{n}=\frac{1}{n(n+1)}$ .

See Solution

Problem 21505

Approximate the area under $f(x)=2x+3$ from $x=0$ to $x=2$ using $n=4$ right rectangles. What is $R_{4}$ ? A. 15 B. 11 C. 17 D. 13

See Solution

Problem 21506

Rewrite $\int \frac{e^{x}}{x^{2}} d x$ using integration by parts with $d v=\frac{1}{x^{2}} d x$ , $u=e^{x}$ . Compare summation by parts to integration by parts. Apply it to $\sum_{n=5}^{12} \frac{2^{n}}{n(n+1)}$ with $u_{n}=2^{n}$ , $(\Delta v)_{n}=\frac{1}{n(n+1)}$ .

See Solution

Problem 21507

Select a possible graph of the twice-differentiable function $y=f(x)$ based on these derivative properties.

See Solution

Problem 21508

Determine where the function $y=x^{4}+8 x^{3}-72 x^{2}+4$ is concave down. Choose the correct interval.

See Solution

Problem 21509

A train's position is $s(t)=\frac{130}{t}$ for $1 \leq t \leq 6$ .
(a) Graph $s(t)$ . (b) Calculate average velocity from $t=1$ to $t=6$ : $\square$ (round to the nearest tenth).

See Solution

Problem 21510

Approximate the area under $f(x)=2x+3$ from $x=0$ to $x=2$ using $n=4$ right rectangles. Compute $R_{4}$ .

See Solution

Problem 21511

Find the critical numbers of the function $f(\theta) = 18 \cos \theta + 9 \sin^2 \theta$ .

See Solution

Problem 21512

Calculate the integral: $\int(2 x^{5}-7 x^{3}+9) \, dx$ . Choose the correct answer from the options provided.

See Solution

Problem 21513

Find the rate of radius increase of a balloon inflating at 800 cm³/min when the radius is 30 cm. $V=\frac{4}{3} \pi r^{3}$

See Solution

Problem 21514

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for $x=\cos 2 t, y=\sin ^{2} t$ and the Cartesian equation with its domain.

See Solution

Problem 21515

Calculate the average rate of change of $f(x)=-2x^{2}+10$ between $x=3$ and $x=8$ . Simplify your result.

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Problem 21516

Calculate the integral: $\int 9 x^{-5} dx$ . Choose the correct answer from the options provided.

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Problem 21517

Evaluate the series $\sum_{n=1}^{\infty} \cos (n \pi)$ : find $S_{k}$ , check convergence/divergence, and use a test.

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Problem 21518

A ball dropped from a 400 ft building has height $s=400-16 t^{2}$ . Find its velocity when it hits the ground. Options: $-128 \mathrm{ft} / \mathrm{sec}$ , $-160 \mathrm{ft} / \mathrm{sec}$ , $-192 \mathrm{ft} / \mathrm{sec}$ , $-98 \mathrm{ft} / \mathrm{sec}$ .

See Solution

Problem 21519

A Midwest town's population dropped from 215,000 to 211,000 in 4 years. Find the population after another 4 years.

See Solution

Problem 21520

Find the derivative $\frac{d y}{d x}$ for the equation $e^{2 x}=\sin (x+7 y)$ .

See Solution

Problem 21521

Calculate the average value of $y=2 x^{4}$ from $x=-2$ to $x=2$ . Options: A. 0 B. $\frac{128}{5}$ C. $\frac{16}{5}$ D. $\frac{32}{5}$

See Solution

Problem 21522

Evaluate the integral: $\int_{0}^{1} x e^{-x^{2}} dx$ and $\int_{0}^{\frac{e-1}{2 e}} \frac{\ln(e^{2}-1)}{\frac{e^{2}-1}{2 e^{2}}} e^{\frac{e(e^{2}-1)}{2}}$ .

See Solution

Problem 21523

Evaluate the integral from 2 to 5: $\int_{2}^{5} 3 x d x=\square$ (Type an integer or a simplified fraction.)

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Problem 21524

Evaluate the integral from 0 to 1: $\int_{0}^{1} x e^{-x^{2}} d x$ .

See Solution

Problem 21525

Find the radius of convergence for the series $\sum(4 x)^{k}$ and test endpoints for the interval of convergence.

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Problem 21526

Find the indefinite integral: $\int \sin^{2} x \cos x \, dx$ . Use $\mathrm{C}$ for the constant.

See Solution

Problem 21527

Find the indefinite integral: $\int 6 x \, dx = \square$

See Solution

Problem 21528

Evaluate the integral from 1 to 2: $\int_{1}^{2}(2 x+6) d x = \square($ Simplify your answer.)

See Solution

Problem 21529

Strontium-90 decays as $A(t)=A_{0} e^{-0.0244 t}$ . Given 400g, find decay rate, amount after 40 years, time for 100g, and half-life.

See Solution

Problem 21530

Find the radius and interval of convergence for the series $\sum_{k=0}^{\infty} \frac{(3 x-5)^{k}}{k !}$ . Radius $R=\square$ .

See Solution

Problem 21531

Evaluate the integral from 0 to b: $\int_{0}^{b} 8 e^{x} dx$ . What is the result?

See Solution

Problem 21532

Strontium-90 decays as $A(t)=A_{0} e^{-0.0244 t}$ . Given 400g, find: (a) decay rate, (b) amount after 40 years, (c) time for 100g left, (d) half-life.

See Solution

Problem 21533

Find the average rate of change of height $h(t)=-3 \cos t+8$ from $t=0$ to $t=3$ seconds. Show your work.

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Problem 21534

Evaluate the integral from 0 to 16 of $3 \sqrt{x} \, dx$ . What is the result? A. 288 B. 192 C. 128 D. 24

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Problem 21535

Calculate the integral $\int(\sec (y) \tan (y)-\csc (y) \cot (y)) d y$ and verify by differentiating. Use $C$ for the constant.

See Solution

Problem 21536

Calculate the indefinite integral and use $C$ for the constant: $\int x(1-7 x^{2})^{4} dx$

See Solution

Problem 21537

Calculate the integral of $5(t^{2}-5t-2) \, dt$ . What is the result?

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Problem 21538

Find the radius of convergence for the series $\sum(3 k x)^{k}$ and test endpoints for the interval of convergence.

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Problem 21539

Find the derivative of the function defined by $(x)=(2x+7)(x^2-3)$ .

See Solution

Problem 21540

Find the tangent line equation for the function $f(x)=-9 x^{4}+13 x^{2}-4$ at the point $(1,0)$ . $y=$

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Problem 21541

Find the radius and interval of convergence for the series $\sum_{k=1}^{\infty} \sin^{k}\left(\frac{6}{5 k^{4}}\right) x^{k}$ .

See Solution

Problem 21542

Find the radius and interval of convergence for the series $\sum_{k=4}^{\infty} \frac{4^{k}(x-2)^{k}}{k}$ . Radius: $R=\square$ .

See Solution

Problem 21543

Calculate the average value of $y=2 x^{4}$ from $x=-2$ to $x=2$ . Options: A. $\frac{16}{5}$ B. 0 C. $\frac{32}{5}$ D. $\frac{128}{5}$

See Solution

Problem 21544

Calculate the integral: $\int(5+x^{3})(4-x^{2}) dx$ . Choose the correct answer from the options provided.

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Problem 21545

Find where the function $h(x)=-x^{3}+4x^{2}$ is increasing or decreasing and identify local extreme values.

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Problem 21546

Calculate the integral of $5 e^{x}-\frac{1}{x}$ with respect to $x$ .

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Problem 21547

Evaluate the integral from -1 to 1 of $(3 x^{2}-8 x)$ . What is the result? A. 2 B. 12 C. 7 D. -7

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Problem 21548

Find the radius and interval of convergence for the series $\sum_{k=0}^{\infty}\left(\frac{x}{3}\right)^{k}$ . Radius: $R=\square$ .

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Problem 21549

Evaluate the integral from 0 to b: $\int_{0}^{b} 9 x^{8} d x$ . What is the result? A. $\frac{1}{9} b^{9}$ B. $b^{7}$ C. $9 b^{9}$ D. $b^{9}$

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Problem 21550

Calculate the one-sided limit: $\lim _{x \rightarrow-6^{+}} \frac{1}{x+6}$ . If it doesn't exist, write DNE.

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Problem 21551

Find the slope of $y=(6x+2)^{2}$ at the point $(0,4)$ . Calculate $y'(0)$ .

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Problem 21552

Find the radius and interval of convergence for the series $\sum_{k=3}^{\infty} \frac{3^{k}(x-5)^{k}}{k}$ .

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Problem 21553

Evaluate the integral from -3 to -1 for $x^{2}+x+4$ . What is the result? A. 12.67 B. 20.33 C. 12.50 D. 13.17

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Problem 21554

Find the power series for $g$ from $f$ by differentiation or integration. Determine the interval of convergence. $g(x)=\frac{14}{(1-14 x)^{2}}, f(x)=\frac{1}{1-14 x}$

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Problem 21555

A ball dropped from 20 m bounces to $\frac{2}{3}$ of its height. Find the total distance traveled with infinite bounces.

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Problem 21556

Find the power series for $g$ by using $f$ . Given $g(x)=\frac{14}{(1-14 x)^{2}}$ , $f(x)=\frac{1}{1-14 x}$ .

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Problem 21557

Find the power series for $g(x)=\ln(1-11x)$ centered at 0 from the options given.

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Problem 21558

Find the power series for $f(x)=\frac{1}{1+x^{11}}$ centered at 0 and its interval of convergence. Which is correct? A. $\sum_{k=0}^{\infty}(-x^{11})^{k}$ B. $\sum_{k=0}^{\infty} 11 x^{k}$ C. $\sum_{k=0}^{\infty}(-11 x)^{k}$ D. $\sum_{k=0}^{\infty} x^{11 k}$

See Solution

Problem 21559

How long to triple \$15,000 at 1.75\% continuous interest?

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Problem 21560

Find the interval of convergence for the series: $\sum_{k=1}^{\infty} \frac{(-1)^{k}(x-5)^{2 k+8}}{(k+4) !}$ .

See Solution

Problem 21561

Find $f(x+h)$ and the simplified difference quotient for $f(x)=x^{2}-4x$ .

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Problem 21562

Find the distance fallen after $t$ seconds using $D(t)=16t^{2}$ . Evaluate $D(2)$ , $D(5)$ , and the average rate from 2 to 5.

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Problem 21563

Soit la fonction $f(x)=\sin(2x)+1$ sur $[0, \frac{\pi}{2}]$ . Calculez $f'(x)$ , $f''(x)$ , et tracez le graphe.

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Problem 21564

Find the distance fallen after $t$ seconds using $D(t)=16 t^{2}$ . Calculate $D(2)$ , $D(5)$ , and average rate from 2 to 5.

See Solution

Problem 21565

Find the derivative of the function $y(x) = (x e^{x})^{3}$ .

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Problem 21566

Concert tickets sold out in 9 hours. Given $L(t)$ values, find:
(a) Estimate $L'(5.5)$ . Show work and units. (b) Minimum times $L'(t) = 0$ for $0 \leq t \leq 9$ ? Explain. (c) Is there a time in $[1,4]$ where $L'(t) = -10$ ? Justify.

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Problem 21567

Find the indefinite integral $\int\left(x^{2}+2\right)^{9} 2 x \, dx$ . Use $u=x^{2}+2$ .

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Problem 21568

1. Find the max revenue for $p(x)=2800-70x$ , then sketch $R(x)=xp(x)$ and find break-even points for $C(x)=420x+4480$ .

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Problem 21569

A bacteria colony grows as $B(t)=85 e^{0.049 t}$ . Find initial size, growth rate, size after 9 days, time to 137g, and doubling time.

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Problem 21570

Find the limit as $x$ approaches 1 for the expression $\frac{x-1}{\sqrt{x}-1}$ .

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Problem 21571

Soit $g(x)=x-\cos (x)$ sur $[0,2 \pi]$ . Trouvez $g^{\prime}(x)$ , $g^{\prime \prime}(x)$ , et analysez la fonction et son graphe.

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Problem 21572

A charge $q_{1} = 2.0 \times 10^{-6} \mathrm{C}$ is fixed. Find the speed of charge $q_{2} = 8.0 \times 10^{-6} \mathrm{C}$ at $0.50 \mathrm{m}$ .

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Problem 21573

Find the values of $x$ for which the series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{3^{n} \cdot n}(x-1)^{n}$ converges.

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Problem 21574

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{1}{x+1}$ , where $h \neq 0$ , and simplify.

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Problem 21575

Find the derivative of $y = [f(x) + 3g(x)] g(x)$ .

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Problem 21576

Differentiate the function $h(t)=\frac{t^{2}-3 t+2}{t+5}$ with respect to $t$ .

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Problem 21577

Find the tangent line of $y=f(x)$ at $x=1$ for $f(x)=\frac{4}{x}-\frac{2}{\sqrt{x}}+\frac{3}{x^{2}}$ .

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Problem 21578

Find the indefinite integral: $\int \sqrt[8]{x^{9}} d x$ (Use $C$ for the constant of integration.)

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Problem 21579

Hill's function models oxygen binding to hemoglobin: $f(P)=\frac{P^{n}}{k^{n}+P^{n}}$ . Find $f^{\prime}(P)$ and show $f^{\prime}(P)>0$ for $P>0$ .

See Solution

Problem 21580

Find the indefinite integral and include the constant $C$ . Calculate $\int(u+1)(5u+4) du$ .

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Problem 21581

Evaluate the integral from 0 to 2 of $(2x - 6)(4x^2 + 2)$ .

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Problem 21582

Differentiate the function $f(x)=\frac{5}{(7-5 x^{2})^{4}}$ with respect to $x$ .

See Solution

Problem 21583

Find the derivative $f'(P)$ of Hill's function $f(P)=\frac{P^{n}}{k^{n}+P^{n}}$ and show $f'(P)>0$ for $P>0$ .

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Problem 21584

Given the acceleration $a(t)=2t+4$ and initial velocity $v(0)=-5$ , find $v(t)$ and the distance traveled from $t=0$ to $t=3$ .

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Problem 21585

Find the velocity, speed, and acceleration of $\overrightarrow{\mathrm{r}}(t)=t^{2} \hat{\mathrm{i}}+1/t^{2} \hat{\mathrm{j}}$ at $t=1$ .

See Solution

Problem 21586

Differentiate the function $f(x)=\frac{3}{(1-2 x^{2})^{4}}$ with respect to $x$ .

See Solution

Problem 21587

Evaluate the integral using the substitution $u=2x$ : $\int \cos(2x) \, dx$ . Use $C$ for the constant of integration.

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Problem 21588

Find the function $f(x)$ given that $f^{\prime}(x)=\frac{1}{x}+3$ and $f(e)=5$ .

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Problem 21589

Evaluate the integral using the substitution $u=\sin(\theta)$ . Include constant $C$ . $\int \sin^{2}(\theta) \cos(\theta) d\theta$

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Problem 21590

Find the general antiderivative of $x^{6}$ .

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Problem 21591

Evaluate the integral: $\int(7-5 x)^{6} d x$ (Use $C$ for the constant of integration.)

See Solution

Problem 21592

Evaluate the integral from 0 to 1: $\int_{0}^{1} x^{5} e^{-x^{6}} d x$ .

See Solution

Problem 21593

Evaluate the integral from 9 to 10: $\int_{9}^{10} x \sqrt{x-9} \, dx$ .

See Solution

Problem 21594

Chewing frequency $c$ is related to body mass $M(t)=1+2\sqrt{t}$ by $c=kM^{-0.128}$ . Find $\frac{d c}{d t}$ .

See Solution

Problem 21595

Evaluate the integral: $\int e^{x} \sqrt{7+e^{x}} \, dx + C$ .

See Solution

Problem 21596

Evaluate the integral: $\int \frac{d x}{2 x+5}$ , using absolute values and $C$ for the constant of integration.

See Solution

Problem 21597

Druzinsky (1993) found that chewing frequency $c=kM^{-0.128}$ . Given $M(t)=1+2\sqrt{t}$ , find $\frac{d c}{d t}$ . Also, $L=rM^{0.312}$ , find $\frac{d c}{d L}$ .

See Solution

Problem 21598

Calculate the integral: $\int(4 x^{11}-7 x^{2}+4) \, dx$

See Solution

Problem 21599

A ball is thrown horizontally from a 92.0 m cliff at 19.8 m/s. Find (a) horizontal distance and (b) speed before impact.

See Solution

Problem 21600

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=-2 x^{2}+6 x-7$ , where $h \neq 0$ .

See Solution

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Calculus

Problem 21501

Find the turning points of the curve defined by x=3tx=3tx=3t and y=12t−t3y=12t-t^3y=12t−t3.

Problem 21502

Find the absolute min and max of f(t)=t25−t2f(t) = t \sqrt{25 - t^2}f(t)=t25−t2​ on the interval [−1,5][-1, 5][−1,5].

Problem 21503

Find the absolute maximum of f(x)=x3−x2−x+1f(x)=x^{3}-x^{2}-x+1f(x)=x3−x2−x+1 on [0,2][0,2][0,2].

Problem 21504

Problem 21505

Approximate the area under f(x)=2x+3f(x)=2x+3f(x)=2x+3 from x=0x=0x=0 to x=2x=2x=2 using n=4n=4n=4 right rectangles. What is R4R_{4}R4​? A. 15 B. 11 C. 17 D. 13

Problem 21506

Problem 21507

Select a possible graph of the twice-differentiable function y=f(x)y=f(x)y=f(x) based on these derivative properties.

Problem 21508

Determine where the function y=x4+8x3−72x2+4y=x^{4}+8 x^{3}-72 x^{2}+4y=x4+8x3−72x2+4 is concave down. Choose the correct interval.

Problem 21509

A train's position is s(t)=130ts(t)=\frac{130}{t}s(t)=t130​ for 1≤t≤61 \leq t \leq 61≤t≤6. (a) Graph s(t)s(t)s(t). (b) Calculate average velocity from t=1t=1t=1 to t=6t=6t=6: □\square□ (round to the nearest tenth).

Problem 21510

Approximate the area under f(x)=2x+3f(x)=2x+3f(x)=2x+3 from x=0x=0x=0 to x=2x=2x=2 using n=4n=4n=4 right rectangles. Compute R4R_{4}R4​.

Problem 21511

Find the critical numbers of the function f(θ)=18cos⁡θ+9sin⁡2θf(\theta) = 18 \cos \theta + 9 \sin^2 \thetaf(θ)=18cosθ+9sin2θ.

Problem 21512

Calculate the integral: ∫(2x5−7x3+9) dx\int(2 x^{5}-7 x^{3}+9) \, dx∫(2x5−7x3+9)dx. Choose the correct answer from the options provided.

Problem 21513

Find the rate of radius increase of a balloon inflating at 800 cm³/min when the radius is 30 cm. V=43πr3V=\frac{4}{3} \pi r^{3}V=34​πr3

Problem 21514

Find dy dx\frac{\mathrm{d} y}{\mathrm{~d} x} dxdy​ for x=cos⁡2t,y=sin⁡2tx=\cos 2 t, y=\sin ^{2} tx=cos2t,y=sin2t and the Cartesian equation with its domain.

Problem 21515

Calculate the average rate of change of f(x)=−2x2+10f(x)=-2x^{2}+10f(x)=−2x2+10 between x=3x=3x=3 and x=8x=8x=8. Simplify your result.

Problem 21516

Calculate the integral: ∫9x−5dx\int 9 x^{-5} dx∫9x−5dx. Choose the correct answer from the options provided.

Problem 21517

Evaluate the series ∑n=1∞cos⁡(nπ)\sum_{n=1}^{\infty} \cos (n \pi)∑n=1∞​cos(nπ): find SkS_{k}Sk​, check convergence/divergence, and use a test.

Problem 21518

Problem 21519

A Midwest town's population dropped from 215,000 to 211,000 in 4 years. Find the population after another 4 years.

Problem 21520

Find the derivative dydx\frac{d y}{d x}dxdy​ for the equation e2x=sin⁡(x+7y)e^{2 x}=\sin (x+7 y)e2x=sin(x+7y).

Problem 21521

Calculate the average value of y=2x4y=2 x^{4}y=2x4 from x=−2x=-2x=−2 to x=2x=2x=2. Options: A. 0 B. 1285\frac{128}{5}5128​ C. 165\frac{16}{5}516​ D. 325\frac{32}{5}532​

Problem 21522

Evaluate the integral: ∫01xe−x2dx\int_{0}^{1} x e^{-x^{2}} dx∫01​xe−x2dx and ∫0e−12eln⁡(e2−1)e2−12e2ee(e2−1)2\int_{0}^{\frac{e-1}{2 e}} \frac{\ln(e^{2}-1)}{\frac{e^{2}-1}{2 e^{2}}} e^{\frac{e(e^{2}-1)}{2}}∫02ee−1​​2e2e2−1​ln(e2−1)​e2e(e2−1)​.

Problem 21523

Evaluate the integral from 2 to 5: ∫253xdx=□\int_{2}^{5} 3 x d x=\square∫25​3xdx=□ (Type an integer or a simplified fraction.)

Problem 21524

Evaluate the integral from 0 to 1: ∫01xe−x2dx\int_{0}^{1} x e^{-x^{2}} d x∫01​xe−x2dx.

Problem 21525

Find the radius of convergence for the series ∑(4x)k\sum(4 x)^{k}∑(4x)k and test endpoints for the interval of convergence.

Problem 21526

Find the indefinite integral: ∫sin⁡2xcos⁡x dx\int \sin^{2} x \cos x \, dx∫sin2xcosxdx. Use C\mathrm{C}C for the constant.

Problem 21527

Find the indefinite integral: ∫6x dx=□\int 6 x \, dx = \square∫6xdx=□

Problem 21528

Evaluate the integral from 1 to 2: ∫12(2x+6)dx=□(\int_{1}^{2}(2 x+6) d x = \square(∫12​(2x+6)dx=□( Simplify your answer.)

Problem 21529

Strontium-90 decays as A(t)=A0e−0.0244tA(t)=A_{0} e^{-0.0244 t}A(t)=A0​e−0.0244t. Given 400g, find decay rate, amount after 40 years, time for 100g, and half-life.

Problem 21530

Find the radius and interval of convergence for the series ∑k=0∞(3x−5)kk!\sum_{k=0}^{\infty} \frac{(3 x-5)^{k}}{k !}∑k=0∞​k!(3x−5)k​. Radius R=□R=\squareR=□.

Problem 21531

Evaluate the integral from 0 to b: ∫0b8exdx\int_{0}^{b} 8 e^{x} dx∫0b​8exdx. What is the result?

Problem 21532

Strontium-90 decays as A(t)=A0e−0.0244tA(t)=A_{0} e^{-0.0244 t}A(t)=A0​e−0.0244t. Given 400g, find: (a) decay rate, (b) amount after 40 years, (c) time for 100g left, (d) half-life.

Problem 21533

Find the average rate of change of height h(t)=−3cos⁡t+8h(t)=-3 \cos t+8h(t)=−3cost+8 from t=0t=0t=0 to t=3t=3t=3 seconds. Show your work.

Problem 21534

Evaluate the integral from 0 to 16 of 3x dx3 \sqrt{x} \, dx3x​dx. What is the result? A. 288 B. 192 C. 128 D. 24

Problem 21535

Calculate the integral ∫(sec⁡(y)tan⁡(y)−csc⁡(y)cot⁡(y))dy\int(\sec (y) \tan (y)-\csc (y) \cot (y)) d y∫(sec(y)tan(y)−csc(y)cot(y))dy and verify by differentiating. Use CCC for the constant.

Problem 21536

Calculate the indefinite integral and use CCC for the constant: ∫x(1−7x2)4dx\int x(1-7 x^{2})^{4} dx∫x(1−7x2)4dx

Problem 21537

Calculate the integral of 5(t2−5t−2) dt5(t^{2}-5t-2) \, dt5(t2−5t−2)dt. What is the result?

Problem 21538

Find the radius of convergence for the series ∑(3kx)k\sum(3 k x)^{k}∑(3kx)k and test endpoints for the interval of convergence.

Problem 21539

Find the derivative of the function defined by (x)=(2x+7)(x2−3)(x)=(2x+7)(x^2-3)(x)=(2x+7)(x2−3).

Problem 21540

Find the tangent line equation for the function f(x)=−9x4+13x2−4f(x)=-9 x^{4}+13 x^{2}-4f(x)=−9x4+13x2−4 at the point (1,0)(1,0)(1,0). y=y= y=

Problem 21541

Find the radius and interval of convergence for the series ∑k=1∞sin⁡k(65k4)xk\sum_{k=1}^{\infty} \sin^{k}\left(\frac{6}{5 k^{4}}\right) x^{k}∑k=1∞​sink(5k46​)xk.

Find the turning points of the curve defined by $x=3t$ and $y=12t-t^3$ .

Find the absolute min and max of $f(t) = t \sqrt{25 - t^2}$ on the interval $[-1, 5]$ .

Find the absolute maximum of $f(x)=x^{3}-x^{2}-x+1$ on $[0,2]$ .

Approximate the area under $f(x)=2x+3$ from $x=0$ to $x=2$ using $n=4$ right rectangles. What is $R_{4}$ ? A. 15 B. 11 C. 17 D. 13

Select a possible graph of the twice-differentiable function $y=f(x)$ based on these derivative properties.

Determine where the function $y=x^{4}+8 x^{3}-72 x^{2}+4$ is concave down. Choose the correct interval.

A train's position is $s(t)=\frac{130}{t}$ for $1 \leq t \leq 6$ .
(a) Graph $s(t)$ . (b) Calculate average velocity from $t=1$ to $t=6$ : $\square$ (round to the nearest tenth).

Approximate the area under $f(x)=2x+3$ from $x=0$ to $x=2$ using $n=4$ right rectangles. Compute $R_{4}$ .

Find the critical numbers of the function $f(\theta) = 18 \cos \theta + 9 \sin^2 \theta$ .

Calculate the integral: $\int(2 x^{5}-7 x^{3}+9) \, dx$ . Choose the correct answer from the options provided.

Find the rate of radius increase of a balloon inflating at 800 cm³/min when the radius is 30 cm. $V=\frac{4}{3} \pi r^{3}$

Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ for $x=\cos 2 t, y=\sin ^{2} t$ and the Cartesian equation with its domain.

Calculate the average rate of change of $f(x)=-2x^{2}+10$ between $x=3$ and $x=8$ . Simplify your result.

Calculate the integral: $\int 9 x^{-5} dx$ . Choose the correct answer from the options provided.

Evaluate the series $\sum_{n=1}^{\infty} \cos (n \pi)$ : find $S_{k}$ , check convergence/divergence, and use a test.

Find the derivative $\frac{d y}{d x}$ for the equation $e^{2 x}=\sin (x+7 y)$ .

Calculate the average value of $y=2 x^{4}$ from $x=-2$ to $x=2$ . Options: A. 0 B. $\frac{128}{5}$ C. $\frac{16}{5}$ D. $\frac{32}{5}$

Evaluate the integral: $\int_{0}^{1} x e^{-x^{2}} dx$ and $\int_{0}^{\frac{e-1}{2 e}} \frac{\ln(e^{2}-1)}{\frac{e^{2}-1}{2 e^{2}}} e^{\frac{e(e^{2}-1)}{2}}$ .

Evaluate the integral from 2 to 5: $\int_{2}^{5} 3 x d x=\square$ (Type an integer or a simplified fraction.)

Evaluate the integral from 0 to 1: $\int_{0}^{1} x e^{-x^{2}} d x$ .

Find the radius of convergence for the series $\sum(4 x)^{k}$ and test endpoints for the interval of convergence.

Find the indefinite integral: $\int \sin^{2} x \cos x \, dx$ . Use $\mathrm{C}$ for the constant.

Find the indefinite integral: $\int 6 x \, dx = \square$

Evaluate the integral from 1 to 2: $\int_{1}^{2}(2 x+6) d x = \square($ Simplify your answer.)

Strontium-90 decays as $A(t)=A_{0} e^{-0.0244 t}$ . Given 400g, find decay rate, amount after 40 years, time for 100g, and half-life.

Find the radius and interval of convergence for the series $\sum_{k=0}^{\infty} \frac{(3 x-5)^{k}}{k !}$ . Radius $R=\square$ .

Evaluate the integral from 0 to b: $\int_{0}^{b} 8 e^{x} dx$ . What is the result?

Strontium-90 decays as $A(t)=A_{0} e^{-0.0244 t}$ . Given 400g, find: (a) decay rate, (b) amount after 40 years, (c) time for 100g left, (d) half-life.

Find the average rate of change of height $h(t)=-3 \cos t+8$ from $t=0$ to $t=3$ seconds. Show your work.

Evaluate the integral from 0 to 16 of $3 \sqrt{x} \, dx$ . What is the result? A. 288 B. 192 C. 128 D. 24

Calculate the integral $\int(\sec (y) \tan (y)-\csc (y) \cot (y)) d y$ and verify by differentiating. Use $C$ for the constant.

Calculate the indefinite integral and use $C$ for the constant: $\int x(1-7 x^{2})^{4} dx$

Calculate the integral of $5(t^{2}-5t-2) \, dt$ . What is the result?

Find the radius of convergence for the series $\sum(3 k x)^{k}$ and test endpoints for the interval of convergence.

Find the derivative of the function defined by $(x)=(2x+7)(x^2-3)$ .

Find the tangent line equation for the function $f(x)=-9 x^{4}+13 x^{2}-4$ at the point $(1,0)$ . $y=$

Find the radius and interval of convergence for the series $\sum_{k=1}^{\infty} \sin^{k}\left(\frac{6}{5 k^{4}}\right) x^{k}$ .

Find the radius and interval of convergence for the series $\sum_{k=4}^{\infty} \frac{4^{k}(x-2)^{k}}{k}$ . Radius: $R=\square$ .

Calculate the average value of $y=2 x^{4}$ from $x=-2$ to $x=2$ . Options: A. $\frac{16}{5}$ B. 0 C. $\frac{32}{5}$ D. $\frac{128}{5}$

Calculate the integral: $\int(5+x^{3})(4-x^{2}) dx$ . Choose the correct answer from the options provided.

Find where the function $h(x)=-x^{3}+4x^{2}$ is increasing or decreasing and identify local extreme values.

Calculate the integral of $5 e^{x}-\frac{1}{x}$ with respect to $x$ .

Evaluate the integral from -1 to 1 of $(3 x^{2}-8 x)$ . What is the result? A. 2 B. 12 C. 7 D. -7

Find the radius and interval of convergence for the series $\sum_{k=0}^{\infty}\left(\frac{x}{3}\right)^{k}$ . Radius: $R=\square$ .

Evaluate the integral from 0 to b: $\int_{0}^{b} 9 x^{8} d x$ . What is the result? A. $\frac{1}{9} b^{9}$ B. $b^{7}$ C. $9 b^{9}$ D. $b^{9}$

Calculate the one-sided limit: $\lim _{x \rightarrow-6^{+}} \frac{1}{x+6}$ . If it doesn't exist, write DNE.

Find the slope of $y=(6x+2)^{2}$ at the point $(0,4)$ . Calculate $y'(0)$ .

Find the radius and interval of convergence for the series $\sum_{k=3}^{\infty} \frac{3^{k}(x-5)^{k}}{k}$ .

Evaluate the integral from -3 to -1 for $x^{2}+x+4$ . What is the result? A. 12.67 B. 20.33 C. 12.50 D. 13.17

Find the power series for $g$ from $f$ by differentiation or integration. Determine the interval of convergence. $g(x)=\frac{14}{(1-14 x)^{2}}, f(x)=\frac{1}{1-14 x}$

A ball dropped from 20 m bounces to $\frac{2}{3}$ of its height. Find the total distance traveled with infinite bounces.

Find the power series for $g$ by using $f$ . Given $g(x)=\frac{14}{(1-14 x)^{2}}$ , $f(x)=\frac{1}{1-14 x}$ .

Find the power series for $g(x)=\ln(1-11x)$ centered at 0 from the options given.

Find the interval of convergence for the series: $\sum_{k=1}^{\infty} \frac{(-1)^{k}(x-5)^{2 k+8}}{(k+4) !}$ .

Find $f(x+h)$ and the simplified difference quotient for $f(x)=x^{2}-4x$ .

Find the distance fallen after $t$ seconds using $D(t)=16t^{2}$ . Evaluate $D(2)$ , $D(5)$ , and the average rate from 2 to 5.

Soit la fonction $f(x)=\sin(2x)+1$ sur $[0, \frac{\pi}{2}]$ . Calculez $f'(x)$ , $f''(x)$ , et tracez le graphe.

Find the distance fallen after $t$ seconds using $D(t)=16 t^{2}$ . Calculate $D(2)$ , $D(5)$ , and average rate from 2 to 5.

Find the derivative of the function $y(x) = (x e^{x})^{3}$ .

Find the indefinite integral $\int\left(x^{2}+2\right)^{9} 2 x \, dx$ . Use $u=x^{2}+2$ .

1. Find the max revenue for $p(x)=2800-70x$ , then sketch $R(x)=xp(x)$ and find break-even points for $C(x)=420x+4480$ .

A bacteria colony grows as $B(t)=85 e^{0.049 t}$ . Find initial size, growth rate, size after 9 days, time to 137g, and doubling time.

Find the limit as $x$ approaches 1 for the expression $\frac{x-1}{\sqrt{x}-1}$ .

Soit $g(x)=x-\cos (x)$ sur $[0,2 \pi]$ . Trouvez $g^{\prime}(x)$ , $g^{\prime \prime}(x)$ , et analysez la fonction et son graphe.

A charge $q_{1} = 2.0 \times 10^{-6} \mathrm{C}$ is fixed. Find the speed of charge $q_{2} = 8.0 \times 10^{-6} \mathrm{C}$ at $0.50 \mathrm{m}$ .

Find the values of $x$ for which the series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{3^{n} \cdot n}(x-1)^{n}$ converges.

Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{1}{x+1}$ , where $h \neq 0$ , and simplify.