Calculus

Problem 22501

Find the slope of the tangent line for $g(x)=4 \sin ^{3} x$ at $x=\frac{\pi}{4}$ .

See Solution

Problem 22502

A scientist has 100 grams of radium (half-life 1690 years). What is the average amount after 900 years? Answer: $\square$ grams.

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Problem 22503

A scientist places 50 grams of radium (half-life 1690 years) in a vault. What is the average radium amount in 1400 years? Average: $\square$ grams (round to two decimal places).

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Problem 22504

Résoudre l'intégrale $\int x^{4} e^{-\frac{1}{2} x} d x$ par intégration par parties, puis trouver l'aire sous $y=x^{4} e^{-\frac{1}{2} x}$ sur $[-1, 1]$ .

See Solution

Problem 22505

Approximate the volume change of a cylinder with radius $r=22 \mathrm{~cm}$ as height drops from $h=13$ cm to $h=12.9$ cm using $V(h)=\pi r^2 h$ . Find $\Delta V \approx \square \mathrm{cm}^{3}$ in terms of $\pi$ .

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Problem 22506

Find the dimensions of a rectangular tank with a square base and volume $10,976 \mathrm{ft}^{3}$ that minimizes surface area.

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Problem 22507

Solve the differential equation: $\frac{d y}{d x}=\frac{2 x}{10 y}$ .

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Problem 22508

Find the Taylor polynomials $p_{1}, p_{2}, p_{3}, p_{4}$ at $a=0$ for $f(x)=3 \cos(-3x)$ .

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Problem 22509

Find the value of $\int_{1}^{4}\left[3 g(x) - 4 f(x) - \sqrt{9 - (x - 1)^{2}}\right] \, dx$ given the integrals.

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Problem 22510

Find $g^{\prime}(8)$ given $g(x)=f^{-1}(x)$ , $f(6)=8$ , $f(8)=2$ , $f^{\prime}(6)=-3$ , and $f^{\prime}(8)=4$ .

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Problem 22511

Given $f(x)=6 x^{2}$ , find the linearization $L(x)$ at $a=4$ , approximate $6(4.1)^{2}$ , and find the difference from the actual value.

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Problem 22512

Find the linear and quadratic approximating polynomials for $f(x)=e^{-x}$ at $a=0$ and estimate $e^{-0.2}$ .

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Problem 22513

Evaluate the following integrals exactly, using substitution for c) and d): a) $\int(2 x+5)^{2}+\frac{3}{x} d x$ b) $\int_{1}^{5} \frac{3}{x^{2}} d x$ c) $\int_{0}^{3} 3 x(4 x^{2}-5)^{3} d x$ d) $\int \frac{e^{x}}{e^{x}-3} d x$

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Problem 22514

Given $f(x)=6 x^{2}$ , find the linearization $L(x)$ at $a=4$ , approximate $6(4.1)^{2}$ , and find the difference from the actual value.

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Problem 22515

Approximate the change in pressure from $z=7 \mathrm{~km}$ to $z=7.03 \mathrm{~km}$ using $P(z)=1000 e^{-\frac{z}{10}}$ . Answer: $\square$ .

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Problem 22516

Use the Ratio Test to check if the series $\sum_{k=1}^{\infty} \frac{5 k^{2}}{4^{k}}$ converges or diverges. Choose A, B, or C and fill in $r=\square$ .

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Problem 22517

Find the second derivative $\frac{d^{2}y}{dx^{2}}$ given $5y^{2} + 3 = x^{2}$ .

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Problem 22518

Determine the half-life of a radioactive element with decay model $A(t)=A_{0} e^{-0.0274 t}$ .

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Problem 22519

(a) Find dimensions of a pen using 1500 m of fencing for max area against a barn. (b) Minimize fence for 4 pens of 100 m² each.

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Problem 22520

Calculate the doubling time for an investment with continuous compounding at $6\%$ interest.

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Problem 22521

How long to triple an investment with continuous compounding at $5\%$ interest?

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Problem 22522

Find the time for \$ 7,000 to grow to \$ 21,000 at a 5\% continuous compound interest. Round to the nearest tenth of a year.

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Problem 22523

Find $f(-2)$ for the continuous function $f(x)=\frac{x^{3}+8}{x+2}$ when $x \neq -2$ .

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Problem 22524

Estimate the volume change of a cylinder with radius $r=21 \mathrm{~cm}$ as height changes from $h=13 \mathrm{~cm}$ to $h=12.9 \mathrm{~cm}$ . Use $\Delta V \approx \square \mathrm{cm}^{3}$ in terms of $\pi$ .

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Problem 22525

Find $\frac{d^{2} y}{d x^{2}}$ for the function $y=e^{x^{4}}$ .

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Problem 22526

Find the number of years for electricity consumption to double with a continuous growth rate of $10\%$ .

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Problem 22527

Find the years until electricity consumption doubles with a continuous increase of $7\%$ per year.

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Problem 22528

Coût de fabrication de $x$ bicyclettes et $y$ tricycles : $C(x, y)=20,000+60x+20y+50\sqrt{xy}$ . Trouvez les coûts marginaux et moyens.

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Problem 22529

Approximate the change in atmospheric pressure as altitude rises from $z=6 \mathrm{~km}$ to $z=6.02 \mathrm{~km}$ using $P(z)=1000 e^{-\frac{z}{10}}$ . The change is approximately $\square$ .

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Problem 22530

Find the slope of the tangent to the curve $f(x)=(4x+7)^{\frac{1}{3}}$ at $x=5$ , rounded to 1 decimal place.

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Problem 22531

The radius of an oil slick expands at $2 \mathrm{~m/min}$ . Find the area increase rate when radius is $25 \mathrm{~m}$ and after $3 \mathrm{~min}$ .

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Problem 22532

Déterminez le coût marginal pour fabriquer une bicyclette, donné par $C(x, y)=20,000+60x+20y+50\sqrt{xy}$ .

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Problem 22533

Find the maximum number of flu-infected students modeled by $P(t)=\frac{360}{1+39 e^{-0.3 t}}$ .

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Problem 22534

Let $f(x)=5-x^{2}$ and $a=1$ . Are linear approximations near $a$ under or over estimates? Compute $f^{\prime \prime}(1)$ .

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Problem 22535

Given $f(x)=5-x^{2}$ and $a=1$ , find the linear approximation $L(x)$ at $a$ . Also, compare $L$ and $f$ on a graph. Are approximations under or over?

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Problem 22536

Find the second derivative $f^{\prime \prime}(-1)$ for the function $f(x)=x^{3}+\frac{5}{x}$ .

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Problem 22537

Let $f(x)=5-x^{2}$ and $L(x)=6-2x$ . Graph both functions and determine if linear approximations at $a=1$ are under/overestimates. Compute $f''(1)$ .

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Problem 22538

The line $x+y=5$ is tangent to $y=f(x)$ at $x=2$ . Find $f(2)$ and $f'(2)$ .

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Problem 22539

Déterminez le coût marginal d'un tricycle à partir de la fonction de coût $C(x, y)=20,000+60x+20y+50\sqrt{xy}$ .

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Problem 22540

(a) Find dimensions for a pen with $500 \mathrm{~m}$ of fencing to maximize area. Sides perpendicular: $\square \mathrm{m}$ , parallel: $\square \mathrm{m}$ .
(b) For 4 adjacent pens each with area $400 \mathrm{~m}^{2}$ , find dimensions to minimize fencing. Perpendicular sides: $\square \mathrm{m}$ , parallel sides: $\square \mathrm{m}$ .

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Problem 22541

Find $\frac{d^{2} y}{d x^{2}}$ at the point (2,3) for the equation $x^{2}+y^{2}=13$ .

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Problem 22542

Find the integral $\int x \cdot \sqrt{5 x^{2}-4} \, dx$ and simplify with $+C$ for the constant.

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Problem 22543

Find the instantaneous rate of change of $y=\frac{x-3}{x+3}$ at $x=3$ , where $x \neq -3$ .

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Problem 22544

Find the dimensions of a square-based tank with volume $5,324 \mathrm{ft}^{3}$ that minimizes surface area. Height: $\square \mathrm{ft}$ , base: $\square \mathrm{ft}$ .

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Problem 22545

Find the linearization $L(x)$ of $f(x)=6x^2$ at $a=1$ , approximate $6(1.1)^2$ , and calculate the difference from the actual value.

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Problem 22546

Find the average rate of change of $f(x)=\sqrt{x}$ from $x_1=81$ to $x_2=121$ . What is it?

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Problem 22547

Find the limit: $\lim _{x \rightarrow \infty} \frac{(\ln x)^{2}}{x}$ using L'Hospital's rule.

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Problem 22548

Coût de fabrication $C(x, y)=20,000+60x+20y+50\sqrt{xy}$ . Trouvez les coûts marginaux et les dérivées secondes.

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Problem 22549

Calculate the integral from 5 to 10 of $x^{2}$ with respect to $x$ : $\int_{5}^{10} x^{2} d x$ .

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Problem 22550

Déterminez le coût moyen marginal d'une bicyclette pour $x=5$ et $y=5$ dans $C(x, y)=20,000+60x+20y+50\sqrt{xy}$ .

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Problem 22551

How much work is needed to lift a $650-1 b$ piano and a cable weighing $0.15 \mathrm{lb} / \mathrm{ft}$ to 320 feet?

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Problem 22552

Find the object's velocity when its acceleration is zero, given $s(t)=6 \sin t+\frac{3}{2} t^{2}+8$ .

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Problem 22553

Graph and find the volume of the solid formed by revolving the area between $y=\sqrt{x}$ and $y=4$ around the y-axis.

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Problem 22554

A cube's side length increases at $3 \mathrm{~cm/s}$ . Find the volume change rate when the side length is $5 \mathrm{~cm}$ .

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Problem 22555

Find the function $F(x)$ with $F'(x)=e^{2x}+\frac{5}{\sqrt{x}}$ and $F(0)=\frac{7}{2}$ .

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Problem 22556

A sphere's radius increases at $3 \mathrm{~cm/s}$ . Find the volume change rate when the radius is $5 \mathrm{~cm}$ .

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Problem 22557

Find the derivative of $y$ where $y = \ln(2x^3 - 4x^2 + 2x - 1)$ .

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Problem 22558

Find the instantaneous rate of change of $y=\sin x+x e^{x}+6$ at $x=5$ .

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Problem 22559

Find the acceleration of an object at time $t=\frac{\pi}{2}$ , given its position $s(t)=\sin(3t)-\cos(4t)$ .

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Problem 22560

Find the change in the angle of elevation of a camera 2000 ft away from a rocket at $s(t)=50 t^{2}$ after 10 seconds.

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Problem 22561

Find the derivative of $f(\ln x)$ where $f(x)=e^{x}+x^{2}$ .

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Problem 22562

A running back starts at $3 \mathrm{~m/s}$ and accelerates at $1.5 \mathrm{~m/s}^2$ over $75 \mathrm{~m}$ . Find final velocity.

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Problem 22563

Find $f^{\prime}(e)$ given that $f(x)^{\prime}=\sin u$ , $u=v-\frac{1}{v^{2}}$ , and $v=\ln x$ .

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Problem 22564

Find the derivative of $\ln \left(2 x^{2}-4\right)^{4}$ with respect to $x$ .

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Problem 22565

Graph the function $f(x)=-2x^2-5$ on $[0,1]$ and find its average value. What is the average value?

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Problem 22566

Find the general antiderivative of the function: $\int(5 x^{14}-7 x+6) \, dx = \square$

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Problem 22567

Find the average value of the function $f(t)=(t-10)^{2}$ on the interval $[0,15]$ . Answer: av(f) $=\square$ .

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Problem 22568

Evaluate the integral $\int_{-2}^{0}(9x-x^{2})dx$ using the limit definition of a definite integral. Answer: $\square$

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Problem 22569

Evaluate the integral $\int_{-2}^{0}(9x-8x^{2})dx$ using the limit definition of integrals. Answer: $\square$ (integer or fraction).

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Problem 22570

Find the average cost function $\bar{C}(x)=\frac{C(x)}{x}$ , average cost for 100 lbs, and minimize average cost.

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Problem 22571

Calculate the average rate of change of $f(x)=5 x^{2}-4$ over the interval $[5, b]$ as a function of $b$ .

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Problem 22572

Evaluate the integral from -2 to 3: $\int_{-2}^{3} x^{3} d x=\square$ (Simplify your answer.)

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Problem 22573

Evaluate the integral $\int_{-2}^{3} x^{3} d x$ using the Riemann sum. Choose the correct option below.

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Problem 22574

Estimate the area under $f(x)=3 x^{2}$ from $x=0$ to $x=12$ using lower and upper sums with 2 and 4 rectangles.

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Problem 22575

Find the derivative of the function $h(n)=n^{3} \sqrt[3]{n^{2}}$ , i.e., calculate $h^{\prime}(n)$ .

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Problem 22576

Estimate the area under $f(x)=\frac{5}{x}$ from $x=\frac{1}{5}$ to $x=1$ using 2 and 4 rectangles (midpoint rule).

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Problem 22577

Find $\frac{d y}{d x}$ using implicit differentiation for the equation $x \cos (5 x + 4 y) = y \sin x$ .

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Problem 22578

Find the derivative of $T=\sqrt{\arccos (\phi x)}$ , i.e., calculate $T^{\prime}$ .

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Problem 22579

A logistic growth function $f(t)=\frac{114,000}{1+4200 e^{-t}}$ models flu cases. Find: a) initial cases, b) cases after 4 weeks, c) limiting population.

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Problem 22580

Find the derivative of $T=\sqrt{\arccos (4 x)}$ , i.e., calculate $T^{\prime}$ .

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Problem 22581

Find the value of $k$ for which the piecewise function $h(x)$ is continuous at $x=1$ given $\lim_{x \to 1} f(x)=2$ .

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Problem 22582

Déterminez les dimensions et l'aire du rectangle d'aire maximale inscrit entre l'axe des $x$ et $y=\frac{4}{x^{2}+4}$ .

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Problem 22583

Find the horizontal asymptote of the function $f(x)=\mathrm{e}^{2 \mathrm{x}}$ .

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Problem 22584

Estimate the distance traveled in 10 sec using left and right endpoint values of velocity: $0, 5, 11, 18, 24, 36, 22, 15, 8, 3, 0$ cm/sec.

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Problem 22585

Calculate the average of $f(x)=x^{3}$ over the interval $[-3,3]$ and plot it on the graph.

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Problem 22586

Find the derivative of $\tan(e^{-2 x})$ .

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Problem 22587

Calculate the average of $f(x)=x^{3}-x$ on the interval $[0,2]$ and plot it with the function.

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Problem 22588

Find the derivative of $(7x + 3)(x^2 - 2)^4$ with respect to $x$ .

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Problem 22589

Invest \$11,122 at 5.9% interest, compounded continuously. Find the function, balances after 1, 2, 5, 10 years, and doubling time.

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Problem 22590

Find the derivative of $e^{\sin^{2} x}$ with respect to $x$ .

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Problem 22591

Evaluate the integral $\int_{0}^{1} \frac{x+3}{x^{2}-4} d x$ using the Fundamental Theorem of Calculus.

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Problem 22592

Find points on the graph of $f(x)=x^{3}-3x^{2}+x+4$ where the tangent line's slope is 10.

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Problem 22593

Find the limit as $x$ approaches infinity for the expression $\frac{4 a x^{4}-2 b x^{3}+c x-1}{x^{4}+b-2}$ .

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Problem 22594

Bestimme die Tangentensteigung der Funktion $f(x)=\frac{1}{50} x^{2}$ bei $x=20$ . Erreicht das Fahrzeug mit $78\%$ die Markierungsstange?

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Problem 22595

Find the average rate of change of $f(x)$ from $x_{1}=-7.8$ to $x_{2}=4.9$ for $f(x)=6.4 x^{3}-2.1 x^{2}-3.8 x+9.5$ . Round to the nearest hundredth.

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Problem 22596

Find the critical numbers of $g(x)=4-\frac{2}{x}+\frac{9}{x^{2}}$ and classify each as max, min, or neither.

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Problem 22597

You start at 1 on a number line, moving to $1/2$ and then halving each step. Will you reach 0? Explain why or why not.

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Problem 22598

You're at 1 on a number line, moving to 0 by halving your step each time. Will you reach 0? Discuss your thoughts.

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Problem 22599

Find the average rate of change of $g(x)=-x^{2}-6x+14$ from $x=-4$ to $x=0$ .

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Problem 22600

Find the average rate of change of $h(x)=-x^{2}-x+5$ from $x=-6$ to $x=2$ .

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Calculus

Problem 22501

Find the slope of the tangent line for g(x)=4sin⁡3xg(x)=4 \sin ^{3} xg(x)=4sin3x at x=π4x=\frac{\pi}{4}x=4π​.

Problem 22502

A scientist has 100 grams of radium (half-life 1690 years). What is the average amount after 900 years? Answer: □\square□ grams.

Problem 22503

A scientist places 50 grams of radium (half-life 1690 years) in a vault. What is the average radium amount in 1400 years? Average: □\square□ grams (round to two decimal places).

Problem 22504

Résoudre l'intégrale ∫x4e−12xdx\int x^{4} e^{-\frac{1}{2} x} d x∫x4e−21​xdx par intégration par parties, puis trouver l'aire sous y=x4e−12xy=x^{4} e^{-\frac{1}{2} x}y=x4e−21​x sur [−1,1][-1, 1][−1,1].

Problem 22505

Approximate the volume change of a cylinder with radius r=22 cmr=22 \mathrm{~cm}r=22 cm as height drops from h=13h=13h=13 cm to h=12.9h=12.9h=12.9 cm using V(h)=πr2hV(h)=\pi r^2 hV(h)=πr2h. Find ΔV≈□cm3\Delta V \approx \square \mathrm{cm}^{3}ΔV≈□cm3 in terms of π\piπ.

Problem 22506

Find the dimensions of a rectangular tank with a square base and volume 10,976ft310,976 \mathrm{ft}^{3}10,976ft3 that minimizes surface area.

Problem 22507

Solve the differential equation: dydx=2x10y\frac{d y}{d x}=\frac{2 x}{10 y}dxdy​=10y2x​.

Problem 22508

Find the Taylor polynomials p1,p2,p3,p4p_{1}, p_{2}, p_{3}, p_{4}p1​,p2​,p3​,p4​ at a=0a=0a=0 for f(x)=3cos⁡(−3x)f(x)=3 \cos(-3x)f(x)=3cos(−3x).

Problem 22509

Find the value of ∫14[3g(x)−4f(x)−9−(x−1)2] dx\int_{1}^{4}\left[3 g(x) - 4 f(x) - \sqrt{9 - (x - 1)^{2}}\right] \, dx∫14​[3g(x)−4f(x)−9−(x−1)2​]dx given the integrals.

Problem 22510

Find g′(8)g^{\prime}(8)g′(8) given g(x)=f−1(x)g(x)=f^{-1}(x)g(x)=f−1(x), f(6)=8f(6)=8f(6)=8, f(8)=2f(8)=2f(8)=2, f′(6)=−3f^{\prime}(6)=-3f′(6)=−3, and f′(8)=4f^{\prime}(8)=4f′(8)=4.

Problem 22511

Given f(x)=6x2f(x)=6 x^{2}f(x)=6x2, find the linearization L(x)L(x)L(x) at a=4a=4a=4, approximate 6(4.1)26(4.1)^{2}6(4.1)2, and find the difference from the actual value.

Problem 22512

Find the linear and quadratic approximating polynomials for f(x)=e−xf(x)=e^{-x}f(x)=e−x at a=0a=0a=0 and estimate e−0.2e^{-0.2}e−0.2.

Problem 22513

Problem 22514

Given f(x)=6x2f(x)=6 x^{2}f(x)=6x2, find the linearization L(x)L(x)L(x) at a=4a=4a=4, approximate 6(4.1)26(4.1)^{2}6(4.1)2, and find the difference from the actual value.

Problem 22515

Approximate the change in pressure from z=7 kmz=7 \mathrm{~km}z=7 km to z=7.03 kmz=7.03 \mathrm{~km}z=7.03 km using P(z)=1000e−z10P(z)=1000 e^{-\frac{z}{10}}P(z)=1000e−10z​. Answer: □\square□.

Problem 22516

Use the Ratio Test to check if the series ∑k=1∞5k24k\sum_{k=1}^{\infty} \frac{5 k^{2}}{4^{k}}∑k=1∞​4k5k2​ converges or diverges. Choose A, B, or C and fill in r=□r=\squarer=□.

Problem 22517

Find the second derivative d2ydx2 \frac{d^{2}y}{dx^{2}} dx2d2y​ given 5y2+3=x2 5y^{2} + 3 = x^{2} 5y2+3=x2.

Problem 22518

Determine the half-life of a radioactive element with decay model A(t)=A0e−0.0274tA(t)=A_{0} e^{-0.0274 t}A(t)=A0​e−0.0274t.

Problem 22519

(a) Find dimensions of a pen using 1500 m of fencing for max area against a barn. (b) Minimize fence for 4 pens of 100 m² each.

Problem 22520

Calculate the doubling time for an investment with continuous compounding at 6%6\%6% interest.

Problem 22521

How long to triple an investment with continuous compounding at 5%5\%5% interest?

Problem 22522

Find the time for \$ 7,000 to grow to \$ 21,000 at a 5\% continuous compound interest. Round to the nearest tenth of a year.

Problem 22523

Find f(−2)f(-2)f(−2) for the continuous function f(x)=x3+8x+2f(x)=\frac{x^{3}+8}{x+2}f(x)=x+2x3+8​ when x≠−2x \neq -2x=−2.

Problem 22524

Estimate the volume change of a cylinder with radius r=21 cmr=21 \mathrm{~cm}r=21 cm as height changes from h=13 cmh=13 \mathrm{~cm}h=13 cm to h=12.9 cmh=12.9 \mathrm{~cm}h=12.9 cm. Use ΔV≈□cm3\Delta V \approx \square \mathrm{cm}^{3}ΔV≈□cm3 in terms of π\piπ.

Problem 22525

Find d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ for the function y=ex4y=e^{x^{4}}y=ex4.

Problem 22526

Find the number of years for electricity consumption to double with a continuous growth rate of 10%10\%10%.

Problem 22527

Find the years until electricity consumption doubles with a continuous increase of 7%7\%7% per year.

Problem 22528

Coût de fabrication de xxx bicyclettes et yyy tricycles : C(x,y)=20,000+60x+20y+50xyC(x, y)=20,000+60x+20y+50\sqrt{xy}C(x,y)=20,000+60x+20y+50xy​. Trouvez les coûts marginaux et moyens.

Problem 22529

Approximate the change in atmospheric pressure as altitude rises from z=6 kmz=6 \mathrm{~km}z=6 km to z=6.02 kmz=6.02 \mathrm{~km}z=6.02 km using P(z)=1000e−z10P(z)=1000 e^{-\frac{z}{10}}P(z)=1000e−10z​. The change is approximately □\square□.

Problem 22530

Find the slope of the tangent to the curve f(x)=(4x+7)13f(x)=(4x+7)^{\frac{1}{3}}f(x)=(4x+7)31​ at x=5x=5x=5, rounded to 1 decimal place.

Problem 22531

The radius of an oil slick expands at 2 m/min2 \mathrm{~m/min}2 m/min. Find the area increase rate when radius is 25 m25 \mathrm{~m}25 m and after 3 min3 \mathrm{~min}3 min.

Problem 22532

Déterminez le coût marginal pour fabriquer une bicyclette, donné par C(x,y)=20,000+60x+20y+50xyC(x, y)=20,000+60x+20y+50\sqrt{xy}C(x,y)=20,000+60x+20y+50xy​.

Problem 22533

Find the maximum number of flu-infected students modeled by P(t)=3601+39e−0.3tP(t)=\frac{360}{1+39 e^{-0.3 t}}P(t)=1+39e−0.3t360​.

Problem 22534

Let f(x)=5−x2f(x)=5-x^{2}f(x)=5−x2 and a=1a=1a=1. Are linear approximations near aaa under or over estimates? Compute f′′(1)f^{\prime \prime}(1)f′′(1).

Problem 22535

Given f(x)=5−x2f(x)=5-x^{2}f(x)=5−x2 and a=1a=1a=1, find the linear approximation L(x)L(x)L(x) at aaa. Also, compare LLL and fff on a graph. Are approximations under or over?

Problem 22536

Find the second derivative f′′(−1)f^{\prime \prime}(-1)f′′(−1) for the function f(x)=x3+5xf(x)=x^{3}+\frac{5}{x}f(x)=x3+x5​.

Problem 22537

Let f(x)=5−x2f(x)=5-x^{2}f(x)=5−x2 and L(x)=6−2xL(x)=6-2xL(x)=6−2x. Graph both functions and determine if linear approximations at a=1a=1a=1 are under/overestimates. Compute f′′(1)f''(1)f′′(1).

Problem 22538

The line x+y=5x+y=5x+y=5 is tangent to y=f(x)y=f(x)y=f(x) at x=2x=2x=2. Find f(2)f(2)f(2) and f′(2)f'(2)f′(2).

Problem 22539

Déterminez le coût marginal d'un tricycle à partir de la fonction de coût C(x,y)=20,000+60x+20y+50xyC(x, y)=20,000+60x+20y+50\sqrt{xy}C(x,y)=20,000+60x+20y+50xy​.

Problem 22540

Problem 22541

Find the slope of the tangent line for $g(x)=4 \sin ^{3} x$ at $x=\frac{\pi}{4}$ .

A scientist has 100 grams of radium (half-life 1690 years). What is the average amount after 900 years? Answer: $\square$ grams.

A scientist places 50 grams of radium (half-life 1690 years) in a vault. What is the average radium amount in 1400 years? Average: $\square$ grams (round to two decimal places).

Résoudre l'intégrale $\int x^{4} e^{-\frac{1}{2} x} d x$ par intégration par parties, puis trouver l'aire sous $y=x^{4} e^{-\frac{1}{2} x}$ sur $[-1, 1]$ .

Approximate the volume change of a cylinder with radius $r=22 \mathrm{~cm}$ as height drops from $h=13$ cm to $h=12.9$ cm using $V(h)=\pi r^2 h$ . Find $\Delta V \approx \square \mathrm{cm}^{3}$ in terms of $\pi$ .

Find the dimensions of a rectangular tank with a square base and volume $10,976 \mathrm{ft}^{3}$ that minimizes surface area.

Solve the differential equation: $\frac{d y}{d x}=\frac{2 x}{10 y}$ .

Find the Taylor polynomials $p_{1}, p_{2}, p_{3}, p_{4}$ at $a=0$ for $f(x)=3 \cos(-3x)$ .

Find the value of $\int_{1}^{4}\left[3 g(x) - 4 f(x) - \sqrt{9 - (x - 1)^{2}}\right] \, dx$ given the integrals.

Find $g^{\prime}(8)$ given $g(x)=f^{-1}(x)$ , $f(6)=8$ , $f(8)=2$ , $f^{\prime}(6)=-3$ , and $f^{\prime}(8)=4$ .

Given $f(x)=6 x^{2}$ , find the linearization $L(x)$ at $a=4$ , approximate $6(4.1)^{2}$ , and find the difference from the actual value.

Find the linear and quadratic approximating polynomials for $f(x)=e^{-x}$ at $a=0$ and estimate $e^{-0.2}$ .

Given $f(x)=6 x^{2}$ , find the linearization $L(x)$ at $a=4$ , approximate $6(4.1)^{2}$ , and find the difference from the actual value.

Approximate the change in pressure from $z=7 \mathrm{~km}$ to $z=7.03 \mathrm{~km}$ using $P(z)=1000 e^{-\frac{z}{10}}$ . Answer: $\square$ .

Use the Ratio Test to check if the series $\sum_{k=1}^{\infty} \frac{5 k^{2}}{4^{k}}$ converges or diverges. Choose A, B, or C and fill in $r=\square$ .

Find the second derivative $\frac{d^{2}y}{dx^{2}}$ given $5y^{2} + 3 = x^{2}$ .

Determine the half-life of a radioactive element with decay model $A(t)=A_{0} e^{-0.0274 t}$ .

Calculate the doubling time for an investment with continuous compounding at $6\%$ interest.

How long to triple an investment with continuous compounding at $5\%$ interest?

Find $f(-2)$ for the continuous function $f(x)=\frac{x^{3}+8}{x+2}$ when $x \neq -2$ .

Estimate the volume change of a cylinder with radius $r=21 \mathrm{~cm}$ as height changes from $h=13 \mathrm{~cm}$ to $h=12.9 \mathrm{~cm}$ . Use $\Delta V \approx \square \mathrm{cm}^{3}$ in terms of $\pi$ .

Find $\frac{d^{2} y}{d x^{2}}$ for the function $y=e^{x^{4}}$ .

Find the number of years for electricity consumption to double with a continuous growth rate of $10\%$ .

Find the years until electricity consumption doubles with a continuous increase of $7\%$ per year.

Coût de fabrication de $x$ bicyclettes et $y$ tricycles : $C(x, y)=20,000+60x+20y+50\sqrt{xy}$ . Trouvez les coûts marginaux et moyens.

Approximate the change in atmospheric pressure as altitude rises from $z=6 \mathrm{~km}$ to $z=6.02 \mathrm{~km}$ using $P(z)=1000 e^{-\frac{z}{10}}$ . The change is approximately $\square$ .

Find the slope of the tangent to the curve $f(x)=(4x+7)^{\frac{1}{3}}$ at $x=5$ , rounded to 1 decimal place.

The radius of an oil slick expands at $2 \mathrm{~m/min}$ . Find the area increase rate when radius is $25 \mathrm{~m}$ and after $3 \mathrm{~min}$ .

Déterminez le coût marginal pour fabriquer une bicyclette, donné par $C(x, y)=20,000+60x+20y+50\sqrt{xy}$ .

Find the maximum number of flu-infected students modeled by $P(t)=\frac{360}{1+39 e^{-0.3 t}}$ .

Let $f(x)=5-x^{2}$ and $a=1$ . Are linear approximations near $a$ under or over estimates? Compute $f^{\prime \prime}(1)$ .

Given $f(x)=5-x^{2}$ and $a=1$ , find the linear approximation $L(x)$ at $a$ . Also, compare $L$ and $f$ on a graph. Are approximations under or over?

Find the second derivative $f^{\prime \prime}(-1)$ for the function $f(x)=x^{3}+\frac{5}{x}$ .

Let $f(x)=5-x^{2}$ and $L(x)=6-2x$ . Graph both functions and determine if linear approximations at $a=1$ are under/overestimates. Compute $f''(1)$ .

The line $x+y=5$ is tangent to $y=f(x)$ at $x=2$ . Find $f(2)$ and $f'(2)$ .

Déterminez le coût marginal d'un tricycle à partir de la fonction de coût $C(x, y)=20,000+60x+20y+50\sqrt{xy}$ .

Find $\frac{d^{2} y}{d x^{2}}$ at the point (2,3) for the equation $x^{2}+y^{2}=13$ .

Find the integral $\int x \cdot \sqrt{5 x^{2}-4} \, dx$ and simplify with $+C$ for the constant.

Find the instantaneous rate of change of $y=\frac{x-3}{x+3}$ at $x=3$ , where $x \neq -3$ .

Find the dimensions of a square-based tank with volume $5,324 \mathrm{ft}^{3}$ that minimizes surface area. Height: $\square \mathrm{ft}$ , base: $\square \mathrm{ft}$ .

Find the linearization $L(x)$ of $f(x)=6x^2$ at $a=1$ , approximate $6(1.1)^2$ , and calculate the difference from the actual value.

Find the average rate of change of $f(x)=\sqrt{x}$ from $x_1=81$ to $x_2=121$ . What is it?

Find the limit: $\lim _{x \rightarrow \infty} \frac{(\ln x)^{2}}{x}$ using L'Hospital's rule.

Coût de fabrication $C(x, y)=20,000+60x+20y+50\sqrt{xy}$ . Trouvez les coûts marginaux et les dérivées secondes.

Calculate the integral from 5 to 10 of $x^{2}$ with respect to $x$ : $\int_{5}^{10} x^{2} d x$ .

Déterminez le coût moyen marginal d'une bicyclette pour $x=5$ et $y=5$ dans $C(x, y)=20,000+60x+20y+50\sqrt{xy}$ .

How much work is needed to lift a $650-1 b$ piano and a cable weighing $0.15 \mathrm{lb} / \mathrm{ft}$ to 320 feet?

Find the object's velocity when its acceleration is zero, given $s(t)=6 \sin t+\frac{3}{2} t^{2}+8$ .

Graph and find the volume of the solid formed by revolving the area between $y=\sqrt{x}$ and $y=4$ around the y-axis.

A cube's side length increases at $3 \mathrm{~cm/s}$ . Find the volume change rate when the side length is $5 \mathrm{~cm}$ .

Find the function $F(x)$ with $F'(x)=e^{2x}+\frac{5}{\sqrt{x}}$ and $F(0)=\frac{7}{2}$ .

A sphere's radius increases at $3 \mathrm{~cm/s}$ . Find the volume change rate when the radius is $5 \mathrm{~cm}$ .

Find the derivative of $y$ where $y = \ln(2x^3 - 4x^2 + 2x - 1)$ .

Find the instantaneous rate of change of $y=\sin x+x e^{x}+6$ at $x=5$ .

Find the acceleration of an object at time $t=\frac{\pi}{2}$ , given its position $s(t)=\sin(3t)-\cos(4t)$ .

Find the change in the angle of elevation of a camera 2000 ft away from a rocket at $s(t)=50 t^{2}$ after 10 seconds.

Find the derivative of $f(\ln x)$ where $f(x)=e^{x}+x^{2}$ .

A running back starts at $3 \mathrm{~m/s}$ and accelerates at $1.5 \mathrm{~m/s}^2$ over $75 \mathrm{~m}$ . Find final velocity.

Find $f^{\prime}(e)$ given that $f(x)^{\prime}=\sin u$ , $u=v-\frac{1}{v^{2}}$ , and $v=\ln x$ .

Find the derivative of $\ln \left(2 x^{2}-4\right)^{4}$ with respect to $x$ .

Graph the function $f(x)=-2x^2-5$ on $[0,1]$ and find its average value. What is the average value?

Find the general antiderivative of the function: $\int(5 x^{14}-7 x+6) \, dx = \square$

Find the average value of the function $f(t)=(t-10)^{2}$ on the interval $[0,15]$ . Answer: av(f) $=\square$ .

Evaluate the integral $\int_{-2}^{0}(9x-x^{2})dx$ using the limit definition of a definite integral. Answer: $\square$

Evaluate the integral $\int_{-2}^{0}(9x-8x^{2})dx$ using the limit definition of integrals. Answer: $\square$ (integer or fraction).

Find the average cost function $\bar{C}(x)=\frac{C(x)}{x}$ , average cost for 100 lbs, and minimize average cost.

Calculate the average rate of change of $f(x)=5 x^{2}-4$ over the interval $[5, b]$ as a function of $b$ .

Evaluate the integral from -2 to 3: $\int_{-2}^{3} x^{3} d x=\square$ (Simplify your answer.)

Evaluate the integral $\int_{-2}^{3} x^{3} d x$ using the Riemann sum. Choose the correct option below.

Estimate the area under $f(x)=3 x^{2}$ from $x=0$ to $x=12$ using lower and upper sums with 2 and 4 rectangles.