Calculus

Problem 25401

Find the derivative $P^{\prime}(t)$ of $P(t)=170+15t-t^{2}$ , and calculate $P(5)$ and $P^{\prime}(5)$ .

See Solution

Problem 25402

Differentiate $g(N)=r N\left(1-\frac{N}{K}\right)$ with respect to $N$ .

See Solution

Problem 25403

Utilisez la diffërentielle pour approximer $\cos 91^{\circ}$ .

See Solution

Problem 25404

Find the tangent line equation for $h(x)=x \sqrt{4+2 x}$ at $x=6$ .

See Solution

Problem 25405

Find the maximum height and time to reach it for $h(t)=-4.9 t^{2}+24 t+8$ .

See Solution

Problem 25406

Find the area between $C'(t)=5$ and $R'(t)=9 e^{-0.4 t}$ from $t=0$ to the game's useful life. What is the useful life? $t=\square$ years.

See Solution

Problem 25407

Find the derivative $f^{\prime}(x)$ for the function $f(x)=x^{2} \sin ^{5}(2x)$ .

See Solution

Problem 25408

Find $f^{\prime}(2)$ if $f(x)=(h(x))^{2}$ , with $g(2)=4$ , $g^{\prime}(2)=-3$ , $h(2)=-2$ , and $h^{\prime}(2)=5$ .

See Solution

Problem 25409

Given $f(-1)=1$ and $\frac{d y}{d x}=-x^{2}-x y+y^{2}-1$ , which statement about $f$ at $x=-1$ is true? (A), (B), (C), or (D)?

See Solution

Problem 25410

Differentiate $R(T)=\frac{2 \pi^{5}}{19} \frac{k^{7}}{c^{5} h^{3}} T^{6}$ with constants $k, c, h > 0$ .

See Solution

Problem 25411

A metal pan at $145^{\circ} \mathrm{F}$ is in a freezer at $0^{\circ} \mathrm{F}$ . Find its temp after 15 min using $T(t)=T_0+(T_1-T_0)e^{-kt}$ .

See Solution

Problem 25412

Determine if the series $-\frac{20}{7}+\frac{80}{49}-\frac{320}{343}+\frac{1280}{2401} \cdots$ converges or diverges. If it converges, find the sum.

See Solution

Problem 25413

Determine if the series $-\frac{20}{7}+\frac{80}{49}-\frac{320}{343}+\frac{1280}{2401} \ldots$ converges or diverges. If it converges, find the sum.

See Solution

Problem 25414

Find the values of $p$ where demand is elastic for $x = 2704 - 4p^2$ using $PED = \frac{dQ}{dP} \times \frac{P}{Q}$ .

See Solution

Problem 25415

Determine if the integral $\int_{0}^{+\infty} e^{-x^{2}} d x$ is convergent or divergent.

See Solution

Problem 25416

Find the tangent line to $f(x)=a x^{4}$ at $x=1$ , where $a$ is a positive constant.

See Solution

Problem 25417

Find the percentage rate of change of $f(x)=5700-3x^2$ at $x=25$ . Answer in the form $\square \%$ .

See Solution

Problem 25418

Find the tangent line to $f(x)=a x^{4}$ at $x=1$ , where $a$ is a positive constant.

See Solution

Problem 25419

Find the relative rate of change of $f(x) = 6x^2 - \ln x$ at $x = 4$ . Answer: $\square$ .

See Solution

Problem 25420

Find the function $f$ such that $\int_{0}^{x} f(t) dt = 3 \cos x + 9x - 3$ . What is $f(x)$ ? $f(x) = \square$

See Solution

Problem 25421

Find the elasticity of demand $E(p)$ using the price-demand equation $x=f(p)=17,500-325 p$ .

See Solution

Problem 25422

Find the derivative $\frac{d y}{d x}$ for the equation $\tan y=5 x^{2}+2$ .

See Solution

Problem 25423

Find the radius and interval of convergence for the series $\sum_{n=1}^{\infty}\left[\frac{2 \cdot 4 \cdot 6 \cdots 2 n}{3 \cdot 5 \cdot 7 \cdots(2 n+1)}\right] x^{2 n+1}$ .

See Solution

Problem 25424

Find the elasticity of demand $E(p)$ using the price-demand equation $x=f(p)=4800-3 p^{2}$ .

See Solution

Problem 25425

Find $\frac{d y}{d x}$ when $x=4$ for the equation $x^{2}+2xy-8=0$ . A) $\frac{3}{2}$ B) $-\frac{3}{2}$ C) $-\frac{3}{4}$ D) $-\frac{5}{4}$

See Solution

Problem 25426

Find points where $f(x)=\frac{\pi}{8} \sin x$ equals its average value on $[0, \pi]$ . Answer: $x=\square$ .

See Solution

Problem 25427

Find points on the curve $y=x^{3}+3x+3$ where the tangent line is parallel to $5x-y=-2$ .

See Solution

Problem 25428

Soit $f(x) = \begin{cases} 3x^2 - 1 & \text{si } x < 1 \\ \frac{7x^2 + 1}{x} & \text{si } x \geq 1 \end{cases}$ .
a) Trouver $f'(x)$ pour $x \neq 1$ .
b) Calculer $\lim_{x \to 1^{+}} f'(x)$ et $\lim_{x \to 1^{-}} f'(x)$ .
c) Vérifier si $f$ est dérivable en $x=1$ .
d) Identifier les nombres critiques de $f$ .

See Solution

Problem 25429

Find the limit as $x$ approaches $-\infty$ for $\frac{5-7x}{9x+8}$ .

See Solution

Problem 25430

Evaluate the integral $\int 2 x\left(x^{2}+7\right)^{3} d x$ using the substitution $u=x^{2}+7$ . Rewrite in terms of $u$ .

See Solution

Problem 25431

Find the new limits of integration for $u=x^{2}+9$ in $\int_{3}^{9} f(x) d x$ . New lower limit is $\square$ .

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Problem 25432

Find the relative rate of change for robberies in 1999 using $r(t)=10.5-3.3 \ln t$ . Round to the nearest hundredth.

See Solution

Problem 25433

Evaluate the integral using $u=x^{2}+5$ : $\int 2 x\left(x^{2}+5\right)^{5} d x=\int(\square) d u$

See Solution

Problem 25434

Find the derivative: $\frac{d}{d x}\left(\int_{1}^{6 x^{2}-1} \frac{\ln t}{\sqrt{t^{2}+1}} d t\right)$ .

See Solution

Problem 25435

Find the relative max and min of $f(x, y) = x^{2} + y^{2} + 16x - 12y$ .

See Solution

Problem 25436

Evaluate the integral $\int 2 x\left(x^{2}+5\right)^{5} d x$ using the substitution $u=x^{2}+5$ . What is the result?

See Solution

Problem 25437

Find the limit: $\lim _{x \rightarrow 25} \frac{x-25}{\sqrt{x}-5}$ . Simplify your answer.

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Problem 25438

A business plans to buy equipment generating \ $13,000/year for 7 years at a 6.6\% continuous interest rate. Find the present value. Future value: \$\square$ (round to nearest dollar).

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Problem 25439

Find $p$ values for elastic and inelastic demand using $x=f(p)=\sqrt{624-8p}$ . Demand is inelastic for $p$ in the interval $\square$ .

See Solution

Problem 25440

Determine the relative max and min of the function $f(x, y) = 6x^{2} - 7y^{2}$ .

See Solution

Problem 25441

Find the relative max and min of the function $f(x, y) = e^{8x^2 + 2y^2 + 6}$ .

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Problem 25442

Calculate the producers' surplus at price $\bar{p}=\$ 70$ for the supply equation $p=S(x)=15+0.1x+0.0003x^2$ .

See Solution

Problem 25443

Evaluate the integral: $\int 2 x\left(x^{2}-7\right)^{102} d x=\square$

See Solution

Problem 25444

Calculate the integral: $\int \frac{4 x^{2}}{\sqrt{9-16 x^{3}}} d x=\square$

See Solution

Problem 25445

Evaluate the integral $\int \sqrt{6 x+5} \, dx$ using the substitution $u=ax+b$ .

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Problem 25446

Find an antiderivative of $\sin 20 x$ and verify by differentiating. What is $\int \sin 20 x \, dx = \square$ ?

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Problem 25447

Rewrite the integral as $\int(\square) du$ using the substitution $u = x^{2} - 7$ .

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Problem 25448

Find the limit: $\lim _{x \rightarrow \infty} \frac{-5}{2^{x^{2}+1}}$ .

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Problem 25449

Find the position of a particle with acceleration $a(t)=6t-4$ , given $v(3)=25$ and $x(1)=8$ . What is $x(t)$ ?

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Problem 25450

Find the integral of $\frac{1}{(2x-1)^{2}}$ with respect to $x$ .

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Problem 25451

Find the max population density from $P(x,y) = -20x^2 - 20y^2 + 440x + 280y + 200$ . Where does it occur?

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Problem 25452

Calculate the integral: $\int x^{7}(x^{8}+4)^{4} \, dx$

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Problem 25453

Find the integral of $\int \cos x \csc ^{7} x \, dx$ .

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Problem 25454

Find the integral: $\int \sec 8w \tan 8w \, dw = \square$

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Problem 25455

Find the partial derivatives $\frac{\partial z}{\partial x}$ , $\frac{\partial z}{\partial y}$ , and evaluate them at $(-3,4)$ for $z=-5 x^{3}+y^{2}+7 x y$ .

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Problem 25456

Calculate the integral $\int_{0}^{\pi / 6} \cos 9 x \, dx = \square$ (Provide the exact answer.)

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Problem 25457

La fonction $k(x)$ a-t-elle un point de rebroussement ou anguleux ? Justifiez votre réponse.

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Problem 25458

Evaluate the integral: $\int_{2}^{3} \frac{2 x}{\left(x^{2}+1\right)^{3}} d x=\square$

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Problem 25459

Find the partial derivatives $f_{x}$ and $f_{y}$ for $f(x, y)=y \ln (3 x+8 y)$ .

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Problem 25460

Find $f_{x}, f_{y}, f_{x}(3,-3)$ , and $f_{y}(-2,-1)$ for $f(x, y)=\sqrt{x^{2}+y^{2}}$ .

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Problem 25461

Calculate the integral $\int 3 \sin^{2} x \cos x \, dx = \square$ . What is the exact answer?

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Problem 25462

Find the partial derivatives $f_{x}$ and $f_{y}$ for the function $f(x, y)=3 x \ln(xy)$ .

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Problem 25463

Rewrite the integral using $u$ : $\int_{2}^{3} \frac{2 x}{(x^{2}+1)^{3}} dx = \int_{5}^{\square}(\square) du$

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Problem 25464

Find the limit: $\lim _{x \rightarrow-2} e^{\left(\frac{x^{2}-4}{x^{2}+7 x+10}\right)}$

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Problem 25465

Find all second order derivatives for $r(x, y)=\frac{xy}{7x+6y}$ . What is $r_{xx}(x, y)=\square$ ?

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Problem 25466

Find the local max and min of $f(x)=x^{3}-27x$ using the first derivative test. What is $f^{\prime}(x)$ ? $f^{\prime}(x)=\square$

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Problem 25467

Find the partial derivatives $\frac{\partial z}{\partial x}$ , $\frac{\partial z}{\partial y}$ , and evaluate them at $(3,0)$ for $z=-8 x^{3}-6 y^{2}-7 x y$ .

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Problem 25468

Find the point $a$ , value $f(a)$ , and slope of $f(x)=x^{2}$ at $(a, f(a))$ given the tangent line $y=4x-4$ .

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Problem 25469

Find the slope of the curve $y=x^{3}$ at the point $(4,64)$ . The slope is $\square$ .

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Problem 25470

Find the relative maximum and minimum points of the function $f(x)=-x^{3}-3 x^{2}+3$ using the first-derivative test.

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Problem 25471

Find the marginal profit $\frac{d P}{d n}$ at $n=25$ given $P(q)=500 \sqrt{q}+10 q$ and $q(n)=10,000 n-100 n^{2}-n^{3}$ .

See Solution

Problem 25472

Find $\frac{dy}{dx}$ for the equation $5x + 5 = \ln(xy)$ .

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Problem 25473

A stone is thrown up; what is its acceleration when its velocity is zero at the top? Answer using $a = -g$ , where $g$ is gravity.

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Problem 25474

Find $\frac{d y}{d x}$ for the equation $5 x^{2}+2 y^{3}=4$ .

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Problem 25475

How much work (in Joules) is needed to pump water from a circular pool (diameter $14 \mathrm{~m}$ , depth $3 \mathrm{~m}$ )? Use $g = 9.8 \frac{\mathrm{m}}{\mathrm{s}^{2}}$ and water density $1000 \frac{\mathrm{kg}}{\mathrm{m}^{3}}$ .

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Problem 25476

Find dimensions that maximize the printed area of a poster with total area $19440 \mathrm{~cm}^{2}$ and margins of $10 \mathrm{~cm}$ (top/bottom) and $6 \mathrm{~cm}$ (sides).

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Problem 25477

Estimate the integral $\int_{0}^{1} 5 e^{-x^{2}} d x$ using Simpson's Rule with $\dot{n}=4$ . Round to three decimals.

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Problem 25478

Solve the integral: $\int \frac{x^{2}}{(x^{3}+3)^{3}} dx$ .

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Problem 25479

Estimate the integral $\int_{0}^{1} \sqrt{2+x^{2}} \, dx$ using Simpson's Rule with $n=4$ , rounding to three decimal places.

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Problem 25480

Evaluate the line integral $\int_{C} x \, ds$ for the curve $\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}$ , $0 \leq t \leq 1$ .

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Problem 25481

Find the average value $f_{\text{ave}}$ of $f(x)=\sqrt{16-x^{2}}$ from 0 to 4, then find $c$ where $f(c)=f_{\text{ave}}$ .

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Problem 25482

Find the limits for the following expressions: 9. $\lim _{x \rightarrow 2} 16$ , 10. $\lim _{x \rightarrow 3} 2 x$ , 11. $\lim _{x \rightarrow-5}(t^{2}-5)$ , 12. $\lim _{t \rightarrow 1 / 3}(5 t-7)$ , 13. $\lim _{x \rightarrow-1}(x^{3}-3 x^{2}-2 x+1)$ , 14. $\lim _{r \rightarrow 9} \frac{4 r-3}{11}$ , 15. $\lim _{t \rightarrow-3} \frac{t-2}{t+5}$ , 16. $\lim _{x \rightarrow-6} \frac{x^{2}+6}{x-6}$ , 17. $\lim _{h \rightarrow 0} \frac{h}{h^{2}-7 h+1}$ , 18. $\lim _{h \rightarrow 0} \frac{h^{3}-3 h^{2}-4}{h^{2}+1}$ , 19. $\lim _{p \rightarrow 4} \sqrt{p^{2}+p+5}$ , 20. $\lim _{y \rightarrow 15} \sqrt{y+3}$ , 21. $\lim _{x \rightarrow-2} \frac{x^{2}+2 x}{x+2}$ , 22. $\lim _{x \rightarrow-1} \frac{x+1}{x+1}$ , 23. $\lim _{x \rightarrow 2} \frac{x^{2}-x-2}{x-2}$ , 24. $\lim _{t \rightarrow 0} \frac{t^{3}+3 t^{2}}{t^{3}-4 t^{2}}$ , 25. $\lim _{x \rightarrow-1} \frac{x^{2}+2 x+1}{x+1}$ , 26. $\lim _{t \rightarrow 2} \frac{t^{2}-4}{t-2}$ , 27. $\lim _{x \rightarrow 3} \frac{x-3}{x^{2}-9}$ , 28. $\lim _{x \rightarrow 0} \frac{x^{2}-2 x}{x}$ , 29. $\lim _{x \rightarrow 4} \frac{x^{2}-9 x+20}{x^{2}-3 x-4}$ , 30. $\lim _{x \rightarrow-2} \frac{x^{4}-16}{x^{2}-x-6}$ , 31. $\lim _{x \rightarrow 2} \frac{3 x^{2}-x-10}{x^{2}+5 x-14}$ , 32. $\lim _{x \rightarrow-4} \frac{x^{2}+2 x-8}{x^{2}+5 x+4}$ , 33. $\lim _{h \rightarrow 0} \frac{(2+h)^{2}-2^{2}}{h}$ , 34. $\lim _{x \rightarrow 0} \frac{(x+2)^{2}-4}{x}$ .

See Solution

Problem 25483

Differentiate $y=7^{4x+2}$ with respect to $x$ .

See Solution

Problem 25484

Find the derivative of $y=(f(x)+15 g(x)) g(x)$ with respect to $x$ .

See Solution

Problem 25485

Differentiate with respect to $x$ :
32. $y=7^{4x+2}$
33. $y=\ln(10-6x^{3})$

See Solution

Problem 25486

Differentiate $y=7^{4x+2}$ with respect to $x$ .

See Solution

Problem 25487

Solve the equation $y'' - 2y' - 3y = -9t + 12$ with $y(0) = 1$ and $y'(0) = 16$ . Find $y(t)$ .

See Solution

Problem 25488

Differentiate $y=7^{4x+2}$ with respect to $x$ .

See Solution

Problem 25489

Calculate the integral $\int_{0}^{\frac{\pi}{2}} \tan ^{3} x \, dx$ .

See Solution

Problem 25490

Evaluate the Riemann sum of $f(x)$ from $[-2,4]$ using 3 equal-width rectangles and midpoints.

See Solution

Problem 25491

Find the derivative of $y=(2 x)^{\ln (6 x)}$ using logarithmic differentiation.

See Solution

Problem 25492

Differentiate $y=\ln(10-6x^3)$ with respect to $x$ .

See Solution

Problem 25493

Find the limit: $\lim _{x \rightarrow 4} \frac{x^{2}-9 x+20}{x^{2}-3 x-4}$

See Solution

Problem 25494

Find the limits: 27. $\lim _{x \rightarrow 3} \frac{x-3}{x^{2}-9}$ , 29. $\lim _{x \rightarrow 4} \frac{x^{2}-9 x+20}{x^{2}-3 x-4}$ , 31. $\lim _{x \rightarrow 2} \frac{3 x^{2}-x-10}{x^{2}+5 x-14}$ , 33. $\lim _{h \rightarrow 0} \frac{(2+h)^{2}-2^{2}}{h}$ .

See Solution

Problem 25495

Find the limits:
25. $\lim _{x \rightarrow-1} \frac{x^{2}+2 x+1}{x+1}$
27. $\lim _{x \rightarrow 3} \frac{x-3}{x^{2}-9}$
29. $\lim _{x \rightarrow 4} \frac{x^{2}-9 x+20}{x^{2}-3 x-4}$
31. $\lim _{x \rightarrow 2} \frac{3 x^{2}-x-10}{x^{2}+5 x-14}$
33. $\lim _{h \rightarrow 0} \frac{(2+h)^{2}-2^{2}}{h}$

See Solution

Problem 25496

Evaluate the integral from 0 to $\sqrt{\ln (5)}$ of $x e^{x^{2}}$ dx. What is the result?

See Solution

Problem 25497

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

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Problem 25498

Find $c$ where $f'(c)=0$ for $f(x)=e^{1-x^{2}}$ and check for local extremum. Also, find the tangent line to $f(x)=a x^{4}$ at $x=1$ .

See Solution

Problem 25499

Differentiate the function $f(x)=\frac{1}{(x^{4}+1)^{3}}$ .

See Solution

Problem 25500

Find limits as $h \rightarrow 0$ for given functions and evaluate $\lim _{x \rightarrow 6} \frac{\sqrt{x-2}-2}{x-6}$ .

See Solution

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Calculus

Problem 25401

Find the derivative P′(t)P^{\prime}(t)P′(t) of P(t)=170+15t−t2P(t)=170+15t-t^{2}P(t)=170+15t−t2, and calculate P(5)P(5)P(5) and P′(5)P^{\prime}(5)P′(5).

Problem 25402

Differentiate g(N)=rN(1−NK)g(N)=r N\left(1-\frac{N}{K}\right)g(N)=rN(1−KN​) with respect to NNN.

Problem 25403

Utilisez la diffërentielle pour approximer cos⁡91∘\cos 91^{\circ}cos91∘.

Problem 25404

Find the tangent line equation for h(x)=x4+2xh(x)=x \sqrt{4+2 x}h(x)=x4+2x​ at x=6x=6x=6.

Problem 25405

Find the maximum height and time to reach it for h(t)=−4.9t2+24t+8h(t)=-4.9 t^{2}+24 t+8h(t)=−4.9t2+24t+8.

Problem 25406

Find the area between C′(t)=5C'(t)=5C′(t)=5 and R′(t)=9e−0.4tR'(t)=9 e^{-0.4 t}R′(t)=9e−0.4t from t=0t=0t=0 to the game's useful life. What is the useful life? t=□t=\squaret=□ years.

Problem 25407

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=x2sin⁡5(2x)f(x)=x^{2} \sin ^{5}(2x)f(x)=x2sin5(2x).

Problem 25408

Find f′(2)f^{\prime}(2)f′(2) if f(x)=(h(x))2f(x)=(h(x))^{2}f(x)=(h(x))2, with g(2)=4g(2)=4g(2)=4, g′(2)=−3g^{\prime}(2)=-3g′(2)=−3, h(2)=−2h(2)=-2h(2)=−2, and h′(2)=5h^{\prime}(2)=5h′(2)=5.

Problem 25409

Given f(−1)=1f(-1)=1f(−1)=1 and dydx=−x2−xy+y2−1\frac{d y}{d x}=-x^{2}-x y+y^{2}-1dxdy​=−x2−xy+y2−1, which statement about fff at x=−1x=-1x=−1 is true? (A), (B), (C), or (D)?

Problem 25410

Differentiate R(T)=2π519k7c5h3T6R(T)=\frac{2 \pi^{5}}{19} \frac{k^{7}}{c^{5} h^{3}} T^{6}R(T)=192π5​c5h3k7​T6 with constants k,c,h>0k, c, h > 0k,c,h>0.

Problem 25411

A metal pan at 145∘F145^{\circ} \mathrm{F}145∘F is in a freezer at 0∘F0^{\circ} \mathrm{F}0∘F. Find its temp after 15 min using T(t)=T0+(T1−T0)e−ktT(t)=T_0+(T_1-T_0)e^{-kt}T(t)=T0​+(T1​−T0​)e−kt.

Problem 25412

Determine if the series −207+8049−320343+12802401⋯-\frac{20}{7}+\frac{80}{49}-\frac{320}{343}+\frac{1280}{2401} \cdots−720​+4980​−343320​+24011280​⋯ converges or diverges. If it converges, find the sum.

Problem 25413

Determine if the series −207+8049−320343+12802401…-\frac{20}{7}+\frac{80}{49}-\frac{320}{343}+\frac{1280}{2401} \ldots−720​+4980​−343320​+24011280​… converges or diverges. If it converges, find the sum.

Problem 25414

Find the values of p p p where demand is elastic for x=2704−4p2 x = 2704 - 4p^2 x=2704−4p2 using PED=dQdP×PQ PED = \frac{dQ}{dP} \times \frac{P}{Q} PED=dPdQ​×QP​.

Problem 25415

Determine if the integral ∫0+∞e−x2dx\int_{0}^{+\infty} e^{-x^{2}} d x∫0+∞​e−x2dx is convergent or divergent.

Problem 25416

Find the tangent line to f(x)=ax4f(x)=a x^{4}f(x)=ax4 at x=1x=1x=1, where aaa is a positive constant.

Problem 25417

Find the percentage rate of change of f(x)=5700−3x2f(x)=5700-3x^2f(x)=5700−3x2 at x=25x=25x=25. Answer in the form □%\square \%□%.

Problem 25418

Find the tangent line to f(x)=ax4f(x)=a x^{4}f(x)=ax4 at x=1x=1x=1, where aaa is a positive constant.

Problem 25419

Find the relative rate of change of f(x)=6x2−ln⁡xf(x) = 6x^2 - \ln xf(x)=6x2−lnx at x=4x = 4x=4. Answer: □\square□.

Problem 25420

Find the function fff such that ∫0xf(t)dt=3cos⁡x+9x−3\int_{0}^{x} f(t) dt = 3 \cos x + 9x - 3∫0x​f(t)dt=3cosx+9x−3. What is f(x)f(x)f(x)? f(x)=□ f(x) = \square f(x)=□

Problem 25421

Find the elasticity of demand E(p)E(p)E(p) using the price-demand equation x=f(p)=17,500−325px=f(p)=17,500-325 px=f(p)=17,500−325p.

Problem 25422

Find the derivative dydx\frac{d y}{d x}dxdy​ for the equation tan⁡y=5x2+2\tan y=5 x^{2}+2tany=5x2+2.

Problem 25423

Find the radius and interval of convergence for the series ∑n=1∞[2⋅4⋅6⋯2n3⋅5⋅7⋯(2n+1)]x2n+1\sum_{n=1}^{\infty}\left[\frac{2 \cdot 4 \cdot 6 \cdots 2 n}{3 \cdot 5 \cdot 7 \cdots(2 n+1)}\right] x^{2 n+1}n=1∑∞​[3⋅5⋅7⋯(2n+1)2⋅4⋅6⋯2n​]x2n+1.

Problem 25424

Find the elasticity of demand E(p)E(p)E(p) using the price-demand equation x=f(p)=4800−3p2x=f(p)=4800-3 p^{2}x=f(p)=4800−3p2.

Problem 25425

Find dydx\frac{d y}{d x}dxdy​ when x=4x=4x=4 for the equation x2+2xy−8=0x^{2}+2xy-8=0x2+2xy−8=0. A) 32\frac{3}{2}23​ B) −32-\frac{3}{2}−23​ C) −34-\frac{3}{4}−43​ D) −54-\frac{5}{4}−45​

Problem 25426

Find points where f(x)=π8sin⁡xf(x)=\frac{\pi}{8} \sin xf(x)=8π​sinx equals its average value on [0,π][0, \pi][0,π]. Answer: x=□x=\squarex=□.

Problem 25427

Find points on the curve y=x3+3x+3y=x^{3}+3x+3y=x3+3x+3 where the tangent line is parallel to 5x−y=−25x-y=-25x−y=−2.

Problem 25428

Problem 25429

Find the limit as xxx approaches −∞-\infty−∞ for 5−7x9x+8\frac{5-7x}{9x+8}9x+85−7x​.

Problem 25430

Evaluate the integral ∫2x(x2+7)3dx\int 2 x\left(x^{2}+7\right)^{3} d x∫2x(x2+7)3dx using the substitution u=x2+7u=x^{2}+7u=x2+7. Rewrite in terms of uuu.

Problem 25431

Find the new limits of integration for u=x2+9u=x^{2}+9u=x2+9 in ∫39f(x)dx\int_{3}^{9} f(x) d x∫39​f(x)dx. New lower limit is □\square□.

Problem 25432

Find the relative rate of change for robberies in 1999 using r(t)=10.5−3.3ln⁡tr(t)=10.5-3.3 \ln tr(t)=10.5−3.3lnt. Round to the nearest hundredth.

Problem 25433

Evaluate the integral using u=x2+5u=x^{2}+5u=x2+5: ∫2x(x2+5)5dx=∫(□)du\int 2 x\left(x^{2}+5\right)^{5} d x=\int(\square) d u∫2x(x2+5)5dx=∫(□)du

Problem 25434

Find the derivative: ddx(∫16x2−1ln⁡tt2+1dt)\frac{d}{d x}\left(\int_{1}^{6 x^{2}-1} \frac{\ln t}{\sqrt{t^{2}+1}} d t\right)dxd​(∫16x2−1​t2+1​lnt​dt).

Problem 25435

Find the relative max and min of f(x,y)=x2+y2+16x−12yf(x, y) = x^{2} + y^{2} + 16x - 12yf(x,y)=x2+y2+16x−12y.

Problem 25436

Evaluate the integral ∫2x(x2+5)5dx\int 2 x\left(x^{2}+5\right)^{5} d x∫2x(x2+5)5dx using the substitution u=x2+5u=x^{2}+5u=x2+5. What is the result?

Problem 25437

Find the limit: lim⁡x→25x−25x−5\lim _{x \rightarrow 25} \frac{x-25}{\sqrt{x}-5}limx→25​x​−5x−25​. Simplify your answer.

Problem 25438

Problem 25439

Find ppp values for elastic and inelastic demand using x=f(p)=624−8px=f(p)=\sqrt{624-8p}x=f(p)=624−8p​. Demand is inelastic for ppp in the interval □\square□.

Problem 25440

Determine the relative max and min of the function f(x,y)=6x2−7y2f(x, y) = 6x^{2} - 7y^{2}f(x,y)=6x2−7y2.

Problem 25441

Find the derivative $P^{\prime}(t)$ of $P(t)=170+15t-t^{2}$ , and calculate $P(5)$ and $P^{\prime}(5)$ .

Differentiate $g(N)=r N\left(1-\frac{N}{K}\right)$ with respect to $N$ .

Utilisez la diffërentielle pour approximer $\cos 91^{\circ}$ .

Find the tangent line equation for $h(x)=x \sqrt{4+2 x}$ at $x=6$ .

Find the maximum height and time to reach it for $h(t)=-4.9 t^{2}+24 t+8$ .

Find the area between $C'(t)=5$ and $R'(t)=9 e^{-0.4 t}$ from $t=0$ to the game's useful life. What is the useful life? $t=\square$ years.

Find the derivative $f^{\prime}(x)$ for the function $f(x)=x^{2} \sin ^{5}(2x)$ .

Find $f^{\prime}(2)$ if $f(x)=(h(x))^{2}$ , with $g(2)=4$ , $g^{\prime}(2)=-3$ , $h(2)=-2$ , and $h^{\prime}(2)=5$ .

Given $f(-1)=1$ and $\frac{d y}{d x}=-x^{2}-x y+y^{2}-1$ , which statement about $f$ at $x=-1$ is true? (A), (B), (C), or (D)?

Differentiate $R(T)=\frac{2 \pi^{5}}{19} \frac{k^{7}}{c^{5} h^{3}} T^{6}$ with constants $k, c, h > 0$ .

A metal pan at $145^{\circ} \mathrm{F}$ is in a freezer at $0^{\circ} \mathrm{F}$ . Find its temp after 15 min using $T(t)=T_0+(T_1-T_0)e^{-kt}$ .

Determine if the series $-\frac{20}{7}+\frac{80}{49}-\frac{320}{343}+\frac{1280}{2401} \cdots$ converges or diverges. If it converges, find the sum.

Determine if the series $-\frac{20}{7}+\frac{80}{49}-\frac{320}{343}+\frac{1280}{2401} \ldots$ converges or diverges. If it converges, find the sum.

Find the values of $p$ where demand is elastic for $x = 2704 - 4p^2$ using $PED = \frac{dQ}{dP} \times \frac{P}{Q}$ .

Determine if the integral $\int_{0}^{+\infty} e^{-x^{2}} d x$ is convergent or divergent.

Find the tangent line to $f(x)=a x^{4}$ at $x=1$ , where $a$ is a positive constant.

Find the percentage rate of change of $f(x)=5700-3x^2$ at $x=25$ . Answer in the form $\square \%$ .

Find the tangent line to $f(x)=a x^{4}$ at $x=1$ , where $a$ is a positive constant.

Find the relative rate of change of $f(x) = 6x^2 - \ln x$ at $x = 4$ . Answer: $\square$ .

Find the function $f$ such that $\int_{0}^{x} f(t) dt = 3 \cos x + 9x - 3$ . What is $f(x)$ ? $f(x) = \square$

Find the elasticity of demand $E(p)$ using the price-demand equation $x=f(p)=17,500-325 p$ .

Find the derivative $\frac{d y}{d x}$ for the equation $\tan y=5 x^{2}+2$ .

Find the radius and interval of convergence for the series $\sum_{n=1}^{\infty}\left[\frac{2 \cdot 4 \cdot 6 \cdots 2 n}{3 \cdot 5 \cdot 7 \cdots(2 n+1)}\right] x^{2 n+1}$ .

Find the elasticity of demand $E(p)$ using the price-demand equation $x=f(p)=4800-3 p^{2}$ .

Find $\frac{d y}{d x}$ when $x=4$ for the equation $x^{2}+2xy-8=0$ . A) $\frac{3}{2}$ B) $-\frac{3}{2}$ C) $-\frac{3}{4}$ D) $-\frac{5}{4}$

Find points where $f(x)=\frac{\pi}{8} \sin x$ equals its average value on $[0, \pi]$ . Answer: $x=\square$ .

Find points on the curve $y=x^{3}+3x+3$ where the tangent line is parallel to $5x-y=-2$ .

Find the limit as $x$ approaches $-\infty$ for $\frac{5-7x}{9x+8}$ .

Evaluate the integral $\int 2 x\left(x^{2}+7\right)^{3} d x$ using the substitution $u=x^{2}+7$ . Rewrite in terms of $u$ .

Find the new limits of integration for $u=x^{2}+9$ in $\int_{3}^{9} f(x) d x$ . New lower limit is $\square$ .

Find the relative rate of change for robberies in 1999 using $r(t)=10.5-3.3 \ln t$ . Round to the nearest hundredth.

Evaluate the integral using $u=x^{2}+5$ : $\int 2 x\left(x^{2}+5\right)^{5} d x=\int(\square) d u$

Find the derivative: $\frac{d}{d x}\left(\int_{1}^{6 x^{2}-1} \frac{\ln t}{\sqrt{t^{2}+1}} d t\right)$ .

Find the relative max and min of $f(x, y) = x^{2} + y^{2} + 16x - 12y$ .

Evaluate the integral $\int 2 x\left(x^{2}+5\right)^{5} d x$ using the substitution $u=x^{2}+5$ . What is the result?

Find the limit: $\lim _{x \rightarrow 25} \frac{x-25}{\sqrt{x}-5}$ . Simplify your answer.

Find $p$ values for elastic and inelastic demand using $x=f(p)=\sqrt{624-8p}$ . Demand is inelastic for $p$ in the interval $\square$ .

Determine the relative max and min of the function $f(x, y) = 6x^{2} - 7y^{2}$ .

Find the relative max and min of the function $f(x, y) = e^{8x^2 + 2y^2 + 6}$ .

Calculate the producers' surplus at price $\bar{p}=\$ 70$ for the supply equation $p=S(x)=15+0.1x+0.0003x^2$ .

Evaluate the integral: $\int 2 x\left(x^{2}-7\right)^{102} d x=\square$

Calculate the integral: $\int \frac{4 x^{2}}{\sqrt{9-16 x^{3}}} d x=\square$

Evaluate the integral $\int \sqrt{6 x+5} \, dx$ using the substitution $u=ax+b$ .

Find an antiderivative of $\sin 20 x$ and verify by differentiating. What is $\int \sin 20 x \, dx = \square$ ?

Rewrite the integral as $\int(\square) du$ using the substitution $u = x^{2} - 7$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{-5}{2^{x^{2}+1}}$ .

Find the position of a particle with acceleration $a(t)=6t-4$ , given $v(3)=25$ and $x(1)=8$ . What is $x(t)$ ?

Find the integral of $\frac{1}{(2x-1)^{2}}$ with respect to $x$ .

Find the max population density from $P(x,y) = -20x^2 - 20y^2 + 440x + 280y + 200$ . Where does it occur?

Calculate the integral: $\int x^{7}(x^{8}+4)^{4} \, dx$

Find the integral of $\int \cos x \csc ^{7} x \, dx$ .

Find the integral: $\int \sec 8w \tan 8w \, dw = \square$

Find the partial derivatives $\frac{\partial z}{\partial x}$ , $\frac{\partial z}{\partial y}$ , and evaluate them at $(-3,4)$ for $z=-5 x^{3}+y^{2}+7 x y$ .

Calculate the integral $\int_{0}^{\pi / 6} \cos 9 x \, dx = \square$ (Provide the exact answer.)

La fonction $k(x)$ a-t-elle un point de rebroussement ou anguleux ? Justifiez votre réponse.

Evaluate the integral: $\int_{2}^{3} \frac{2 x}{\left(x^{2}+1\right)^{3}} d x=\square$

Find the partial derivatives $f_{x}$ and $f_{y}$ for $f(x, y)=y \ln (3 x+8 y)$ .

Find $f_{x}, f_{y}, f_{x}(3,-3)$ , and $f_{y}(-2,-1)$ for $f(x, y)=\sqrt{x^{2}+y^{2}}$ .

Calculate the integral $\int 3 \sin^{2} x \cos x \, dx = \square$ . What is the exact answer?

Find the partial derivatives $f_{x}$ and $f_{y}$ for the function $f(x, y)=3 x \ln(xy)$ .

Rewrite the integral using $u$ : $\int_{2}^{3} \frac{2 x}{(x^{2}+1)^{3}} dx = \int_{5}^{\square}(\square) du$

Find the limit: $\lim _{x \rightarrow-2} e^{\left(\frac{x^{2}-4}{x^{2}+7 x+10}\right)}$

Find all second order derivatives for $r(x, y)=\frac{xy}{7x+6y}$ . What is $r_{xx}(x, y)=\square$ ?

Find the local max and min of $f(x)=x^{3}-27x$ using the first derivative test. What is $f^{\prime}(x)$ ? $f^{\prime}(x)=\square$

Find the partial derivatives $\frac{\partial z}{\partial x}$ , $\frac{\partial z}{\partial y}$ , and evaluate them at $(3,0)$ for $z=-8 x^{3}-6 y^{2}-7 x y$ .

Find the point $a$ , value $f(a)$ , and slope of $f(x)=x^{2}$ at $(a, f(a))$ given the tangent line $y=4x-4$ .

Find the slope of the curve $y=x^{3}$ at the point $(4,64)$ . The slope is $\square$ .

Find the relative maximum and minimum points of the function $f(x)=-x^{3}-3 x^{2}+3$ using the first-derivative test.

Find the marginal profit $\frac{d P}{d n}$ at $n=25$ given $P(q)=500 \sqrt{q}+10 q$ and $q(n)=10,000 n-100 n^{2}-n^{3}$ .

Find $\frac{dy}{dx}$ for the equation $5x + 5 = \ln(xy)$ .

A stone is thrown up; what is its acceleration when its velocity is zero at the top? Answer using $a = -g$ , where $g$ is gravity.

Find $\frac{d y}{d x}$ for the equation $5 x^{2}+2 y^{3}=4$ .