Calculus

Problem 19801

Find the dimensions of a box with volume $296352 \mathrm{~cm}^{3}$ that minimizes surface area.
1. Surface area formula in terms of $x$ : $A(x)=$
2. Derivative $A^{\prime}(x)$ : $A^{\prime}(x)=$
3. Solve $A^{\prime}(x)=0$ : $A^{\prime}(x)=0 \text{ when } x=$
4. Second derivative $A^{\prime \prime}(x)$ : $A^{\prime \prime}(x)=$
Evaluate $A^{\prime \prime}(x)$ at the $x$ -value found earlier.

See Solution

Problem 19802

Find the average cost of producing 500 units given the marginal cost function $\frac{d c}{d q}=0.003 q^{2}-0.6 q+90$ and fixed costs of \ $9000. Average cost at$ q=500$ is \$\square.

See Solution

Problem 19803

Evaluate the surface integral: $\iint_{S} \mathbf{v} \cdot d \mathbf{S} = \int_{0}^{\pi / 2} \int_{0}^{2 \pi} 8 \sin (\varphi) \cos ^{2}(\varphi) d \theta d \varphi$ .

See Solution

Problem 19804

Calculate the integral: $\int 14 x^{3} e^{14 x^{2}} d x$

See Solution

Problem 19805

Use the Limit Comparison Test to check if the series $\sum_{n=1}^{\infty} \frac{2 n^{1 / 5}}{\sqrt{5 n^{3}+7 n+7}}$ converges or diverges.

See Solution

Problem 19806

Calculate the surface integral $\iint_{S} \mathbf{F} \cdot d \mathbf{S}$ using the divergence theorem for $\mathbf{F}(x, y, z)=x^{4} \mathbf{i}-x^{3} z^{2} \mathbf{j}+4 x y^{2} z \mathbf{k}$ , where $S$ is bounded by $x^{2}+y^{2}=1$ , $z=x+4$ , and $z=0$ .

See Solution

Problem 19807

Find the derivative $d y$ of the function $y=e^{5 \sqrt{x}+1}$ .

See Solution

Problem 19808

Find dy for the equation y = 2x√(12 - x³).

See Solution

Problem 19809

Find the derivative $\frac{d y}{d x}$ for the function $y=\tan^{-1}(\ln(2x^3))$ .

See Solution

Problem 19810

How much longer does it take for Alexa's \ $8,000 at$ 3 \frac{5}{8} \% $continuous interest to double compared to Malik's \$8,000 at$ 3 \frac{3}{4} \%$ quarterly?

See Solution

Problem 19811

Calculate the integral: $\int_{0}^{6}\left(18 s^{2}+\frac{216(1-s)}{3}-e^{6}+1\right) d s$ .

See Solution

Problem 19812

Find the value of $a$ that makes the function $g(x)=\left\{\begin{array}{ll} a x & \text { if } x<0 \\ x^{2}-15 x & \text { if } x \geq 0 \end{array}\right.$ differentiable for all $x$ .

See Solution

Problem 19813

Calculate the integral of the function $(4t + 2)e^{t^2 + t}$ with respect to $t$ : $\int(4 t + 2)e^{t^{2}+t} dt$ .

See Solution

Problem 19814

How long to triple an investment with continuous compounding at $15\%$ ?

See Solution

Problem 19815

Evaluate the double integral: $\int_{0}^{6} \int_{0}^{6}\left(-e^{r}+r^{2}(1-s)+r s^{2}\right) d r d s$

See Solution

Problem 19816

Find the 971st derivative of $\cos x$ .

See Solution

Problem 19817

Find $\frac{d y}{d t}$ given $x^{2}+2 y^{2}+2 y=16$ , $\frac{d x}{d t}=5$ , $x=2$ , and $y=-3$ .

See Solution

Problem 19818

Evaluate the surface integral $\iint_{S} x^{2} y z d S$ for the plane $z=1+4x+2y$ over the rectangle $0 \leq x \leq 4$ , $0 \leq y \leq 2$ .

See Solution

Problem 19819

Given a fluid flow in $\mathbb{R}^{3}$ with velocity $\bar{q}(x, y, z, t)=\left\langle\sqrt{y} t^{2}, x e^{-t} \sin (y t), \cos (2 x)\right\rangle$ , find:
A. Acceleration in the Eulerian frame. B. Acceleration in the Lagrangian frame. C. Acceleration in the Eulerian frame at $(0.5, 0.5, 0.5, 0.5)$ . D. Acceleration in the Lagrangian frame at $(0.5, 0.5, 0.5, 0.5)$ . E. Lagrangian acceleration behavior as $t$ increases with $(0.5, 0.5, 0.5)$ .

See Solution

Problem 19820

Find the derivative of $y=3 x^{2}-7 x+10$ . Options: A. $6 x-7$ B. $6 x+10$ C. $-7 x$ D. $3 x+10$

See Solution

Problem 19821

Evaluate the surface integral $\iint_{S} \mathbf{F} \cdot d \mathbf{S}$ for $\mathbf{F}(x, y, z)=x^{4} \mathbf{i}-x^{3} z^{2} \mathbf{j}+4 x y^{2} z \mathbf{k}$ over the surface bounded by $x^{2}+y^{2}=1$ , $z=x+4$ , and $z=0$ .

See Solution

Problem 19822

Find the velocity and acceleration of the particle at $t=3$ sec, given $s=\sqrt{7+6t}$ .

See Solution

Problem 19823

Find the general antiderivative of $f(x)=3 x^{2}+\frac{1}{2}$ and the particular one with $F(1)=1$ .

See Solution

Problem 19824

Find the area under $f(x)=x^{2} e^{x}$ from $0$ to $6$ using right endpoints as a limit: $A=\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \left(\square_{x}\right)$ .

See Solution

Problem 19825

Find the area under $f(x)=x^{2} e^{x}$ from $0$ to $6$ using right endpoints as a limit: $A=\lim_{n \to \infty} \sum_{i=1}^{n}(\square_{x})$ .

See Solution

Problem 19826

A lighthouse is $600 \mathrm{~m}$ from shore, rotating its beam 4 times/min. Find the speed of the beam at point $Q$ ( $350 \mathrm{~m}$ from $P$ ).
Let $x$ be distance from $P$ to the beam and $\theta$ be the angle. Relate $x$ and $\theta$ .
Find the related rates equation: $\frac{d x}{d t}=\left(\square \frac{d \theta}{d t}\right.$
Calculate speed at point $Q$ and round to the nearest integer.

See Solution

Problem 19827

Find a region with area equal to the limit: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{3}{n} \sqrt{1+\frac{3 i}{n}}$ .

See Solution

Problem 19828

Find the general antiderivative of $h(x)=3 x^{5}-10 x^{4}+x^{2}+7$ .

See Solution

Problem 19829

How long will it take for a drink cooling from $182^{\circ} \mathrm{F}$ to $93^{\circ} \mathrm{F}$ at $72^{\circ} \mathrm{F}$ ?

See Solution

Problem 19830

Find the antiderivative of $v(x)=3(x+6)(2 x+1)$ and check your answer by differentiating it.

See Solution

Problem 19831

Find the sum of the series: $\sum_{i=1}^{\infty}\left(-\frac{2}{5}\right) \cdot\left(-\frac{4}{7}\right)^{i-1}$ .

See Solution

Problem 19832

Trouver la dérivée de la fonction $g(x)=\ln (1+\cos x)$ pour $x \in]-\pi, \pi[$ .

See Solution

Problem 19833

Analyze the series $S=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(n-1)}{n(n+3)}$ for absolute or conditional convergence.

See Solution

Problem 19834

Find integers $b$ such that the series $\sum_{n=1}^{\infty} \frac{(-1)^{n} n^{b}(2 n) !}{(n+1) !}$ converges. Specify tests used.

See Solution

Problem 19835

Find integers $b$ such that the series $\sum_{n=1}^{\infty} \frac{(-1)^{n} n^{b} (2n)!}{(n+1)!}$ converges. Specify the tests used.

See Solution

Problem 19836

Determine if the series $\sum_{n=1}^{\infty}\left(\frac{3 n-5}{4 n+2}\right)^{n}$ converges or diverges, and state the tests used.

See Solution

Problem 19837

Find the limit: $\lim _{x \rightarrow \infty}(\sqrt{x+4}-\sqrt{x+49})=\square$ (Hint: Use the conjugate.)

See Solution

Problem 19838

Calculate the average rate of change of the function $g(x)=9 x^{3}-5$ over the interval $[-3,3]$ .

See Solution

Problem 19839

Find the derivative $f'(1)$ for the function $f(x)=\cos(\ln(x^5))$ .

See Solution

Problem 19840

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

See Solution

Problem 19841

Find the tangent line equation for $y=16 \sqrt{x}$ at (16, 64). Sketch both the curve and the tangent line. The equation is $\square$ .

See Solution

Problem 19842

Evaluate the triple integral: $\int_{0}^{\pi} \int_{0}^{\frac{\pi}{6}} \int_{2 \sec \varphi}^{6} \rho^{2} \sin \varphi d \rho d \varphi d \theta$ .

See Solution

Problem 19843

Find the derivative of the integral: $\frac{d}{d x} \int_{1}^{4 / x} \cos ^{2}(t) d t$ .

See Solution

Problem 19844

Find the domain of $y=\frac{9-e^{x}}{3+e^{x}}$ and use limits to determine the asymptotes. Domain: ($, ).

See Solution

Problem 19845

Find the slope of the graph of $f(x)=x^{2}+1$ at the point $(-2,5)$ and the tangent line equation. $m=\square$

See Solution

Problem 19846

Find the tangent line equation for the curve $y=6x^3$ at the point (1,6). Sketch both the curve and line.

See Solution

Problem 19847

Find the area between the curves $y=x^{2}-5x+3$ and $y=-x^{2}+5x-5$ for $x$ in $[0,4]$ .

See Solution

Problem 19848

Find the function formula for the integral: $\int_{4}^{x^{2}}(t+7) dt =$

See Solution

Problem 19849

Find the slope of $g(x)=\frac{3 x}{x-2}$ at $(4,6)$ and the tangent line equation at that point.

See Solution

Problem 19850

Find the slope of $f(x)=\sqrt{2 x}$ at (2,2) and the equation of the tangent line at that point.

See Solution

Problem 19851

Find the slope of $f(x)=\sqrt{3x}$ at (3,3) and the equation of the tangent line at that point.

See Solution

Problem 19852

Find the slope of the curve $y=\frac{1}{x-3}$ at $x=5$ . The slope is $\square$ .

See Solution

Problem 19853

Find the point $(x, y)$ where the graph of $y=9 x^{2}+6 x-2$ has a horizontal tangent line.

See Solution

Problem 19854

Calculate the area between the curves $y=x^{2}-5x+3$ and $y=-x^{2}+5x-5$ for $x$ in $[0,4]$ .

See Solution

Problem 19855

Find the area between $R(t)=100+10t$ and $C(t)=61+5t$ from $t=0$ to $t=5$ . What does this area represent?

See Solution

Problem 19856

Find the slope of the curve $y=-9 x^{2}$ at the point $(-5,-225)$ . What is the slope?

See Solution

Problem 19857

Use the Integral Test to check if the series $\sum_{n=1}^{\infty} \frac{1}{(3 n-1)^{2}}$ converges or diverges.

See Solution

Problem 19858

Graph $y=5 x^{\frac{2}{5}}$ and find where vertical tangents occur. Confirm with limit calculations: $\lim_{h \to 0} \frac{f(x_0+h)-f(x_0)}{h}$ .

See Solution

Problem 19859

Find the formula for the integral function: $\int_{-2}^{x}(20 t^{2}-19 t) dt$ .

See Solution

Problem 19860

Find the slope of $f(x)=\sqrt{3x+7}$ at $(6,5)$ and the tangent line equation. Slope is $\square$ .

See Solution

Problem 19861

Find and simplify the difference quotient for $f(x)=x^{2}+5$ : $\frac{f(x+h)-f(x)}{h}$ .

See Solution

Problem 19862

Find the derivative of $p(\theta)=\sqrt{7 \theta}$ and calculate $p^{\prime}(1)$ , $p^{\prime}(7)$ , $p^{\prime}\left(\frac{3}{7}\right)$ .

See Solution

Problem 19863

Find the derivative $\frac{d y}{d x}$ for $y = -2 x^{3}$ . What is $\frac{d y}{d x} = \square$ ?

See Solution

Problem 19864

Find the derivative $\frac{dy}{dx}$ for $y = -5x^{3/2}$ . What is $\frac{dy}{dx} = \square$ ?

See Solution

Problem 19865

Differentiate $f(x)=4x+\frac{5}{x}$ and find the slope of the tangent line at $x=5$ .

See Solution

Problem 19866

Calculate the area between the curves $y=x^{2}-4x+2$ and $y=-x^{2}+4x-4$ from $x=0$ to $x=3$ .

See Solution

Problem 19867

Calculate the derivative of $f(x)=2+x^{2}$ and find $f^{\prime}(-6)$ , $f^{\prime}(0)$ , $f^{\prime}(4)$ .

See Solution

Problem 19868

Differentiate $w=g(z)=7+\sqrt{25-z}$ and find the tangent line at $(21,9)$ . Derivative is $\square$ .

See Solution

Problem 19869

Find the derivative $\frac{d s}{d t}$ for the function $s=\frac{t}{6 t+1}$ .

See Solution

Problem 19870

Find the derivative at $\theta=1$ for $r=\frac{5}{\sqrt{10-\theta}}$ . What is $\left.\frac{d r}{d \theta}\right|_{\theta=1}$ ?

See Solution

Problem 19871

计算曲线 $y=\sin x$ 在点 $\left(\frac{\pi}{2}, 1\right)$ 的曲率。

See Solution

Problem 19872

Find the derivative of $f(x)=\frac{2}{x+8}$ using $f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$ . $f^{\prime}(x)=\square$

See Solution

Problem 19873

Analyze a "downward W shape" graph to find:
(a) Open intervals where the function increases. (b) Open intervals where the function decreases. (c) Open intervals where the function is constant.

See Solution

Problem 19874

Find the derivative of the function $f(x)=1+\sqrt{3 x}$ using $f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$ .

See Solution

Problem 19875

Find the derivative of $f(x)=2+\sqrt{3x}$ using $f^{\prime}(x)=\lim_{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$ . $f^{\prime}(x)=\square$

See Solution

Problem 19876

Is the function $f(x)=\begin{cases} 3x-1 & x \geq 0 \\ x^2+5x-1 & x<0 \end{cases}$ differentiable at $x=0$ ?

See Solution

Problem 19877

Find the first and second derivatives of $r=\frac{1}{2 s}-\frac{7}{5 s^{4}}$ . What is $\frac{d r}{d s}$ ?

See Solution

Problem 19878

Find the first and second derivatives of $y=\frac{4 x^{3}}{3}-x+5 e^{x}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 19879

Find the integral of $(3-2x)^{100}$ with respect to $x$ : $\int(3-2 x)^{100} \mathrm{~d} x$ .

See Solution

Problem 19880

Does $f(x)=2 x^{3}+3 x^{2}-12 x+4$ have an absolute maximum on $[0,2]$ ? Justify your answer.

See Solution

Problem 19881

求不定积分： (1) $\int e^{5 x} d x$ (2) $\int(3-2 x)^{100} \mathrm{~d} x$

See Solution

Problem 19882

Find the first and second derivatives of $s=4 t^{3}-3 t^{4}$ . What is $\frac{\mathrm{ds}}{\mathrm{dt}}$ ?

See Solution

Problem 19883

Is the function $f(x)=\left\{\begin{array}{ll} 5 x+\tan x, & x \geq 0 \\ 4 x^{2}, & x<0 \end{array}\right.$ differentiable at $x=0$ ?

See Solution

Problem 19884

Find the derivative $f'(x)$ of $f(x)=\frac{x^3}{12}$ . Graph $f(x)$ and $f'(x)$ . Identify where $f'$ is positive, zero, or negative, and relate this to the intervals where $f(x)$ increases or decreases.

See Solution

Problem 19885

Does $f(x)=2 x^{3}+3 x^{2}-12 x+4$ have an absolute max on $[0,2]$ ? Also, find $b$ and $c$ for $b \leq f(x) \leq c$ .

See Solution

Problem 19886

A cone with radius $a=7 \mathrm{~m}$ and height $b=18 \mathrm{~m}$ fills with water at $18 \mathrm{~m}^{3}/\mathrm{min}$ . Find the rise rate when water is $17 \mathrm{~m}$ deep.

See Solution

Problem 19887

Find the derivative of the function $f(x)=\frac{5}{x^{2}+3 x}$ , denoted as $f^{\prime}(x)$ .

See Solution

Problem 19888

求不定积分： $\int \frac{\mathrm{d} x}{\sqrt[3]{2-3 x}}$ 和 $\int \frac{\sin \sqrt{t}}{\sqrt{t}} \mathrm{~d} t$ 。

See Solution

Problem 19889

求不定积分： (8) $\int \tan^{10} x \sec^{2} x \mathrm{~d} x$ ; (10) $\int \frac{\mathrm{d} x}{\mathrm{e}^{x}+\mathrm{e}^{-x}}$ ; (12) $\int \frac{x}{\sqrt{2-3 x^{2}}} \mathrm{~d} x$ ;

See Solution

Problem 19890

求不定积分： $\int \sin 2 x \cos 3 x \mathrm{~d} x$ ; $\int \tan ^{3} x \sec x \mathrm{~d} x$ ; $\int \frac{\mathrm{d} x}{x \ln x \ln \ln x}$ .

See Solution

Problem 19891

求以下不定积分：
1. $\int \cos^{2}(\omega t+\varphi) \sin(\omega t+\varphi) \mathrm{d} t$ ;
2. $\int \frac{2x-1}{\sqrt{1-x^{2}}} \mathrm{d} x$ ;
3. $\int \frac{\sin x+\cos x}{\sqrt[3]{\sin x-\cos x}} \mathrm{d} x$ .

See Solution

Problem 19892

How long will it take for \$7000 to grow to \$50000 at a 10% annual interest rate, compounded continuously?

See Solution

Problem 19893

求不定积分： $\int x^{2} \cos x \mathrm{~d} x$ 。

See Solution

Problem 19894

Find the derivative of the function given by $f^{\prime}(x)=(2 \sqrt{x+1}+x+1)^{\prime}$ .

See Solution

Problem 19895

求不定积分： (26) $\int \frac{\sin x \cos x}{1+\sin ^{4} x} \mathrm{~d} x$ ; (28) $\int \frac{\mathrm{d} x}{\sqrt{\left(a^{2}-x^{2}\right)^{3}}}(a>0)$ ; (30) $\int \frac{x^{2}}{\left(1+x^{2}\right)^{2}} \mathrm{~d} x$ .

See Solution

Problem 19896

Calculate the integral: $\int x^{2}(\ln x)^{2} d x$ .

See Solution

Problem 19897

Find the integrals: 1) $\int x^{2} \ln x \, dx$ and 2) $\int \arcsin x \, dx$ .

See Solution

Problem 19898

Find the derivative of $f(x) = 2 \sqrt{x+1} + x + 1$ .

See Solution

Problem 19899

计算以下不定积分：
2. $\int x^{2} \cos x \mathrm{~d} x$ ；
4. $\int x^{2} \ln x \mathrm{~d} x$ ；
6. $\int \arcsin x \mathrm{~d} x$ ；
8. $\int x \cos \frac{x}{2} \mathrm{~d} x$ ；
10. $\int x \tan ^{2} x \mathrm{~d} x$ ；
12. $\int x \sin x \cos x \mathrm{~d} x$ ；
14. $\int e^{\sqrt[3]{x}} \mathrm{~d} x$ ；
16. $\int \cos (\ln x) d x$ ；
18. $\int \ln \left(x+\sqrt{1+x^{2}}\right) d x$ 。

See Solution

Problem 19900

Déterminez l'ensemble de définition de la fonction $f(x) = \frac{4}{9 - x^{2}}$ .

See Solution

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Calculus

Problem 19801

Problem 19802

Find the average cost of producing 500 units given the marginal cost function dcdq=0.003q2−0.6q+90\frac{d c}{d q}=0.003 q^{2}-0.6 q+90dqdc​=0.003q2−0.6q+90 and fixed costs of \9000.Averagecostat9000. Average cost at 9000.Averagecostatq=500$ is \$\square.

Problem 19803

Evaluate the surface integral: ∬Sv⋅dS=∫0π/2∫02π8sin⁡(φ)cos⁡2(φ)dθdφ\iint_{S} \mathbf{v} \cdot d \mathbf{S} = \int_{0}^{\pi / 2} \int_{0}^{2 \pi} 8 \sin (\varphi) \cos ^{2}(\varphi) d \theta d \varphi∬S​v⋅dS=∫0π/2​∫02π​8sin(φ)cos2(φ)dθdφ.

Problem 19804

Calculate the integral: ∫14x3e14x2dx\int 14 x^{3} e^{14 x^{2}} d x∫14x3e14x2dx

Problem 19805

Use the Limit Comparison Test to check if the series ∑n=1∞2n1/55n3+7n+7\sum_{n=1}^{\infty} \frac{2 n^{1 / 5}}{\sqrt{5 n^{3}+7 n+7}}∑n=1∞​5n3+7n+7​2n1/5​ converges or diverges.

Problem 19806

Problem 19807

Find the derivative dyd ydy of the function y=e5x+1y=e^{5 \sqrt{x}+1}y=e5x​+1.

Problem 19808

Find dy for the equation y = 2x√(12 - x³).

Problem 19809

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=tan⁡−1(ln⁡(2x3))y=\tan^{-1}(\ln(2x^3))y=tan−1(ln(2x3)).

Problem 19810

How much longer does it take for Alexa's \8,000at8,000 at 8,000at3 \frac{5}{8} \%continuousinteresttodoublecomparedtoMalik′s$8,000at continuous interest to double compared to Malik's \$8,000 at continuousinteresttodoublecomparedtoMalik′s$8,000at3 \frac{3}{4} \%$ quarterly?

Problem 19811

Calculate the integral: ∫06(18s2+216(1−s)3−e6+1)ds\int_{0}^{6}\left(18 s^{2}+\frac{216(1-s)}{3}-e^{6}+1\right) d s∫06​(18s2+3216(1−s)​−e6+1)ds.

Problem 19812

Find the value of aaa that makes the function g(x)={ax if x<0x2−15x if x≥0g(x)=\left\{\begin{array}{ll} a x & \text { if } x<0 \\ x^{2}-15 x & \text { if } x \geq 0 \end{array}\right.g(x)={axx2−15x​ if x<0 if x≥0​ differentiable for all xxx.

Problem 19813

Calculate the integral of the function (4t+2)et2+t(4t + 2)e^{t^2 + t}(4t+2)et2+t with respect to ttt: ∫(4t+2)et2+tdt\int(4 t + 2)e^{t^{2}+t} dt∫(4t+2)et2+tdt.

Problem 19814

How long to triple an investment with continuous compounding at 15%15\%15%?

Problem 19815

Evaluate the double integral: ∫06∫06(−er+r2(1−s)+rs2)drds\int_{0}^{6} \int_{0}^{6}\left(-e^{r}+r^{2}(1-s)+r s^{2}\right) d r d s∫06​∫06​(−er+r2(1−s)+rs2)drds

Problem 19816

Find the 971st derivative of cos⁡x\cos xcosx.

Problem 19817

Find dydt\frac{d y}{d t}dtdy​ given x2+2y2+2y=16x^{2}+2 y^{2}+2 y=16x2+2y2+2y=16, dxdt=5\frac{d x}{d t}=5dtdx​=5, x=2x=2x=2, and y=−3y=-3y=−3.

Problem 19818

Evaluate the surface integral ∬Sx2yzdS\iint_{S} x^{2} y z d S∬S​x2yzdS for the plane z=1+4x+2yz=1+4x+2yz=1+4x+2y over the rectangle 0≤x≤40 \leq x \leq 40≤x≤4, 0≤y≤20 \leq y \leq 20≤y≤2.

Problem 19819

Problem 19820

Find the derivative of y=3x2−7x+10y=3 x^{2}-7 x+10y=3x2−7x+10. Options: A. 6x−76 x-76x−7 B. 6x+106 x+106x+10 C. −7x-7 x−7x D. 3x+103 x+103x+10

Problem 19821

Problem 19822

Find the velocity and acceleration of the particle at t=3t=3t=3 sec, given s=7+6ts=\sqrt{7+6t}s=7+6t​.

Problem 19823

Find the general antiderivative of f(x)=3x2+12f(x)=3 x^{2}+\frac{1}{2}f(x)=3x2+21​ and the particular one with F(1)=1F(1)=1F(1)=1.

Problem 19824

Find the area under f(x)=x2exf(x)=x^{2} e^{x}f(x)=x2ex from 000 to 666 using right endpoints as a limit: A=lim⁡n→∞∑i=1n(□x)A=\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \left(\square_{x}\right)A=limn→∞​∑i=1n​(□x​).

Problem 19825

Find the area under f(x)=x2exf(x)=x^{2} e^{x}f(x)=x2ex from 000 to 666 using right endpoints as a limit: A=lim⁡n→∞∑i=1n(□x)A=\lim_{n \to \infty} \sum_{i=1}^{n}(\square_{x})A=limn→∞​∑i=1n​(□x​).

Problem 19826

Problem 19827

Find a region with area equal to the limit: lim⁡n→∞∑i=1n3n1+3in\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{3}{n} \sqrt{1+\frac{3 i}{n}}n→∞lim​i=1∑n​n3​1+n3i​​.

Problem 19828

Find the general antiderivative of h(x)=3x5−10x4+x2+7h(x)=3 x^{5}-10 x^{4}+x^{2}+7h(x)=3x5−10x4+x2+7.

Problem 19829

How long will it take for a drink cooling from 182∘F182^{\circ} \mathrm{F}182∘F to 93∘F93^{\circ} \mathrm{F}93∘F at 72∘F72^{\circ} \mathrm{F}72∘F?

Problem 19830

Find the antiderivative of v(x)=3(x+6)(2x+1)v(x)=3(x+6)(2 x+1)v(x)=3(x+6)(2x+1) and check your answer by differentiating it.

Problem 19831

Find the sum of the series: ∑i=1∞(−25)⋅(−47)i−1\sum_{i=1}^{\infty}\left(-\frac{2}{5}\right) \cdot\left(-\frac{4}{7}\right)^{i-1}∑i=1∞​(−52​)⋅(−74​)i−1.

Problem 19832

Trouver la dérivée de la fonction g(x)=ln⁡(1+cos⁡x)g(x)=\ln (1+\cos x)g(x)=ln(1+cosx) pour x∈]−π,π[x \in]-\pi, \pi[x∈]−π,π[.

Problem 19833

Analyze the series S=∑n=1∞(−1)n+1(n−1)n(n+3)S=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(n-1)}{n(n+3)}S=∑n=1∞​n(n+3)(−1)n+1(n−1)​ for absolute or conditional convergence.

Problem 19834

Find integers bbb such that the series ∑n=1∞(−1)nnb(2n)!(n+1)!\sum_{n=1}^{\infty} \frac{(-1)^{n} n^{b}(2 n) !}{(n+1) !}∑n=1∞​(n+1)!(−1)nnb(2n)!​ converges. Specify tests used.

Problem 19835

Find integers bbb such that the series ∑n=1∞(−1)nnb(2n)!(n+1)!\sum_{n=1}^{\infty} \frac{(-1)^{n} n^{b} (2n)!}{(n+1)!}∑n=1∞​(n+1)!(−1)nnb(2n)!​ converges. Specify the tests used.

Problem 19836

Determine if the series ∑n=1∞(3n−54n+2)n\sum_{n=1}^{\infty}\left(\frac{3 n-5}{4 n+2}\right)^{n}∑n=1∞​(4n+23n−5​)n converges or diverges, and state the tests used.

Problem 19837

Find the limit: lim⁡x→∞(x+4−x+49)=□\lim _{x \rightarrow \infty}(\sqrt{x+4}-\sqrt{x+49})=\squarex→∞lim​(x+4​−x+49​)=□ (Hint: Use the conjugate.)

Problem 19838

Calculate the average rate of change of the function g(x)=9x3−5g(x)=9 x^{3}-5g(x)=9x3−5 over the interval [−3,3][-3,3][−3,3].

Problem 19839

Find the derivative f′(1)f'(1)f′(1) for the function f(x)=cos⁡(ln⁡(x5))f(x)=\cos(\ln(x^5))f(x)=cos(ln(x5)).

Problem 19840

Find the limit: lim⁡x→∞(x2+3x−x2−2x)\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+3 x}-\sqrt{x^{2}-2 x}\right)x→∞lim​(x2+3x​−x2−2x​). Simplify your answer.

Problem 19841

Find the tangent line equation for y=16xy=16 \sqrt{x}y=16x​ at (16, 64). Sketch both the curve and the tangent line. The equation is □\square□.

Problem 19842

Evaluate the triple integral: ∫0π∫0π6∫2sec⁡φ6ρ2sin⁡φdρdφdθ\int_{0}^{\pi} \int_{0}^{\frac{\pi}{6}} \int_{2 \sec \varphi}^{6} \rho^{2} \sin \varphi d \rho d \varphi d \theta∫0π​∫06π​​∫2secφ6​ρ2sinφdρdφdθ.

Find the average cost of producing 500 units given the marginal cost function $\frac{d c}{d q}=0.003 q^{2}-0.6 q+90$ and fixed costs of \ $9000. Average cost at$ q=500$ is \$\square.

Evaluate the surface integral: $\iint_{S} \mathbf{v} \cdot d \mathbf{S} = \int_{0}^{\pi / 2} \int_{0}^{2 \pi} 8 \sin (\varphi) \cos ^{2}(\varphi) d \theta d \varphi$ .

Calculate the integral: $\int 14 x^{3} e^{14 x^{2}} d x$

Use the Limit Comparison Test to check if the series $\sum_{n=1}^{\infty} \frac{2 n^{1 / 5}}{\sqrt{5 n^{3}+7 n+7}}$ converges or diverges.

Find the derivative $d y$ of the function $y=e^{5 \sqrt{x}+1}$ .

Find the derivative $\frac{d y}{d x}$ for the function $y=\tan^{-1}(\ln(2x^3))$ .

How much longer does it take for Alexa's \ $8,000 at$ 3 \frac{5}{8} \% $continuous interest to double compared to Malik's \$8,000 at$ 3 \frac{3}{4} \%$ quarterly?

Calculate the integral: $\int_{0}^{6}\left(18 s^{2}+\frac{216(1-s)}{3}-e^{6}+1\right) d s$ .

Find the value of $a$ that makes the function $g(x)=\left\{\begin{array}{ll} a x & \text { if } x<0 \\ x^{2}-15 x & \text { if } x \geq 0 \end{array}\right.$ differentiable for all $x$ .

Calculate the integral of the function $(4t + 2)e^{t^2 + t}$ with respect to $t$ : $\int(4 t + 2)e^{t^{2}+t} dt$ .

How long to triple an investment with continuous compounding at $15\%$ ?

Evaluate the double integral: $\int_{0}^{6} \int_{0}^{6}\left(-e^{r}+r^{2}(1-s)+r s^{2}\right) d r d s$

Find the 971st derivative of $\cos x$ .

Find $\frac{d y}{d t}$ given $x^{2}+2 y^{2}+2 y=16$ , $\frac{d x}{d t}=5$ , $x=2$ , and $y=-3$ .

Evaluate the surface integral $\iint_{S} x^{2} y z d S$ for the plane $z=1+4x+2y$ over the rectangle $0 \leq x \leq 4$ , $0 \leq y \leq 2$ .

Find the derivative of $y=3 x^{2}-7 x+10$ . Options: A. $6 x-7$ B. $6 x+10$ C. $-7 x$ D. $3 x+10$

Find the velocity and acceleration of the particle at $t=3$ sec, given $s=\sqrt{7+6t}$ .

Find the general antiderivative of $f(x)=3 x^{2}+\frac{1}{2}$ and the particular one with $F(1)=1$ .

Find the area under $f(x)=x^{2} e^{x}$ from $0$ to $6$ using right endpoints as a limit: $A=\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \left(\square_{x}\right)$ .

Find the area under $f(x)=x^{2} e^{x}$ from $0$ to $6$ using right endpoints as a limit: $A=\lim_{n \to \infty} \sum_{i=1}^{n}(\square_{x})$ .

Find a region with area equal to the limit: $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{3}{n} \sqrt{1+\frac{3 i}{n}}$ .

Find the general antiderivative of $h(x)=3 x^{5}-10 x^{4}+x^{2}+7$ .

How long will it take for a drink cooling from $182^{\circ} \mathrm{F}$ to $93^{\circ} \mathrm{F}$ at $72^{\circ} \mathrm{F}$ ?

Find the antiderivative of $v(x)=3(x+6)(2 x+1)$ and check your answer by differentiating it.

Find the sum of the series: $\sum_{i=1}^{\infty}\left(-\frac{2}{5}\right) \cdot\left(-\frac{4}{7}\right)^{i-1}$ .

Trouver la dérivée de la fonction $g(x)=\ln (1+\cos x)$ pour $x \in]-\pi, \pi[$ .

Analyze the series $S=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(n-1)}{n(n+3)}$ for absolute or conditional convergence.

Find integers $b$ such that the series $\sum_{n=1}^{\infty} \frac{(-1)^{n} n^{b}(2 n) !}{(n+1) !}$ converges. Specify tests used.

Find integers $b$ such that the series $\sum_{n=1}^{\infty} \frac{(-1)^{n} n^{b} (2n)!}{(n+1)!}$ converges. Specify the tests used.

Determine if the series $\sum_{n=1}^{\infty}\left(\frac{3 n-5}{4 n+2}\right)^{n}$ converges or diverges, and state the tests used.

Find the limit: $\lim _{x \rightarrow \infty}(\sqrt{x+4}-\sqrt{x+49})=\square$ (Hint: Use the conjugate.)

Calculate the average rate of change of the function $g(x)=9 x^{3}-5$ over the interval $[-3,3]$ .

Find the derivative $f'(1)$ for the function $f(x)=\cos(\ln(x^5))$ .

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

Find the tangent line equation for $y=16 \sqrt{x}$ at (16, 64). Sketch both the curve and the tangent line. The equation is $\square$ .

Evaluate the triple integral: $\int_{0}^{\pi} \int_{0}^{\frac{\pi}{6}} \int_{2 \sec \varphi}^{6} \rho^{2} \sin \varphi d \rho d \varphi d \theta$ .

Find the derivative of the integral: $\frac{d}{d x} \int_{1}^{4 / x} \cos ^{2}(t) d t$ .

Find the domain of $y=\frac{9-e^{x}}{3+e^{x}}$ and use limits to determine the asymptotes. Domain: ($, ).

Find the slope of the graph of $f(x)=x^{2}+1$ at the point $(-2,5)$ and the tangent line equation. $m=\square$

Find the tangent line equation for the curve $y=6x^3$ at the point (1,6). Sketch both the curve and line.

Find the area between the curves $y=x^{2}-5x+3$ and $y=-x^{2}+5x-5$ for $x$ in $[0,4]$ .

Find the function formula for the integral: $\int_{4}^{x^{2}}(t+7) dt =$

Find the slope of $g(x)=\frac{3 x}{x-2}$ at $(4,6)$ and the tangent line equation at that point.

Find the slope of $f(x)=\sqrt{2 x}$ at (2,2) and the equation of the tangent line at that point.

Find the slope of $f(x)=\sqrt{3x}$ at (3,3) and the equation of the tangent line at that point.

Find the slope of the curve $y=\frac{1}{x-3}$ at $x=5$ . The slope is $\square$ .

Find the point $(x, y)$ where the graph of $y=9 x^{2}+6 x-2$ has a horizontal tangent line.

Calculate the area between the curves $y=x^{2}-5x+3$ and $y=-x^{2}+5x-5$ for $x$ in $[0,4]$ .

Find the area between $R(t)=100+10t$ and $C(t)=61+5t$ from $t=0$ to $t=5$ . What does this area represent?

Find the slope of the curve $y=-9 x^{2}$ at the point $(-5,-225)$ . What is the slope?

Use the Integral Test to check if the series $\sum_{n=1}^{\infty} \frac{1}{(3 n-1)^{2}}$ converges or diverges.

Graph $y=5 x^{\frac{2}{5}}$ and find where vertical tangents occur. Confirm with limit calculations: $\lim_{h \to 0} \frac{f(x_0+h)-f(x_0)}{h}$ .

Find the formula for the integral function: $\int_{-2}^{x}(20 t^{2}-19 t) dt$ .

Find the slope of $f(x)=\sqrt{3x+7}$ at $(6,5)$ and the tangent line equation. Slope is $\square$ .

Find and simplify the difference quotient for $f(x)=x^{2}+5$ : $\frac{f(x+h)-f(x)}{h}$ .

Find the derivative of $p(\theta)=\sqrt{7 \theta}$ and calculate $p^{\prime}(1)$ , $p^{\prime}(7)$ , $p^{\prime}\left(\frac{3}{7}\right)$ .

Find the derivative $\frac{d y}{d x}$ for $y = -2 x^{3}$ . What is $\frac{d y}{d x} = \square$ ?

Find the derivative $\frac{dy}{dx}$ for $y = -5x^{3/2}$ . What is $\frac{dy}{dx} = \square$ ?

Differentiate $f(x)=4x+\frac{5}{x}$ and find the slope of the tangent line at $x=5$ .

Calculate the area between the curves $y=x^{2}-4x+2$ and $y=-x^{2}+4x-4$ from $x=0$ to $x=3$ .

Calculate the derivative of $f(x)=2+x^{2}$ and find $f^{\prime}(-6)$ , $f^{\prime}(0)$ , $f^{\prime}(4)$ .

Differentiate $w=g(z)=7+\sqrt{25-z}$ and find the tangent line at $(21,9)$ . Derivative is $\square$ .

Find the derivative $\frac{d s}{d t}$ for the function $s=\frac{t}{6 t+1}$ .

Find the derivative at $\theta=1$ for $r=\frac{5}{\sqrt{10-\theta}}$ . What is $\left.\frac{d r}{d \theta}\right|_{\theta=1}$ ?

计算曲线 $y=\sin x$ 在点 $\left(\frac{\pi}{2}, 1\right)$ 的曲率。

Find the derivative of $f(x)=\frac{2}{x+8}$ using $f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$ . $f^{\prime}(x)=\square$

Analyze a "downward W shape" graph to find:
(a) Open intervals where the function increases. (b) Open intervals where the function decreases. (c) Open intervals where the function is constant.

Find the derivative of the function $f(x)=1+\sqrt{3 x}$ using $f^{\prime}(x)=\lim _{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$ .