Calculus

Problem 801

Find the tangent line to $f(x)=-x^{2}+5$ at $x=5$ and use it to estimate $f(5.1)$ .

See Solution

Problem 802

Find points $(x, y)$ where $\nabla f(x, y)=(0,0)$ for $f(x, y)=\frac{x^{3}}{3}-\frac{y^{3}}{3}+3xy$ and classify them.

See Solution

Problem 803

Evaluate the following limits as $x$ approaches 0:
1. $\lim _{x \rightarrow 0} \frac{(1+x) \sin x-x \cos x}{x^{2}}$
2. $\lim _{x \rightarrow 0} \frac{e^{x^{2}}-\cos x}{x \sin x}$
3. $\lim _{x \rightarrow 0} \frac{\sin x-x e^{x}+x^{2}}{x(\cos x-1)}$
4. $\lim _{x \rightarrow 0}\left\{\frac{1}{\sin ^{2} x}-\frac{1}{x^{2}}\right\}$

See Solution

Problem 804

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

See Solution

Problem 805

Integrate: $\int \frac{2 x+5}{\sqrt{4 x^{2}-20 x+29}} d x$ (15 pts, 1a: 5 pts)

See Solution

Problem 806

Find the salt concentration in a 100-gallon tank after 30 mins, starting with 10 lbs salt and adding brine at $4 \mathrm{gal/min}$ .

See Solution

Problem 807

Calculate the surface area from revolving $x=\frac{1}{3}(y^{2}+2)^{3/2}$ around the x-axis for $1 \leq y \leq 2$ .

See Solution

Problem 808

Evaluate the integral $\int_{1}^{\infty} k x^{-5} \, dx$ .

See Solution

Problem 809

13. Solve $\left[x \sin ^{2}\left(\frac{y}{x}\right)-y\right] dx + x dy = 0$ for $y=\frac{\pi}{4}$ at $x=1$ .
14. Find $\frac{dy}{dx}-\frac{y}{x}+\operatorname{cosec}\left(\frac{y}{x}\right)=0$ for $y=0$ at $x=1$ .

See Solution

Problem 810

Calculate the integral: $\int(1+3 x) x^{2} d x$

See Solution

Problem 811

Calculate the integral $\int 4 x\left(x^{2}-2\right)^{2} d x$ .

See Solution

Problem 812

Calculate the average rate of change of $f(x)=2 x^{4}-3 x$ from $x=-1$ to $x=2$ .

See Solution

Problem 813

Find $\frac{d y}{d x}$ in terms of $x$ given that $x=\sec 3 y$ .

See Solution

Problem 814

Find $\frac{d y}{d x}$ in terms of $t$ for $x=\sin(t)$ and $y=\cos^2(t)$ . Simplify your expression.

See Solution

Problem 815

Find $\frac{d y}{d x}$ in terms of $t$ for the equations $x=\sin^3 t$ and $y=\cos^4 t$ .

See Solution

Problem 816

Find and simplify $\frac{d y}{d x}$ using the parametric equations $x=\sin(t)$ and $y=\cos^2(t)$ .

See Solution

Problem 817

Find $\frac{d y}{d x}$ in terms of $t$ for the equations $x=\sin(t)$ , $y=\cos^2(t)$ .

See Solution

Problem 818

Find the rate of change of sales $S(t) = 10,000 + 2000t - 200t^2$ at $t=0$ , $t=4$ , and $t=8$ weeks.

See Solution

Problem 819

Evaluate the integral: $\int \frac{3 a x}{b^{2}+c^{2} x^{2}} \, dx$

See Solution

Problem 820

Upton Chuck free falls for 2.60 seconds. Calculate his final velocity and the distance fallen. Use $v = gt$ and $d = \frac{1}{2}gt^2$ .

See Solution

Problem 821

Find the integral $\int \tan^{8} x \sec^{4} x \, dx$ .

See Solution

Problem 822

Calculate the double integrals: 1) $\iint_{x^{2}+y^{2} \leq 1} \cos \left(x^{2}+y^{2}\right) d x d y$ and 2) $\int_{0}^{1} \int_{\sqrt{y}}^{1} \frac{\cos (x)}{x^{2}} d x d y$ .

See Solution

Problem 823

Solve on separate sheets with signatures. No calculators. Justify answers.
1. (a) Find the integral: $\int x \arctan (x) d x$ (b) Evaluate: $\int_{0}^{\pi / 2} \frac{\cos (t)}{\sqrt{\sin (t)}} d t$

See Solution

Problem 824

Evaluate the integral $\int_{0}^{1} \frac{x^{4}(1-x)^{4}}{1+x^{2}} d x$ using integration techniques.

See Solution

Problem 825

Monthly sales (in thousands) for a music store are given by $S(t)=\frac{200}{t^{2}+36}$ . Find $S(2)$ , $S'(2)$ , and estimate sales for month 3.

See Solution

Problem 826

Monthly sales for a record album are given by $S(t)=\frac{200 t}{t^{2}+36}$ . Find: a) $S(2)$ , b) $S^{\prime}(2)$ , c) estimate sales in month 3 using (a) and (b).

See Solution

Problem 827

Tính tích phân $I=\int_{AB} (2xy \, dx + x^2 \, dy)$ trên đoạn thẳng $AB: y=x^3-3x^2$ với $A(1,-2)$ , $B(2,-4)$ .

See Solution

Problem 828

Differentiate $x^{3} y^{2}$ with respect to $y$ .

See Solution

Problem 829

Differentiate $x^{3} y^{2}$ with respect to $x$ .

See Solution

Problem 830

Find the free extrema of the function $z=-2x^{2}+22y^{2}-192y+x^{2}y$ .

See Solution

Problem 831

Find the free extremes of the function $z=3 x^{2}-8 y^{2}+64 y-y x^{2}$ .

See Solution

Problem 832

Find $\frac{d y}{d x}$ for the equation $4 e^{-3 x}+e^{4 y}+8=0$ .

See Solution

Problem 833

Differentiate the function $\frac{5 x^{2}-5}{2 x}$ .

See Solution

Problem 834

Differentiate the function $F(x)=\frac{\ln x}{3 x^{2}}-\frac{e^{-9 x}}{\ln x^{2}}$ .

See Solution

Problem 835

Find the second derivative $H^{\prime \prime}(x)$ of the function $H(x)=x e^{x}$ .

See Solution

Problem 836

Evaluate the integrals: $\int_{1 / e}^{\tan x} \frac{t d t}{1+t^{2}} + \int_{1 / e}^{\cot x} \frac{d t}{t(1+t^{2})}$ .

See Solution

Problem 837

Find the derivative $g^{\prime}(x)$ for the function $g(x)=\frac{x^{2}}{1+x}+\frac{e^{x}}{1+x^{2}}$ .

See Solution

Problem 838

Let $f(\theta_{0}, \theta_{1})$ be a smooth function. Which statements about gradient descent minimizing $f$ are true?

See Solution

Problem 839

Check if the function $f(x)$ is continuous at $x=-3$ where $f(x)=\begin{cases} 6+x^{2}, & x \geq-3 \\ 4-3 x, & x<-3 \end{cases}$ .

See Solution

Problem 840

Identify type, order, and degree of these differential equations:
1. $\left(\frac{d^{2} w}{d x^{2}}\right)^{3}+x y \frac{d w}{d x}+w=k$
2. $\frac{d^{3} s}{d t^{3}}-k^{2} t^{2}+a^{4}=0$
3. $y^{\prime \prime}+\left(y^{\prime}\right)^{2}=y$
4. $7\left(\frac{\partial x}{\partial y}\right)^{4}-\left(\frac{\partial x}{\partial y}\right)^{2}+c=\frac{\partial^{2} x}{\partial y^{2}}$
5. $\left(x^{2}+y^{2}\right) d x=2 x^{2} y^{2} d y$
6. $\left(\frac{d^{3} s}{d t^{3}}\right)^{2}+s\left(\frac{d^{2} s}{d t^{2}}\right)^{3}+2 s t=0$
7. $2 x^{3}-8=\frac{d^{4} y}{d x^{4}}$
8. $x^{2} d x+y^{2} d y=\frac{d^{2} y}{d x^{2}}$
9. $\frac{\partial u}{\partial t}=h^{2}\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial u^{2}}\right)$
10. $\left(\frac{\partial u}{\partial v}\right)^{3}+x^{2}\left(\frac{\partial^{2} u}{\partial x}\right)+v^{3}\left(\frac{\partial u}{\partial v}\right)^{3}=\frac{d v}{x}$

See Solution

Problem 841

Solve the DE: $(1-x) y' = y^2$ and determine if the solution is general or particular. Also solve: $a^2 dx = y \sqrt{y^2 - a^2} dy$ .

See Solution

Problem 842

Solve the differential equation $y' = x e^{-y - x^2}$ . Is the solution general or particular?

See Solution

Problem 843

Solve the separable differential equation $x y^{3} dx + (y+1)e^{-x} dy = 0$ and identify the type of solution. Also, solve $y' = x y^{3}$ .

See Solution

Problem 844

Solve these differential equations and identify solutions as general or particular:
1. $3v \frac{dv}{dy}=9$ , initial: $x=x_{0}$ , $v=v_{0}$ .
2. $y'=xy^{3}$ .
3. $\frac{dr}{dt}=-2rt$ , initial: $t=0$ , $r=r_{0}$ .
4. $\frac{dV}{dP}=\frac{-V}{P}$ .
5. $r(1+\ln x)dx+(1+\ln y)dy=0$ .
6. $\frac{dy}{dt}=\sin^{2}x\cos^{3}E$ .

See Solution

Problem 845

An ice sphere melts at $2 \pi$ m³/hr. Find the surface area decrease rate when radius is 5 m. Options: (A) $\frac{4 \pi}{5}$ (B) $40 \pi$ (C) $80 \pi^{2}$ (D) $100 \pi$

See Solution

Problem 846

Find the integral of $\frac{1}{\sqrt[7]{x}}$ with respect to $x$ .

See Solution

Problem 847

Calculate the integral: $\int 2 t \sqrt{3 t^{2}-1} \, dt$

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Problem 848

Evaluate the integral: $\int \frac{\ln \theta}{\theta} d \theta$

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Problem 849

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

See Solution

Problem 850

Evaluate $\int F \cdot dr$ for $\bar{f}=x y \bar{i}-z \hat{j}+x^{2} \bar{k}$ along $\alpha: x=t^{2}, y=2t, z=t^{3}$ from $t=0$ to $t=1$ . Ans: $\frac{51}{70}$

See Solution

Problem 851

Evaluate $\int F \cdot d r$ for $\bar{F}=x y \bar{i}-z j+x^{2} \bar{k}$ along the curve $x=t^{2}, y=2t, z=t^{3}$ from $t=0$ to $t=1$ .

See Solution

Problem 852

Evaluate $\int F \cdot d r$ for $\bar{F}=x y \bar{i}-z j+x^{2} \bar{k}$ along the curve $x=t^{2}$ , $y=2t$ , $z=t^{3}$ from $t=0$ to $t=1$ . Answer: $\frac{51}{70}$ .

See Solution

Problem 853

Calculate the work done by the force $\bar{f}=\bar{z} \bar{i}+x \bar{j}+y \hat{k}$ along the curve $\bar{r}=\cos t i+\sin t j-t \bar{k}$ from $t=0$ to $t=2 \pi$ .

See Solution

Problem 854

Evaluate $\oint_{C}(y z \, dx + x z \, dy + x y \, dz)$ over a helix defined by $x=a \cos t$ , $y=a \sin t$ , $z=k t$ for $t$ from 0 to $2\pi$ .

See Solution

Problem 855

Evaluate $\oint_{C}(y z \, dl + x z \, dy + l y \, dz)$ for the helix $x=a \cos t, y=a \sin t, z=k t$ as $t$ goes from 0 to $2\pi$ .

See Solution

Problem 856

Evaluate $\oint_{C}(y z d x+x z d y+x y d z)$ for the helix $x=a \cos t, y=a \sin t, z=k t$ as $t$ goes from 0 to $2\pi$ .

See Solution

Problem 857

Evaluate $\oint_{c}(y z d x+x z d y+x y d z)$ along the helix $x=a \cos t, y=a \sin t, z=k t$ for $t$ from 0 to $2 \pi$ .

See Solution

Problem 858

Evaluate $\oint_{c}(y z d x+x z d y+x y d z)$ on the helix $x=a \cos t, y=a \sin t, z=k t$ for $t$ from 0 to $2 \pi$ .

See Solution

Problem 859

Evaluate $\oint(y z d x+x z d y+x y d z)$ along the helix $x=a \cos t, y=a \sin t, z=k t$ for $t$ from 0 to $2 \pi$ .

See Solution

Problem 860

Find $f(x)$ if $\frac{d f}{d x}=3 x^{2}-3-4 e^{2 x}$ and $f(0)=8$ . Choose from options a, b, c, d.

See Solution

Problem 861

Find the limit as $x$ approaches 0 for $\frac{6(1-\cos x)}{x^{2}}$ .

See Solution

Problem 862

What is the derivative of $\cos x$ with respect to $x$ ?

See Solution

Problem 863

Find the integral of $\sec y \tan y$ with respect to $y$ .

See Solution

Problem 864

Calculate the integral $\int (5 e^{w} + \cos w + w^{3}) \, dw$ .

See Solution

Problem 865

Evaluate the integral: $\int \frac{6 y^{8}+12 y^{2}-y^{6}}{3 y^{7}} d y$

See Solution

Problem 866

Find the limit: $\operatorname{Lim}_{x \rightarrow-3} \frac{4 x^{2}+9 x-9}{x^{3}+27}$ .

See Solution

Problem 867

Solve the differential equation: $\frac{dy}{dx} = 3y$ .

See Solution

Problem 868

Evaluate the integral $\int_{0}^{6}(2+5 x) e^{1 / 3 x} d x$ .

See Solution

Problem 869

Find the integral of $\sec^{2} \theta$ with respect to $\theta$ .

See Solution

Problem 870

Calculate the integral of the function: $\int \cos z \, dz$

See Solution

Problem 871

Find the integral of $7^{x}$ with respect to $x$ .

See Solution

Problem 872

Calculate the integral of $\sin \theta$ with respect to $\theta$ : $\int \sin \theta \, d\theta$ .

See Solution

Problem 873

Calculate the integral: $\int \frac{2 b}{b^{2}+7} d b$

See Solution

Problem 874

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{e^{-1 / x^{9}}}{x^{2}}$

See Solution

Problem 875

Solve the differential equation $\frac{d y}{d x}=\frac{3 y-7 x+7}{3 x-7 y-3}$ .

See Solution

Problem 876

Find the limit: $\lim _{x \rightarrow 0} \frac{e^{-\frac{1}{x^{2}}}}{x^{2}}$ .

See Solution

Problem 877

Solve the differential equation: $y' - \frac{2x-1}{x^2-x} y = 1$ .

See Solution

Problem 878

Solve the differential equation: $y' - \frac{x^2 x - 1}{x^2 - x} y = 1$ .

See Solution

Problem 879

Calculate the integral $J=\int_{0}(x+x+y-2y) d x d y$ over the curves $y=x-x^{2}$ and $y=-x$ .

See Solution

Problem 880

Calculate the double integral $J=\int_{0}(x + xy - 2x) \, dx \, dy$ over the region defined by $y=-x-x^2$ and $y=-x$ .

See Solution

Problem 881

Evaluate the double integral $\iint_{D}(x^{2}+4y+x+25x) \, dx \, dy$ over the region $D$ bounded by $y=x$ and $y=1$ , where $0 \leq x, y \leq 1$ .

See Solution

Problem 882

A company sells $Q(x) = 60x - x^2$ units after spending $\$x$ thousand on ads. Find $Q(7)$ and $Q'(7)$ , and interpret.

See Solution

Problem 883

1. Find the volume of the solid with base between $y=x^{2}$ and $x$ -axis, $x=0$ to $x=2$ , with square cross sections.
2. Find the volume of the solid with base between $y=\sec x$ and $x$ -axis, $x=\pi / 4$ to $x=\pi / 3$ , with square cross sections.
3. Find the volume of the solid formed by revolving the region between the curves about the $x$ -axis: (a) $y=\sqrt{25-x^{2}}$ , $y=3$ . (b) $y=9-x^{2}$ , $y=0$ . (c) $x=\sqrt{y}$ , $x=y / 4$ . (d) $y=\sin x$ , $y=\cos x$ , $x=0$ , $x=\pi / 4$ .

See Solution

Problem 884

Find the volume when the area between $y=x^{2}$ and $y=x^{3}$ is revolved around $x=1$ . Also, find the volume for $y=\sqrt{x}$ , $y=0$ , and $x=9$ revolved around $x=9$ .

See Solution

Problem 885

Find the volume of solids formed by revolving regions around axes for the given curves.

See Solution

Problem 886

Calculate the integral of the function: $\int \frac{x+1}{x} \, dx$ .

See Solution

Problem 887

Evaluate the integral $\int x e^{-x^{2}} \, dx$ .

See Solution

Problem 888

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

See Solution

Problem 889

Calculate the integral $\int(1-2 x)^{9} d x$ .

See Solution

Problem 890

Find the integral of $\sec^{2}(2\theta) \, d\theta$ .

See Solution

Problem 891

Simplify the expression $\left(\frac{04 y^{8}}{y^{\prime}}\right)^{\prime}$ .

See Solution

Problem 892

Evaluate the integral: $\int \frac{(\ln x)^{2}}{x} \, dx$

See Solution

Problem 893

Find the integral of $\sin x \sin (\cos x)$ with respect to $x$ : $\int \sin x \sin (\cos x) d x$ .

See Solution

Problem 894

Find the integral of the function $x \sqrt{x+2}$ with respect to $x$ : $\int x \sqrt{x+2} d x$ .

See Solution

Problem 895

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

See Solution

Problem 896

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

See Solution

Problem 897

Evaluate the integral $\int_{0}^{1} \sqrt[3]{1+7 x} \, dx$ .

See Solution

Problem 898

Find the integral of $\sinh ^{2} x \cosh x$ with respect to $x$ : $\int \sinh ^{2} x \cosh x \, dx$ .

See Solution

Problem 899

Calculate the integral: $\int x(2 x+5)^{8} d x$

See Solution

Problem 900

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

See Solution

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Calculus

Problem 801

Find the tangent line to f(x)=−x2+5f(x)=-x^{2}+5f(x)=−x2+5 at x=5x=5x=5 and use it to estimate f(5.1)f(5.1)f(5.1).

Problem 802

Find points (x,y)(x, y)(x,y) where ∇f(x,y)=(0,0)\nabla f(x, y)=(0,0)∇f(x,y)=(0,0) for f(x,y)=x33−y33+3xyf(x, y)=\frac{x^{3}}{3}-\frac{y^{3}}{3}+3xyf(x,y)=3x3​−3y3​+3xy and classify them.

Problem 803

Problem 804

Find the second derivative d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ for the equation y+xy=5\sqrt{y}+x y=5y​+xy=5.

Problem 805

Integrate: ∫2x+54x2−20x+29dx\int \frac{2 x+5}{\sqrt{4 x^{2}-20 x+29}} d x∫4x2−20x+29​2x+5​dx (15 pts, 1a: 5 pts)

Problem 806

Find the salt concentration in a 100-gallon tank after 30 mins, starting with 10 lbs salt and adding brine at 4gal/min4 \mathrm{gal/min}4gal/min.

Problem 807

Calculate the surface area from revolving x=13(y2+2)3/2x=\frac{1}{3}(y^{2}+2)^{3/2}x=31​(y2+2)3/2 around the x-axis for 1≤y≤21 \leq y \leq 21≤y≤2.

Problem 808

Evaluate the integral ∫1∞kx−5 dx\int_{1}^{\infty} k x^{-5} \, dx∫1∞​kx−5dx.

Problem 809

Problem 810

Calculate the integral: ∫(1+3x)x2dx\int(1+3 x) x^{2} d x∫(1+3x)x2dx

Problem 811

Calculate the integral ∫4x(x2−2)2dx\int 4 x\left(x^{2}-2\right)^{2} d x∫4x(x2−2)2dx.

Problem 812

Calculate the average rate of change of f(x)=2x4−3xf(x)=2 x^{4}-3 xf(x)=2x4−3x from x=−1x=-1x=−1 to x=2x=2x=2.

Problem 813

Find dydx\frac{d y}{d x}dxdy​ in terms of xxx given that x=sec⁡3yx=\sec 3 yx=sec3y.

Problem 814

Find dydx\frac{d y}{d x}dxdy​ in terms of ttt for x=sin⁡(t)x=\sin(t)x=sin(t) and y=cos⁡2(t)y=\cos^2(t)y=cos2(t). Simplify your expression.

Problem 815

Find dydx\frac{d y}{d x}dxdy​ in terms of ttt for the equations x=sin⁡3tx=\sin^3 tx=sin3t and y=cos⁡4ty=\cos^4 ty=cos4t.

Problem 816

Find and simplify dydx\frac{d y}{d x}dxdy​ using the parametric equations x=sin⁡(t)x=\sin(t)x=sin(t) and y=cos⁡2(t)y=\cos^2(t)y=cos2(t).

Problem 817

Find dydx\frac{d y}{d x}dxdy​ in terms of ttt for the equations x=sin⁡(t)x=\sin(t)x=sin(t), y=cos⁡2(t)y=\cos^2(t)y=cos2(t).

Problem 818

Find the rate of change of sales S(t)=10,000+2000t−200t2S(t) = 10,000 + 2000t - 200t^2S(t)=10,000+2000t−200t2 at t=0t=0t=0, t=4t=4t=4, and t=8t=8t=8 weeks.

Problem 819

Evaluate the integral: ∫3axb2+c2x2 dx\int \frac{3 a x}{b^{2}+c^{2} x^{2}} \, dx∫b2+c2x23ax​dx

Problem 820

Upton Chuck free falls for 2.60 seconds. Calculate his final velocity and the distance fallen. Use v=gtv = gtv=gt and d=12gt2d = \frac{1}{2}gt^2d=21​gt2.

Problem 821

Find the integral ∫tan⁡8xsec⁡4x dx\int \tan^{8} x \sec^{4} x \, dx∫tan8xsec4xdx.

Problem 822

Calculate the double integrals: 1) ∬x2+y2≤1cos⁡(x2+y2)dxdy\iint_{x^{2}+y^{2} \leq 1} \cos \left(x^{2}+y^{2}\right) d x d y∬x2+y2≤1​cos(x2+y2)dxdy and 2) ∫01∫y1cos⁡(x)x2dxdy\int_{0}^{1} \int_{\sqrt{y}}^{1} \frac{\cos (x)}{x^{2}} d x d y∫01​∫y​1​x2cos(x)​dxdy.

Problem 823

Solve on separate sheets with signatures. No calculators. Justify answers. 1. (a) Find the integral: ∫xarctan⁡(x)dx\int x \arctan (x) d x∫xarctan(x)dx (b) Evaluate: ∫0π/2cos⁡(t)sin⁡(t)dt\int_{0}^{\pi / 2} \frac{\cos (t)}{\sqrt{\sin (t)}} d t∫0π/2​sin(t)​cos(t)​dt

Problem 824

Evaluate the integral ∫01x4(1−x)41+x2dx\int_{0}^{1} \frac{x^{4}(1-x)^{4}}{1+x^{2}} d x∫01​1+x2x4(1−x)4​dx using integration techniques.

Problem 825

Monthly sales (in thousands) for a music store are given by S(t)=200t2+36S(t)=\frac{200}{t^{2}+36}S(t)=t2+36200​. Find S(2)S(2)S(2), S′(2)S'(2)S′(2), and estimate sales for month 3.

Problem 826

Monthly sales for a record album are given by S(t)=200tt2+36S(t)=\frac{200 t}{t^{2}+36}S(t)=t2+36200t​. Find: a) S(2)S(2)S(2), b) S′(2)S^{\prime}(2)S′(2), c) estimate sales in month 3 using (a) and (b).

Problem 827

Tính tích phân I=∫AB(2xy dx+x2 dy)I=\int_{AB} (2xy \, dx + x^2 \, dy)I=∫AB​(2xydx+x2dy) trên đoạn thẳng AB:y=x3−3x2AB: y=x^3-3x^2AB:y=x3−3x2 với A(1,−2)A(1,-2)A(1,−2), B(2,−4)B(2,-4)B(2,−4).

Problem 828

Differentiate x3y2x^{3} y^{2}x3y2 with respect to yyy.

Problem 829

Differentiate x3y2x^{3} y^{2}x3y2 with respect to xxx.

Problem 830

Find the free extrema of the function z=−2x2+22y2−192y+x2yz=-2x^{2}+22y^{2}-192y+x^{2}yz=−2x2+22y2−192y+x2y.

Problem 831

Find the free extremes of the function z=3x2−8y2+64y−yx2z=3 x^{2}-8 y^{2}+64 y-y x^{2}z=3x2−8y2+64y−yx2.

Problem 832

Find dydx\frac{d y}{d x}dxdy​ for the equation 4e−3x+e4y+8=04 e^{-3 x}+e^{4 y}+8=04e−3x+e4y+8=0.

Problem 833

Differentiate the function 5x2−52x\frac{5 x^{2}-5}{2 x}2x5x2−5​.

Problem 834

Differentiate the function F(x)=ln⁡x3x2−e−9xln⁡x2F(x)=\frac{\ln x}{3 x^{2}}-\frac{e^{-9 x}}{\ln x^{2}}F(x)=3x2lnx​−lnx2e−9x​.

Problem 835

Find the second derivative H′′(x)H^{\prime \prime}(x)H′′(x) of the function H(x)=xexH(x)=x e^{x}H(x)=xex.

Problem 836

Evaluate the integrals: ∫1/etan⁡xtdt1+t2+∫1/ecot⁡xdtt(1+t2)\int_{1 / e}^{\tan x} \frac{t d t}{1+t^{2}} + \int_{1 / e}^{\cot x} \frac{d t}{t(1+t^{2})}∫1/etanx​1+t2tdt​+∫1/ecotx​t(1+t2)dt​.

Problem 837

Find the derivative g′(x)g^{\prime}(x)g′(x) for the function g(x)=x21+x+ex1+x2g(x)=\frac{x^{2}}{1+x}+\frac{e^{x}}{1+x^{2}}g(x)=1+xx2​+1+x2ex​.

Problem 838

Let f(θ0,θ1)f(\theta_{0}, \theta_{1})f(θ0​,θ1​) be a smooth function. Which statements about gradient descent minimizing fff are true?

Problem 839

Check if the function f(x)f(x)f(x) is continuous at x=−3x=-3x=−3 where f(x)={6+x2,x≥−34−3x,x<−3f(x)=\begin{cases} 6+x^{2}, & x \geq-3 \\ 4-3 x, & x<-3 \end{cases}f(x)={6+x2,4−3x,​x≥−3x<−3​.

Problem 840

Problem 841

Solve the DE: (1−x)y′=y2(1-x) y' = y^2(1−x)y′=y2 and determine if the solution is general or particular. Also solve: a2dx=yy2−a2dya^2 dx = y \sqrt{y^2 - a^2} dya2dx=yy2−a2​dy.

Find the tangent line to $f(x)=-x^{2}+5$ at $x=5$ and use it to estimate $f(5.1)$ .

Find points $(x, y)$ where $\nabla f(x, y)=(0,0)$ for $f(x, y)=\frac{x^{3}}{3}-\frac{y^{3}}{3}+3xy$ and classify them.

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

Integrate: $\int \frac{2 x+5}{\sqrt{4 x^{2}-20 x+29}} d x$ (15 pts, 1a: 5 pts)

Find the salt concentration in a 100-gallon tank after 30 mins, starting with 10 lbs salt and adding brine at $4 \mathrm{gal/min}$ .

Calculate the surface area from revolving $x=\frac{1}{3}(y^{2}+2)^{3/2}$ around the x-axis for $1 \leq y \leq 2$ .

Evaluate the integral $\int_{1}^{\infty} k x^{-5} \, dx$ .

Calculate the integral: $\int(1+3 x) x^{2} d x$

Calculate the integral $\int 4 x\left(x^{2}-2\right)^{2} d x$ .

Calculate the average rate of change of $f(x)=2 x^{4}-3 x$ from $x=-1$ to $x=2$ .

Find $\frac{d y}{d x}$ in terms of $x$ given that $x=\sec 3 y$ .

Find $\frac{d y}{d x}$ in terms of $t$ for $x=\sin(t)$ and $y=\cos^2(t)$ . Simplify your expression.

Find $\frac{d y}{d x}$ in terms of $t$ for the equations $x=\sin^3 t$ and $y=\cos^4 t$ .

Find and simplify $\frac{d y}{d x}$ using the parametric equations $x=\sin(t)$ and $y=\cos^2(t)$ .

Find $\frac{d y}{d x}$ in terms of $t$ for the equations $x=\sin(t)$ , $y=\cos^2(t)$ .

Find the rate of change of sales $S(t) = 10,000 + 2000t - 200t^2$ at $t=0$ , $t=4$ , and $t=8$ weeks.

Evaluate the integral: $\int \frac{3 a x}{b^{2}+c^{2} x^{2}} \, dx$

Upton Chuck free falls for 2.60 seconds. Calculate his final velocity and the distance fallen. Use $v = gt$ and $d = \frac{1}{2}gt^2$ .

Find the integral $\int \tan^{8} x \sec^{4} x \, dx$ .

Calculate the double integrals: 1) $\iint_{x^{2}+y^{2} \leq 1} \cos \left(x^{2}+y^{2}\right) d x d y$ and 2) $\int_{0}^{1} \int_{\sqrt{y}}^{1} \frac{\cos (x)}{x^{2}} d x d y$ .

Solve on separate sheets with signatures. No calculators. Justify answers.
1. (a) Find the integral: $\int x \arctan (x) d x$ (b) Evaluate: $\int_{0}^{\pi / 2} \frac{\cos (t)}{\sqrt{\sin (t)}} d t$

Evaluate the integral $\int_{0}^{1} \frac{x^{4}(1-x)^{4}}{1+x^{2}} d x$ using integration techniques.

Monthly sales (in thousands) for a music store are given by $S(t)=\frac{200}{t^{2}+36}$ . Find $S(2)$ , $S'(2)$ , and estimate sales for month 3.

Monthly sales for a record album are given by $S(t)=\frac{200 t}{t^{2}+36}$ . Find: a) $S(2)$ , b) $S^{\prime}(2)$ , c) estimate sales in month 3 using (a) and (b).

Tính tích phân $I=\int_{AB} (2xy \, dx + x^2 \, dy)$ trên đoạn thẳng $AB: y=x^3-3x^2$ với $A(1,-2)$ , $B(2,-4)$ .

Differentiate $x^{3} y^{2}$ with respect to $y$ .

Differentiate $x^{3} y^{2}$ with respect to $x$ .

Find the free extrema of the function $z=-2x^{2}+22y^{2}-192y+x^{2}y$ .

Find the free extremes of the function $z=3 x^{2}-8 y^{2}+64 y-y x^{2}$ .

Find $\frac{d y}{d x}$ for the equation $4 e^{-3 x}+e^{4 y}+8=0$ .

Differentiate the function $\frac{5 x^{2}-5}{2 x}$ .

Differentiate the function $F(x)=\frac{\ln x}{3 x^{2}}-\frac{e^{-9 x}}{\ln x^{2}}$ .

Find the second derivative $H^{\prime \prime}(x)$ of the function $H(x)=x e^{x}$ .

Evaluate the integrals: $\int_{1 / e}^{\tan x} \frac{t d t}{1+t^{2}} + \int_{1 / e}^{\cot x} \frac{d t}{t(1+t^{2})}$ .

Find the derivative $g^{\prime}(x)$ for the function $g(x)=\frac{x^{2}}{1+x}+\frac{e^{x}}{1+x^{2}}$ .

Let $f(\theta_{0}, \theta_{1})$ be a smooth function. Which statements about gradient descent minimizing $f$ are true?

Check if the function $f(x)$ is continuous at $x=-3$ where $f(x)=\begin{cases} 6+x^{2}, & x \geq-3 \\ 4-3 x, & x<-3 \end{cases}$ .

Solve the DE: $(1-x) y' = y^2$ and determine if the solution is general or particular. Also solve: $a^2 dx = y \sqrt{y^2 - a^2} dy$ .

Solve the differential equation $y' = x e^{-y - x^2}$ . Is the solution general or particular?

Solve the separable differential equation $x y^{3} dx + (y+1)e^{-x} dy = 0$ and identify the type of solution. Also, solve $y' = x y^{3}$ .

An ice sphere melts at $2 \pi$ m³/hr. Find the surface area decrease rate when radius is 5 m. Options: (A) $\frac{4 \pi}{5}$ (B) $40 \pi$ (C) $80 \pi^{2}$ (D) $100 \pi$

Find the integral of $\frac{1}{\sqrt[7]{x}}$ with respect to $x$ .

Calculate the integral: $\int 2 t \sqrt{3 t^{2}-1} \, dt$

Evaluate the integral: $\int \frac{\ln \theta}{\theta} d \theta$

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

Evaluate $\int F \cdot dr$ for $\bar{f}=x y \bar{i}-z \hat{j}+x^{2} \bar{k}$ along $\alpha: x=t^{2}, y=2t, z=t^{3}$ from $t=0$ to $t=1$ . Ans: $\frac{51}{70}$

Evaluate $\int F \cdot d r$ for $\bar{F}=x y \bar{i}-z j+x^{2} \bar{k}$ along the curve $x=t^{2}, y=2t, z=t^{3}$ from $t=0$ to $t=1$ .

Evaluate $\int F \cdot d r$ for $\bar{F}=x y \bar{i}-z j+x^{2} \bar{k}$ along the curve $x=t^{2}$ , $y=2t$ , $z=t^{3}$ from $t=0$ to $t=1$ . Answer: $\frac{51}{70}$ .

Calculate the work done by the force $\bar{f}=\bar{z} \bar{i}+x \bar{j}+y \hat{k}$ along the curve $\bar{r}=\cos t i+\sin t j-t \bar{k}$ from $t=0$ to $t=2 \pi$ .

Evaluate $\oint_{C}(y z \, dx + x z \, dy + x y \, dz)$ over a helix defined by $x=a \cos t$ , $y=a \sin t$ , $z=k t$ for $t$ from 0 to $2\pi$ .

Evaluate $\oint_{C}(y z \, dl + x z \, dy + l y \, dz)$ for the helix $x=a \cos t, y=a \sin t, z=k t$ as $t$ goes from 0 to $2\pi$ .

Evaluate $\oint_{C}(y z d x+x z d y+x y d z)$ for the helix $x=a \cos t, y=a \sin t, z=k t$ as $t$ goes from 0 to $2\pi$ .

Evaluate $\oint_{c}(y z d x+x z d y+x y d z)$ along the helix $x=a \cos t, y=a \sin t, z=k t$ for $t$ from 0 to $2 \pi$ .

Evaluate $\oint_{c}(y z d x+x z d y+x y d z)$ on the helix $x=a \cos t, y=a \sin t, z=k t$ for $t$ from 0 to $2 \pi$ .

Evaluate $\oint(y z d x+x z d y+x y d z)$ along the helix $x=a \cos t, y=a \sin t, z=k t$ for $t$ from 0 to $2 \pi$ .

Find $f(x)$ if $\frac{d f}{d x}=3 x^{2}-3-4 e^{2 x}$ and $f(0)=8$ . Choose from options a, b, c, d.

Find the limit as $x$ approaches 0 for $\frac{6(1-\cos x)}{x^{2}}$ .

What is the derivative of $\cos x$ with respect to $x$ ?

Find the integral of $\sec y \tan y$ with respect to $y$ .

Calculate the integral $\int (5 e^{w} + \cos w + w^{3}) \, dw$ .

Evaluate the integral: $\int \frac{6 y^{8}+12 y^{2}-y^{6}}{3 y^{7}} d y$

Find the limit: $\operatorname{Lim}_{x \rightarrow-3} \frac{4 x^{2}+9 x-9}{x^{3}+27}$ .

Solve the differential equation: $\frac{dy}{dx} = 3y$ .

Evaluate the integral $\int_{0}^{6}(2+5 x) e^{1 / 3 x} d x$ .

Find the integral of $\sec^{2} \theta$ with respect to $\theta$ .

Calculate the integral of the function: $\int \cos z \, dz$

Find the integral of $7^{x}$ with respect to $x$ .

Calculate the integral of $\sin \theta$ with respect to $\theta$ : $\int \sin \theta \, d\theta$ .

Calculate the integral: $\int \frac{2 b}{b^{2}+7} d b$

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{e^{-1 / x^{9}}}{x^{2}}$