Calculus

Problem 24701

Find all antiderivatives of $H(z)=-7 z^{-8}$ and verify by differentiating.

See Solution

Problem 24702

Find the derivative $f'(a)$ of the function $f(t)=t^{4}-6t$ . What is $f'(a)$ ?

See Solution

Problem 24703

Find all antiderivatives of $G(t)=13$ and verify by differentiating your result.

See Solution

Problem 24704

Given $y=\sqrt{2x+1}$ with $x$ and $y$ as functions of $t$ :
(a) If $dx/dt=15$ , find $dy/dt$ when $x=4$ .
(b) If $dy/dt=3$ , find $dx/dt$ when $x=24$ .

See Solution

Problem 24705

Find the derivative of the function $(3 x^{2} e^{x^{3}})$ .

See Solution

Problem 24706

Find the derivative $f^{\prime}(a)$ for the function $f(x)=\frac{1}{\sqrt{x+9}}$ .

See Solution

Problem 24707

Find the volume of the solid formed by rotating the area between $y=x^2$ , the $x$ -axis, and $x=2$ around $x=2$ .

See Solution

Problem 24708

Evaluate the integral: $\int\left(\frac{6}{\sqrt{x}}+6 \sqrt{x}\right) dx$ .

See Solution

Problem 24709

Calculate the integral of $(12 x+5)^{2}$ with respect to $x$ .

See Solution

Problem 24710

Calculate the integral: $\int \frac{x^{2}}{\left(x^{3}-3\right)^{2}} d x$

See Solution

Problem 24711

Given $v(R)=c R^{2}$ , if $R$ is accurate to $8\%$ , find the speed's percentage error. Choose from A, B, C, or D.

See Solution

Problem 24712

Find the derivative of the function $\left(\frac{1}{\sqrt{3x}}\right)$ .

See Solution

Problem 24713

Find local maxima and minima from $f^{\prime}(x)=(x-2)(x+1)(x+5)$ and identify intervals of increase/decrease.

See Solution

Problem 24714

Find the derivative of the function $f(x) = (x^{2}-4)^{3}(1-x^{3})^{4}$ .

See Solution

Problem 24715

Find the derivative: $y' = \frac{d}{dx}\left( x \arccos x - \sqrt{1-x^{2}} \right)$ .

See Solution

Problem 24716

Differentiate $f(\theta)=\frac{\sec \theta}{9+\sec \theta}$ .

See Solution

Problem 24717

Find the indefinite integral: $\int\left(\sec ^{2} \theta+5\right) d \theta=\square$ . Check by differentiating your result.

See Solution

Problem 24718

You have \$400,000 saved for retirement at 5\% monthly interest. How much can you withdraw monthly for 20 years?

See Solution

Problem 24719

Find the indefinite integral: $\int \frac{4 x^{4}-8 x^{2}}{x} d x$

See Solution

Problem 24720

Find local maxima and minima from $f^{\prime}(x)=(x-2)(x+1)(x+5)$ and identify intervals of increase/decrease.

See Solution

Problem 24721

An arrow shot upward on Mars at $64 \mathrm{~m/s}$ has height $y=64t-1.86t^{2}$ . Find average speed for intervals: (a) (i) $[1,2]$ , (ii) $[1,1.5]$ , (iii) $[1,1.1]$ , (iv) $[1,1.01]$ , (v) $[1,1.001]$ . (b) Estimate speed at $t=1$ .

See Solution

Problem 24722

Find the indefinite integral $\int \sqrt[11]{r^{5}} d r$ and verify by differentiating your result.

See Solution

Problem 24723

Calculate the total amount and interest for continuous compounding on \$ 25,000 at 2.4\% for 8 years.

See Solution

Problem 24724

Use $f(x)=x^{2}-4$ to: (a) find $f'(1)$ , (b) calculate $f(1)$ and tangent line, (c) graph $f(x)$ and tangent line.

See Solution

Problem 24725

Find local maxima/minima and intervals of increase/decrease for $f^{\prime}(x)=(x-2)(x+1)(x+5)$ .

See Solution

Problem 24726

Find the indefinite integral of $\sec^2 x + 7$ and verify by differentiating your result.

See Solution

Problem 24727

Given $f(0)=-3$ and $f^{\prime}(x) \leq 9$ , find the maximum possible value of $f(3)$ .

See Solution

Problem 24728

Find the antiderivative $F$ of $f(x)=x^{5}-4 x^{-3}-4$ with $F(1)=2$ . What is $F(x)=\square$ ?

See Solution

Problem 24729

Find the average velocity of a particle with motion $s=4 \sin \pi t + 4 \cos \pi t$ for intervals and estimate instantaneous velocity at $t=1$ .

See Solution

Problem 24730

Find the antiderivative $F$ of $f(x)=x^{3}-3 x^{-4}-4$ such that $F(1)=1$ .

See Solution

Problem 24731

Find the derivative of $f(x)=x^{2}+5$ at $x=1$ , then find $f(1)$ and the tangent line at $(1, f(1))$ .

See Solution

Problem 24732

Find the antiderivative $F$ of $f(v)=\frac{3}{4} \sec v \tan v$ such that $F(0)=1$ , for $-\frac{\pi}{2}<v<\frac{\pi}{2}$ .

See Solution

Problem 24733

Find the distance the particle travels to the right from $s(t)=3t-2t^{2}$ at $t=4$ , where $t \geq 0$ .

See Solution

Problem 24734

Solve the initial value problem: $g'(x)=9x\left(x^8-\frac{1}{9}\right)$ with $g(1)=3$ . Find $g(x)$ .

See Solution

Problem 24735

Find the derivative of the function $f(x)=e^{\cos x}-\ln (5 x+11)+\sec (3 x)-\arctan (x^{2})+\sinh (\sqrt{x})$ .

See Solution

Problem 24736

Find the $x$ -values where the instantaneous rate of change of $y = \cos(x - \pi)$ is zero in $x \in [0, 2\pi]$ .

See Solution

Problem 24737

Use implicit differentiation on the equation $x^{7}=y^{2}-6$ to find $\frac{d y}{d x}$ .

See Solution

Problem 24738

Calculate $\lim _{x \rightarrow 3} \frac{2 x^{3}-6 x^{2}+x-3}{x-3}$ .

See Solution

Problem 24739

Find the position function $s(t)$ from the velocity $v(t)=8t+7$ with initial position $s(0)=0$ .

See Solution

Problem 24740

Calculate the indefinite integral using substitution or state "IMPOSSIBLE". Use $C$ for the integration constant.
$\int e^{9 x} d x$

See Solution

Problem 24741

Find the derivative of $f(x)=4 \sin x$ using the limit definition of a derivative. No need to simplify.

See Solution

Problem 24742

Find the line equation for the linear approximation of $f(x)=\sec x$ at $a=0$ , estimate $f(0.01)$ , and compute percent error.

See Solution

Problem 24743

Find the position function $s(t)$ from the velocity $v(t)=5 \sqrt{t}$ with initial position $s(0)=2$ .

See Solution

Problem 24744

Find the limit as $x$ approaches $\sqrt{5}$ for the expression $\frac{\sqrt{5}-x}{x^{2}-5}$ .

See Solution

Problem 24745

Find the derivative of the function $f(x)=e^{\cos x}-\ln (5 x+11)+\sec (3 x)-\arctan (x^{2})+\sinh (\sqrt{x})$ .

See Solution

Problem 24746

Find the derivative of the function $f(x)=\left(\frac{6 x^{3}}{\left(x-x^{2}\right)^{4}}\right)$ .

See Solution

Problem 24747

Find the limit as $x$ approaches 2 for the expression $\frac{-3}{(x-2)^{2}}$ .

See Solution

Problem 24748

Find the limit: $\lim _{x \rightarrow \infty} \frac{x^{2}-2 x+5}{x^{3}+2 x^{2}+14}$ . Options: $\frac{5}{14}$ , $\infty$ , 1, 0.

See Solution

Problem 24749

Find the average rate of change of $f(x)=x-3 \sqrt{x}$ from $x=1$ to $x=4$ .

See Solution

Problem 24750

Find the derivative of the function $f(x)=\frac{5}{x+2}$ using the limit definition. No need to simplify.

See Solution

Problem 24751

Evaluate the integral using substitution or state "IMPOSSIBLE": $\int \sqrt[4]{z^{4}+16} z^{3} d z$ . Use $C$ for the constant.

See Solution

Problem 24752

Explain why $\frac{d x}{d t}=k(a-x)(b-x)$ for the reaction $A+B \rightarrow A B$ with concentrations $[A],[B],[A B]$ .

See Solution

Problem 24753

A baseball is tossed at 33 m/s. Height is $h(t)=33t-4.9t^2$ . Find average velocity for intervals [1,1.5], [1,1.25], and [1,1.1].

See Solution

Problem 24754

Find the derivative of $f(x) = 2x^{5}$ using the limit definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ .

See Solution

Problem 24755

Find the derivative of the function $f(x)=\ln(x^{3}+9)-4e^{x/2}-x$ . What is $f'(x)$ ?

See Solution

Problem 24756

Find the limit: $\lim _{x \rightarrow -3} \frac{-3}{(x-2)^{2}}$ .

See Solution

Problem 24757

Evaluate the integral: $\int\left(\frac{1}{e^{x}}+4 \sqrt{x}-\csc ^{2} x+\frac{12}{x^{7}}+\pi\right) d x$

See Solution

Problem 24758

Find the position function from the acceleration $a(t)=-24$ with initial velocity $v(0)=22$ and position $s(0)=0$ .

See Solution

Problem 24759

Find the tangent line equation for $g(x)=x+2 \cos (x)$ at $x=\frac{\pi}{2}$ .

See Solution

Problem 24760

Find the tangent line equation for $g(x)=x+2 \cos (x)$ at $x=\frac{\pi}{2}$ .

See Solution

Problem 24761

A cone's height rises at 8 ft/min, volume is 178 ft³. When radius is 4 ft, find the radius's change rate. Use $V=\frac{1}{3} \pi r^{2} h$ . Round to three decimal places.

See Solution

Problem 24762

Let $f(x)=\left\{\begin{array}{ll} \frac{\sin x}{x}, & 0<x \leq \pi \\ 1, & x=0 \end{array}\right.$ Show $x f(x)=\sin x$ for $0 \leq x \leq \pi$ and find the volume of the solid from revolving the shaded area about the $y$ -axis.

See Solution

Problem 24763

Calculate the growth rate $\frac{dL}{dt}$ for femur length $L$ at $t=15, 20, 30$ weeks using $L=-37.33+3.69t-6.29 \times 10^{-4} t^{3}$ .

See Solution

Problem 24764

Find the derivative $f^{\prime}(x)$ of $f(x)=\frac{x}{x^{2}+3 x-1}$ and simplify your answer.

See Solution

Problem 24765

Calculate the integral: $\int_{0}^{\pi / 3} 4 \frac{\sin u}{\cos ^{2} u} d u$

See Solution

Problem 24766

Find the derivative of $\cos(\sqrt{x})$ with respect to $x$ .

See Solution

Problem 24767

Find the volume of the solid with base $y=28 \sqrt{\cos x}$ on $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and isosceles right triangle cross sections.

See Solution

Problem 24768

Calculate the integral: $\int\left(x^{2}+4\right)^{7} 2 x d x$ using $u=x^{2}+4$ .

See Solution

Problem 24769

Find the $x$ in $[0,2]$ where $f^{\prime}(x)=2 \sin ^{4}(2 x)-3 \sin ^{3}(4 x)$ has a relative minimum. Round to three decimals.

See Solution

Problem 24770

Find the tangent line equation for $f(x)=\frac{7}{x-2}$ at $x=3$ .

See Solution

Problem 24771

Find the volume of a solid with a triangular base at $(0,0),(11,0),(0,11)$ and semicircular cross sections. Set up the integral.

See Solution

Problem 24772

Calculate the growth rate, $\frac{d L}{d t}$ , at $t=10, 20$ , and $25$ weeks for $L=-37.34+3.74 t-6.33 \times 10^{-4} t^{3}$ . What happens as fetus ages?
Write $\frac{\mathrm{dL}}{\mathrm{dt}}=\square$ .

See Solution

Problem 24773

Approximate area under $f(x)=\sqrt{x}$ from $a=5$ to $b=8$ using 6 rectangles. Then find exact area using integral.

See Solution

Problem 24774

Find the volume of a solid with base $y=28 \sqrt{\cos x}$ on $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and isosceles triangle cross sections. Set up the integral for volume.

See Solution

Problem 24775

Find the per capita growth rate from $\frac{\mathrm{dN}}{\mathrm{dt}}=\mathrm{rN}$ . If $r<0$ and $N(0)=20$ , is $N(1)>20$ or $<20$ ?

See Solution

Problem 24776

Find the line equation for the linear approximation of $f(x)=(1331+x)^{-\frac{1}{3}}$ at $a=0$ , estimate $f(0.1)$ , and compute percent error using $100 \cdot \frac{|L(0.1)-\text{exact}|}{|\text{exact}|}$ .

See Solution

Problem 24777

Tyra invests \$5200 at 4.7% interest, compounded continuously. Find the value after 3 years, rounded to the nearest cent.

See Solution

Problem 24778

Given the population size $N(t)$ with $\frac{dN}{dt}=rN$ , find the per capita growth rate and analyze $N(1)$ if $r<0$ and $N(0)=20$ .

See Solution

Problem 24779

Find the volume of a solid with a triangular base $(0,0),(3,0),(0,3)$ and semicircular cross sections. Set up the integral.

See Solution

Problem 24780

Find the value of $c$ for the Mean Value Theorem with $f(x)=\sqrt{x-1}$ on $[2,5]$ . Use a calculator for decimal conversion.

See Solution

Problem 24781

Evaluate the integral $\int_{0}^{\pi / 3}(\cos x+\sec x)^{2} d x$ .

See Solution

Problem 24782

Find the consumers' surplus for the demand function $d(x)=250-0.06 x^{2}$ at demand level $x=40$ .

See Solution

Problem 24783

Find the volume using the shell method for the region bounded by $x=\sqrt{y}, x=-7y$ , and $y=2$ revolved around the $x$ -axis. Volume = $\square$ cubic units.

See Solution

Problem 24784

Find the volume of a solid with a triangular base at $(0,0),(11,0),(0,11)$ and semicircular cross sections. Set up the integral.

See Solution

Problem 24785

Find the size of the squares to cut from a 12-in. tin sheet to maximize the volume of an open-top box.

See Solution

Problem 24786

Find the volume of a solid with a triangular base at $(0,0),(9,0),(0,9)$ and semicircular cross sections. Set up the integral.

See Solution

Problem 24787

Find the integral $\int(\sin 8 y+\cos 5 y) d y$ and verify by differentiating your result.

See Solution

Problem 24788

Find the limit using L'Hôpital's Rule: $\lim _{x \rightarrow \frac{\pi^{+}}{2}} \frac{\cos x}{1-\sin x}$ .

See Solution

Problem 24789

Differentiate $g(t)=t^{6} \cos t$ to find $g^{\prime}(t)$ .

See Solution

Problem 24790

A 25 ft ladder slides down a wall. When it's 10 ft high, how fast is the top moving if the base moves at 3 ft/hour?

See Solution

Problem 24791

Find points on the curve $y=x^{3}+4x+3$ where the tangent is parallel to $6x-y=10$ . Are there multiple points?

See Solution

Problem 24792

Differentiate the function: $g(t)=\frac{t-\sqrt{t}}{t^{1/3}}$ . Find $g'(t)$ .

See Solution

Problem 24793

Approximate the integral $\int_{0}^{\pi}(2 x \sin x) d x$ using 4 equal subintervals and a Right Hand Riemann sum.

See Solution

Problem 24794

Evaluate the integral: $\int \frac{1}{8 x+5} dx$ . Assume $\mathrm{u}>0$ when $\ln \mathrm{u}$ appears.

See Solution

Problem 24795

Calculate the indefinite integral: $\int x \sqrt{12-x^{2}} \, dx$

See Solution

Problem 24796

Differentiate implicitly to find $\frac{d y}{d x}$ for $2 \sqrt{x}-3 \sqrt{y}=1$ and find the slope at (4,1).

See Solution

Problem 24797

Differentiate $y=\frac{x+7}{x^{3}+x-5}$ . Find $y'$ .

See Solution

Problem 24798

Differentiate $\tan(x + 5y) = 5y$ implicitly to find $\frac{dy}{dx}$ .

See Solution

Problem 24799

Calculate the limit: $\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}$ .

See Solution

Problem 24800

Approximate $\int_{0}^{\pi}(2 x \sin x) d x$ using 4 equal subintervals with a Right Hand Riemann sum.

See Solution

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Calculus

Problem 24701

Find all antiderivatives of H(z)=−7z−8H(z)=-7 z^{-8}H(z)=−7z−8 and verify by differentiating.

Problem 24702

Find the derivative f′(a)f'(a)f′(a) of the function f(t)=t4−6tf(t)=t^{4}-6tf(t)=t4−6t. What is f′(a)f'(a)f′(a)?

Problem 24703

Find all antiderivatives of G(t)=13G(t)=13G(t)=13 and verify by differentiating your result.

Problem 24704

Given y=2x+1y=\sqrt{2x+1}y=2x+1​ with xxx and yyy as functions of ttt:(a) If dx/dt=15dx/dt=15dx/dt=15, find dy/dtdy/dtdy/dt when x=4x=4x=4. (b) If dy/dt=3dy/dt=3dy/dt=3, find dx/dtdx/dtdx/dt when x=24x=24x=24.

Problem 24705

Find the derivative of the function (3x2ex3)(3 x^{2} e^{x^{3}})(3x2ex3).

Problem 24706

Find the derivative f′(a)f^{\prime}(a)f′(a) for the function f(x)=1x+9f(x)=\frac{1}{\sqrt{x+9}}f(x)=x+9​1​.

Problem 24707

Find the volume of the solid formed by rotating the area between y=x2y=x^2y=x2, the xxx-axis, and x=2x=2x=2 around x=2x=2x=2.

Problem 24708

Evaluate the integral: ∫(6x+6x)dx\int\left(\frac{6}{\sqrt{x}}+6 \sqrt{x}\right) dx∫(x​6​+6x​)dx.

Problem 24709

Calculate the integral of (12x+5)2(12 x+5)^{2}(12x+5)2 with respect to xxx.

Problem 24710

Calculate the integral: ∫x2(x3−3)2dx\int \frac{x^{2}}{\left(x^{3}-3\right)^{2}} d x∫(x3−3)2x2​dx

Problem 24711

Given v(R)=cR2v(R)=c R^{2}v(R)=cR2, if RRR is accurate to 8%8\%8%, find the speed's percentage error. Choose from A, B, C, or D.

Problem 24712

Find the derivative of the function (13x)\left(\frac{1}{\sqrt{3x}}\right)(3x​1​).

Problem 24713

Find local maxima and minima from f′(x)=(x−2)(x+1)(x+5)f^{\prime}(x)=(x-2)(x+1)(x+5)f′(x)=(x−2)(x+1)(x+5) and identify intervals of increase/decrease.

Problem 24714

Find the derivative of the function f(x)=(x2−4)3(1−x3)4f(x) = (x^{2}-4)^{3}(1-x^{3})^{4}f(x)=(x2−4)3(1−x3)4.

Problem 24715

Find the derivative: y′=ddx(xarccos⁡x−1−x2)y' = \frac{d}{dx}\left( x \arccos x - \sqrt{1-x^{2}} \right)y′=dxd​(xarccosx−1−x2​).

Problem 24716

Differentiate f(θ)=sec⁡θ9+sec⁡θf(\theta)=\frac{\sec \theta}{9+\sec \theta}f(θ)=9+secθsecθ​.

Problem 24717

Find the indefinite integral: ∫(sec⁡2θ+5)dθ=□\int\left(\sec ^{2} \theta+5\right) d \theta=\square∫(sec2θ+5)dθ=□. Check by differentiating your result.

Problem 24718

You have \$400,000 saved for retirement at 5\% monthly interest. How much can you withdraw monthly for 20 years?

Problem 24719

Find the indefinite integral: ∫4x4−8x2xdx\int \frac{4 x^{4}-8 x^{2}}{x} d x∫x4x4−8x2​dx

Problem 24720

Find local maxima and minima from f′(x)=(x−2)(x+1)(x+5)f^{\prime}(x)=(x-2)(x+1)(x+5)f′(x)=(x−2)(x+1)(x+5) and identify intervals of increase/decrease.

Problem 24721

Problem 24722

Find the indefinite integral ∫r511dr\int \sqrt[11]{r^{5}} d r∫11r5​dr and verify by differentiating your result.

Problem 24723

Calculate the total amount and interest for continuous compounding on \$ 25,000 at 2.4\% for 8 years.

Problem 24724

Use f(x)=x2−4f(x)=x^{2}-4f(x)=x2−4 to: (a) find f′(1)f'(1)f′(1), (b) calculate f(1)f(1)f(1) and tangent line, (c) graph f(x)f(x)f(x) and tangent line.

Problem 24725

Find local maxima/minima and intervals of increase/decrease for f′(x)=(x−2)(x+1)(x+5)f^{\prime}(x)=(x-2)(x+1)(x+5)f′(x)=(x−2)(x+1)(x+5).

Problem 24726

Find the indefinite integral of sec⁡2x+7\sec^2 x + 7sec2x+7 and verify by differentiating your result.

Problem 24727

Given f(0)=−3f(0)=-3f(0)=−3 and f′(x)≤9f^{\prime}(x) \leq 9f′(x)≤9, find the maximum possible value of f(3)f(3)f(3).

Problem 24728

Find the antiderivative FFF of f(x)=x5−4x−3−4f(x)=x^{5}-4 x^{-3}-4f(x)=x5−4x−3−4 with F(1)=2F(1)=2F(1)=2. What is F(x)=□F(x)=\squareF(x)=□?

Problem 24729

Find the average velocity of a particle with motion s=4sin⁡πt+4cos⁡πts=4 \sin \pi t + 4 \cos \pi ts=4sinπt+4cosπt for intervals and estimate instantaneous velocity at t=1t=1t=1.

Problem 24730

Find the antiderivative FFF of f(x)=x3−3x−4−4f(x)=x^{3}-3 x^{-4}-4f(x)=x3−3x−4−4 such that F(1)=1F(1)=1F(1)=1.

Problem 24731

Find the derivative of f(x)=x2+5f(x)=x^{2}+5f(x)=x2+5 at x=1x=1x=1, then find f(1)f(1)f(1) and the tangent line at (1,f(1))(1, f(1))(1,f(1)).

Problem 24732

Find the antiderivative FFF of f(v)=34sec⁡vtan⁡vf(v)=\frac{3}{4} \sec v \tan vf(v)=43​secvtanv such that F(0)=1F(0)=1F(0)=1, for −π2<v<π2-\frac{\pi}{2}<v<\frac{\pi}{2}−2π​<v<2π​.

Problem 24733

Find the distance the particle travels to the right from s(t)=3t−2t2s(t)=3t-2t^{2}s(t)=3t−2t2 at t=4t=4t=4, where t≥0t \geq 0t≥0.

Problem 24734

Solve the initial value problem: g′(x)=9x(x8−19)g'(x)=9x\left(x^8-\frac{1}{9}\right)g′(x)=9x(x8−91​) with g(1)=3g(1)=3g(1)=3. Find g(x)g(x)g(x).

Problem 24735

Find the derivative of the function f(x)=ecos⁡x−ln⁡(5x+11)+sec⁡(3x)−arctan⁡(x2)+sinh⁡(x)f(x)=e^{\cos x}-\ln (5 x+11)+\sec (3 x)-\arctan (x^{2})+\sinh (\sqrt{x})f(x)=ecosx−ln(5x+11)+sec(3x)−arctan(x2)+sinh(x​).

Problem 24736

Find the xxx-values where the instantaneous rate of change of y=cos⁡(x−π)y = \cos(x - \pi)y=cos(x−π) is zero in x∈[0,2π]x \in [0, 2\pi]x∈[0,2π].

Problem 24737

Use implicit differentiation on the equation x7=y2−6x^{7}=y^{2}-6x7=y2−6 to find dydx\frac{d y}{d x}dxdy​.

Problem 24738

Calculate lim⁡x→32x3−6x2+x−3x−3\lim _{x \rightarrow 3} \frac{2 x^{3}-6 x^{2}+x-3}{x-3}limx→3​x−32x3−6x2+x−3​.

Problem 24739

Find the position function s(t)s(t)s(t) from the velocity v(t)=8t+7v(t)=8t+7v(t)=8t+7 with initial position s(0)=0s(0)=0s(0)=0.

Problem 24740

Calculate the indefinite integral using substitution or state "IMPOSSIBLE". Use CCC for the integration constant. ∫e9xdx \int e^{9 x} d x ∫e9xdx

Find all antiderivatives of $H(z)=-7 z^{-8}$ and verify by differentiating.

Find the derivative $f'(a)$ of the function $f(t)=t^{4}-6t$ . What is $f'(a)$ ?

Find all antiderivatives of $G(t)=13$ and verify by differentiating your result.

Given $y=\sqrt{2x+1}$ with $x$ and $y$ as functions of $t$ :
(a) If $dx/dt=15$ , find $dy/dt$ when $x=4$ .
(b) If $dy/dt=3$ , find $dx/dt$ when $x=24$ .

Find the derivative of the function $(3 x^{2} e^{x^{3}})$ .

Find the derivative $f^{\prime}(a)$ for the function $f(x)=\frac{1}{\sqrt{x+9}}$ .

Find the volume of the solid formed by rotating the area between $y=x^2$ , the $x$ -axis, and $x=2$ around $x=2$ .

Evaluate the integral: $\int\left(\frac{6}{\sqrt{x}}+6 \sqrt{x}\right) dx$ .

Calculate the integral of $(12 x+5)^{2}$ with respect to $x$ .

Calculate the integral: $\int \frac{x^{2}}{\left(x^{3}-3\right)^{2}} d x$

Given $v(R)=c R^{2}$ , if $R$ is accurate to $8\%$ , find the speed's percentage error. Choose from A, B, C, or D.

Find the derivative of the function $\left(\frac{1}{\sqrt{3x}}\right)$ .

Find local maxima and minima from $f^{\prime}(x)=(x-2)(x+1)(x+5)$ and identify intervals of increase/decrease.

Find the derivative of the function $f(x) = (x^{2}-4)^{3}(1-x^{3})^{4}$ .

Find the derivative: $y' = \frac{d}{dx}\left( x \arccos x - \sqrt{1-x^{2}} \right)$ .

Differentiate $f(\theta)=\frac{\sec \theta}{9+\sec \theta}$ .

Find the indefinite integral: $\int\left(\sec ^{2} \theta+5\right) d \theta=\square$ . Check by differentiating your result.

Find the indefinite integral: $\int \frac{4 x^{4}-8 x^{2}}{x} d x$

Find local maxima and minima from $f^{\prime}(x)=(x-2)(x+1)(x+5)$ and identify intervals of increase/decrease.

Find the indefinite integral $\int \sqrt[11]{r^{5}} d r$ and verify by differentiating your result.

Use $f(x)=x^{2}-4$ to: (a) find $f'(1)$ , (b) calculate $f(1)$ and tangent line, (c) graph $f(x)$ and tangent line.

Find local maxima/minima and intervals of increase/decrease for $f^{\prime}(x)=(x-2)(x+1)(x+5)$ .

Find the indefinite integral of $\sec^2 x + 7$ and verify by differentiating your result.

Given $f(0)=-3$ and $f^{\prime}(x) \leq 9$ , find the maximum possible value of $f(3)$ .

Find the antiderivative $F$ of $f(x)=x^{5}-4 x^{-3}-4$ with $F(1)=2$ . What is $F(x)=\square$ ?

Find the average velocity of a particle with motion $s=4 \sin \pi t + 4 \cos \pi t$ for intervals and estimate instantaneous velocity at $t=1$ .

Find the antiderivative $F$ of $f(x)=x^{3}-3 x^{-4}-4$ such that $F(1)=1$ .

Find the derivative of $f(x)=x^{2}+5$ at $x=1$ , then find $f(1)$ and the tangent line at $(1, f(1))$ .

Find the antiderivative $F$ of $f(v)=\frac{3}{4} \sec v \tan v$ such that $F(0)=1$ , for $-\frac{\pi}{2}<v<\frac{\pi}{2}$ .

Find the distance the particle travels to the right from $s(t)=3t-2t^{2}$ at $t=4$ , where $t \geq 0$ .

Solve the initial value problem: $g'(x)=9x\left(x^8-\frac{1}{9}\right)$ with $g(1)=3$ . Find $g(x)$ .

Find the derivative of the function $f(x)=e^{\cos x}-\ln (5 x+11)+\sec (3 x)-\arctan (x^{2})+\sinh (\sqrt{x})$ .

Find the $x$ -values where the instantaneous rate of change of $y = \cos(x - \pi)$ is zero in $x \in [0, 2\pi]$ .

Use implicit differentiation on the equation $x^{7}=y^{2}-6$ to find $\frac{d y}{d x}$ .

Calculate $\lim _{x \rightarrow 3} \frac{2 x^{3}-6 x^{2}+x-3}{x-3}$ .

Find the position function $s(t)$ from the velocity $v(t)=8t+7$ with initial position $s(0)=0$ .

Calculate the indefinite integral using substitution or state "IMPOSSIBLE". Use $C$ for the integration constant.
$\int e^{9 x} d x$

Find the derivative of $f(x)=4 \sin x$ using the limit definition of a derivative. No need to simplify.

Find the line equation for the linear approximation of $f(x)=\sec x$ at $a=0$ , estimate $f(0.01)$ , and compute percent error.

Find the position function $s(t)$ from the velocity $v(t)=5 \sqrt{t}$ with initial position $s(0)=2$ .

Find the limit as $x$ approaches $\sqrt{5}$ for the expression $\frac{\sqrt{5}-x}{x^{2}-5}$ .

Find the derivative of the function $f(x)=e^{\cos x}-\ln (5 x+11)+\sec (3 x)-\arctan (x^{2})+\sinh (\sqrt{x})$ .

Find the derivative of the function $f(x)=\left(\frac{6 x^{3}}{\left(x-x^{2}\right)^{4}}\right)$ .

Find the limit as $x$ approaches 2 for the expression $\frac{-3}{(x-2)^{2}}$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{x^{2}-2 x+5}{x^{3}+2 x^{2}+14}$ . Options: $\frac{5}{14}$ , $\infty$ , 1, 0.

Find the average rate of change of $f(x)=x-3 \sqrt{x}$ from $x=1$ to $x=4$ .

Find the derivative of the function $f(x)=\frac{5}{x+2}$ using the limit definition. No need to simplify.

Evaluate the integral using substitution or state "IMPOSSIBLE": $\int \sqrt[4]{z^{4}+16} z^{3} d z$ . Use $C$ for the constant.

Explain why $\frac{d x}{d t}=k(a-x)(b-x)$ for the reaction $A+B \rightarrow A B$ with concentrations $[A],[B],[A B]$ .

A baseball is tossed at 33 m/s. Height is $h(t)=33t-4.9t^2$ . Find average velocity for intervals [1,1.5], [1,1.25], and [1,1.1].

Find the derivative of $f(x) = 2x^{5}$ using the limit definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ .

Find the derivative of the function $f(x)=\ln(x^{3}+9)-4e^{x/2}-x$ . What is $f'(x)$ ?

Find the limit: $\lim _{x \rightarrow -3} \frac{-3}{(x-2)^{2}}$ .

Evaluate the integral: $\int\left(\frac{1}{e^{x}}+4 \sqrt{x}-\csc ^{2} x+\frac{12}{x^{7}}+\pi\right) d x$

Find the position function from the acceleration $a(t)=-24$ with initial velocity $v(0)=22$ and position $s(0)=0$ .

Find the tangent line equation for $g(x)=x+2 \cos (x)$ at $x=\frac{\pi}{2}$ .

Find the tangent line equation for $g(x)=x+2 \cos (x)$ at $x=\frac{\pi}{2}$ .

A cone's height rises at 8 ft/min, volume is 178 ft³. When radius is 4 ft, find the radius's change rate. Use $V=\frac{1}{3} \pi r^{2} h$ . Round to three decimal places.

Calculate the growth rate $\frac{dL}{dt}$ for femur length $L$ at $t=15, 20, 30$ weeks using $L=-37.33+3.69t-6.29 \times 10^{-4} t^{3}$ .

Find the derivative $f^{\prime}(x)$ of $f(x)=\frac{x}{x^{2}+3 x-1}$ and simplify your answer.

Calculate the integral: $\int_{0}^{\pi / 3} 4 \frac{\sin u}{\cos ^{2} u} d u$

Find the derivative of $\cos(\sqrt{x})$ with respect to $x$ .

Find the volume of the solid with base $y=28 \sqrt{\cos x}$ on $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and isosceles right triangle cross sections.

Calculate the integral: $\int\left(x^{2}+4\right)^{7} 2 x d x$ using $u=x^{2}+4$ .

Find the $x$ in $[0,2]$ where $f^{\prime}(x)=2 \sin ^{4}(2 x)-3 \sin ^{3}(4 x)$ has a relative minimum. Round to three decimals.

Find the tangent line equation for $f(x)=\frac{7}{x-2}$ at $x=3$ .

Find the volume of a solid with a triangular base at $(0,0),(11,0),(0,11)$ and semicircular cross sections. Set up the integral.

Approximate area under $f(x)=\sqrt{x}$ from $a=5$ to $b=8$ using 6 rectangles. Then find exact area using integral.

Find the volume of a solid with base $y=28 \sqrt{\cos x}$ on $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and isosceles triangle cross sections. Set up the integral for volume.