Calculus

Problem 23401

Calculate the integral $\int_{0}^{6}(3-|3-x|) \, dx$ .

See Solution

Problem 23402

Find the velocity and acceleration at $t=1 \mathrm{~s}$ for $s(t)=t^{2}-3t$ .
$v(1)=\square$

See Solution

Problem 23403

Calculate the integral from 0 to 6 of the function $3 - (3 - x)$ .

See Solution

Problem 23404

Find the slope of the tangent line to the curve $y=x^{2}+6x$ at the point where $x=-3$ using the limit definition.

See Solution

Problem 23405

Tree growth rate $G$ relates to mass $M$ with $G=C M^{0.7}$ . Differentiate to find $\frac{d G}{d t}$ in terms of $\frac{d M}{d t}$ . $\frac{d G}{d t}=\square \frac{d M}{d t}$

See Solution

Problem 23406

Find the velocity $v(1)$ and acceleration $a(1)$ for the position function $s(t)=t^{4}-3t$ at $t=1$ s.

See Solution

Problem 23407

Three carts of mass $1250 \mathrm{~kg}$ move at $15.0 \mathrm{~m/s}$ at D. Find speeds at B ( $[16.6 \mathrm{~m/s}]$ ) and C ( $[10.4 \mathrm{~m/s}]$ ). Heights: A: 16m, B: 0m, C: 8.5m, D: 2.5m.

See Solution

Problem 23408

For the reaction $2 \mathrm{~A} \rightarrow \mathrm{B}$ with a rate constant of $0.853 \mathrm{~s}^{-1}$ , how long until $60\%$ of A remains?

See Solution

Problem 23409

Find the tangent line equation for the curve $y=x^{4}-2x^{3}+5$ at the point where $x=2$ .

See Solution

Problem 23410

A study shows tree growth rate $\mathrm{G}$ depends on mass $M$ : $\mathrm{G}=\mathrm{CM}^{0.7}$ .
(a) Differentiate to find $\frac{d G}{d t}$ in terms of $\frac{d M}{d t}$ : $\frac{\mathrm{dG}}{\mathrm{dt}}=0.7 \mathrm{CM}^{-0.3} \frac{\mathrm{dM}}{\mathrm{dt}}$
Rearrange to express in fractional growth rates: $\frac{\mathrm{dG}}{\mathrm{dt}}=0.7 \mathrm{CM}^{0.7} \cdot \square \frac{\mathrm{dM}}{\mathrm{dt}}$

See Solution

Problem 23411

Evaluate $\int_{0}^{1} x e^{x^{2}} \cos(e^{x^{2}}) \, \mathrm{d} x$ using substitution.

See Solution

Problem 23412

Calculate the integral $\int_{0}^{6} \sqrt{9-(x-3)^{2}} \, dx$ .

See Solution

Problem 23413

Chewing frequency $c$ is related to body mass $M$ by $c=kM^{-0.128}$ .
(a) Given $M(t)=1+2\sqrt{t}$ , find $\frac{dc}{dt}$ .
(b) With jaw length $L=rM^{0.312}$ , find $\frac{dc}{dL}$ .

See Solution

Problem 23414

Find points on the curve $y=\frac{1}{3} x^{3}+\frac{9}{x}$ where the tangent is parallel to $y=8 x+3$ .

See Solution

Problem 23415

Find the volume of the solid formed by rotating the region bounded by $f(x)=-3 x^{2}+12 x+36$ , $x=0$ , and $y=0$ about the $y$ -axis.

See Solution

Problem 23416

Substitute $u=x^{2}+1$ in the integral $\int_{0}^{2} 2 x \ln \left(x^{2}+1\right) d x$ . Which integral results?

See Solution

Problem 23417

Find the tangent line equation to the curve $y=\frac{(x+3)(x-8)}{x}$ at $x=2$ where $\frac{dy}{dx}=1+\frac{24}{x^{2}}$ .

See Solution

Problem 23418

Find the antiderivative of $f(x)=e^{-x}-\cos (x)+\cos (e)$ . Choose the correct option.

See Solution

Problem 23419

Find $F^{\prime}(x)$ for $F(x)=\int_{0}^{x} e^{t^{2}} d t$ . Which option is correct?

See Solution

Problem 23420

Given that $g(x)$ is continuous with critical points at $x=-5$ and $x=-1$ , determine if $x=-5$ is a max, min, or neither.

See Solution

Problem 23421

Is the function $f(x)=\left\{\begin{array}{ll}e^{-x}, & x \leq 0 \\ \cos x, & 0<x \leq 1 \\ x^{2}-1, & 1<x\end{array}\right.$ continuous on $[-1,1]$ ?

See Solution

Problem 23422

Calculate the integral from 0 to 1 of the function $2x - \frac{3}{4}$ . Provide your answer as a decimal.

See Solution

Problem 23423

Find the derivative of $\frac{1}{x^{2}-x+10}$ . What is $\frac{d}{dx} \frac{1}{x^{2}-x+10}$ ?

See Solution

Problem 23424

Find the derivative $\frac{d y}{d x}$ for the function $y=\log _{5}(2 x)$ .

See Solution

Problem 23425

Find the derivative of $(x^{3}+1)^{2}$ . What is $\frac{d}{d x}\left(x^{3}+1\right)^{2}$ ?

See Solution

Problem 23426

Find the limit as $x$ approaches infinity for $\frac{3x + 5}{x - 4}$ . Options: $\infty$ , 0, 3, 4.

See Solution

Problem 23427

Differentiate $xy=\cot(xy)$ implicitly to find $\frac{dy}{dx}$ . Choose from: $\frac{x}{y}$ , $\frac{y}{x}$ , $-\frac{x}{y}$ , $-\frac{y}{x}$ .

See Solution

Problem 23428

Find the integral of $\frac{1}{7 x}$ with respect to $x$ . What is the result?

See Solution

Problem 23429

Calculate the integral from 0 to π/2 of sin(x) dx. Options: 1, 0, -1, π/4.

See Solution

Problem 23430

Find the derivative of the equation $8x^{2}y - e^{y} - 4 = 0$ and solve for $\frac{dy}{dx}$ .

See Solution

Problem 23431

Find the integral of $e^{-x}$ and provide the result.

See Solution

Problem 23432

Find the Riemann sum of $f$ from $[1,7]$ using 3 equal intervals and left endpoints, given $f(1)=2$ , $f(2)=3$ , $f(3)=-1$ , $f(4)=4$ , $f(5)=1$ , $f(6)=3$ , $f(7)=5$ . Options: (a) 0 (b) 4 (c) 10 (d) 14 (e) 20.

See Solution

Problem 23433

Substitute to turn the integrand into a rational function and find the integral:
$\int \frac{5 x^{3}}{\sqrt[3]{x^{2}+1}} d x$

See Solution

Problem 23434

Find $\lim _{x \rightarrow 1} \frac{\ln x}{x-1}$ using L'Hospital's Rule. Options: 0, does not exist, 1, $-1$ .

See Solution

Problem 23435

Evaluate the limit: $\lim _{x \rightarrow 0^{+}} x$ . Choices: 1, limit does not exist, -1, 0.

See Solution

Problem 23436

Calculate the balance from a \$20000 investment at 7.5\% interest compounded continuously for 20 years.

See Solution

Problem 23437

An $80.0 \mathrm{~kg}$ student running at $3.5 \mathrm{~m/s}$ grabs a rope. How high will they swing?

See Solution

Problem 23438

True or False: The general antiderivative of $f(x)=x^{-2}$ is $F(x)=-\frac{1}{x}+C$ .

See Solution

Problem 23439

Calculate the average rate of change of $h(x)=-2x^{2}+6$ between $x=5$ and $x=9$ .

See Solution

Problem 23440

Evaluate: $\lim _{x \rightarrow 0^{+}}(x-\sin x)^{x}$ . Options: 0, 1, e, or limit does not exist.

See Solution

Problem 23441

Evaluate if $\lim _{x \rightarrow 0} e^{x}=1$ is True or False.

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Problem 23442

Two cars approach an intersection: one at $25 \mathrm{~m/s}$ east and the other at $\frac{50}{3} \mathrm{~m/s}$ south. Find their approach rate when one is $200 \mathrm{~m}$ and the other $150 \mathrm{~m}$ from the intersection.

See Solution

Problem 23443

Calculate $L_{4}, R_{4}$ , and their average for the integral $\int_{1}^{5}(x^{2}+5) \, dx$ .

See Solution

Problem 23444

The average velocity of a particle over $[a, b]$ is the slope of the secant line on the position vs. time graph. True or False?

See Solution

Problem 23445

The derivative of $f(x)=x e^{2 x+1}$ uses the product rule. True or False?

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Problem 23446

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{5}{x-9}$ , where $h \neq 0$ . Simplify your answer.

See Solution

Problem 23447

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{x}{x+3}$ and simplify. $\frac{f(x+h)-f(x)}{h}=\square$

See Solution

Problem 23448

Calculate the half-life of a radioactive element with decay model $A(t)=A_{0} e^{-0.0313 t}$ . Round to the nearest tenth.

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Problem 23449

Show that $f''(0) < 0$ for Hill's equation when $m = 1$ and rewrite $f(P)$ with $m = 1$ . Simplify your answer.

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Problem 23450

An employee invests \ $60,000 in two pension plans: one at$ 6\% $semiannually and another at$ 5.25\%$ continuously. Which earns more in 5 years?

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Problem 23451

An employee invests \$60,000 in two plans: 4% quarterly and 3.25% continuously. Find which earns more in 5 years and by how much.

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Problem 23452

Find the correct limit formula for the instantaneous rate of change of $f(x)=x^{2/3}$ at $x=8$ .

See Solution

Problem 23453

Find the slope of the secant line for $f(x)=x^{3}-4x^{2}+5x$ between $x=-1$ and $x=2$ .

See Solution

Problem 23454

Identify $f(x)$ , $a$ , and $L$ in the limit: $\lim _{x \rightarrow 3}(-x+2)=-1$ .

See Solution

Problem 23455

Differentiate $L(x)=L_{\infty}-\left(L_{\infty}-L_{0}\right) e^{-k x}$ and show $\frac{d}{d x} L(x)=k\left(L_{\infty}-L(x)\right)$ .

See Solution

Problem 23456

Find the minimum number of zeros for the continuous function $f(x)$ on $[-1,3]$ using the Intermediate Value Theorem.

See Solution

Problem 23457

Calculate the doubling time for an investment with continuous compounding at a $3\%$ interest rate.

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Problem 23458

Calculate the four second-order partial derivatives of the function $f(x, y)=5 x^{7} y^{4}+8 x^{8} y^{5}$ .

See Solution

Problem 23459

Find the derivative of the function $h(x)=\int_{1}^{\sqrt{x}} \frac{z^{2}}{z^{4}+9} dz$ . What is $h'(x)$ ?

See Solution

Problem 23460

Find the derivative of the function $h(x)=\int_{1}^{\sqrt{x}} \frac{z^{2}}{z^{4}+9} dz$ . What is $h'(x)$ ?

See Solution

Problem 23461

Find the derivative of the function $\frac{4 x^{2}}{e^{x}}$ with respect to $x$ .

See Solution

Problem 23462

Evaluate the integral $\int_{0}^{1} \frac{4 x^{3}}{\sqrt{1-x^{8}}} \mathrm{dx}$ .

See Solution

Problem 23463

Calculate the integral from 1 to 64 of $x^{-5/6}$ with respect to $x$ .

See Solution

Problem 23464

Find values of $k$ for which $\lim_{x \to \infty} \frac{e^{-kx}-7}{e^{-5x}+3}$ exists. What are the limits?

See Solution

Problem 23465

Calculate the Riemann integral $\int_{0}^{8} x^{2} dx$ using $n=250$ subintervals.

See Solution

Problem 23466

Show there exists an interval around 0 where the distance from $f(x)=\sin x$ to $x$ is less than $0.1|x|$ .

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Problem 23467

A 5-meter ladder leans against a wall. If the base moves out at 0.3 m/s and is 1 m from the wall, find the top's falling speed.

See Solution

Problem 23468

Evaluate the integral: $\int_{1 / 2}^{1 / \sqrt{2}} \frac{3}{\sqrt{1-x^{2}}} d x$

See Solution

Problem 23469

Show that $\int_{0}^{2 \sqrt{e \pi}} x^{2} \cos \left(x^{3}\right) \leq 4 e \pi$ .

See Solution

Problem 23470

Let $N=f(t)$ be the population in millions since 2010. Explain: (a) $f^{\prime}(5)=35$ (b) $f^{-1}(695)=15$ (c) $(f^{-1})^{\prime}(695)=\frac{7}{365}$ .

See Solution

Problem 23471

Find $F(v)$ given $f(v)=\frac{3}{7} \sec v \tan v$ and $F(0)=3$ , where $-\frac{\pi}{2}<v<\frac{\pi}{2}$ .

See Solution

Problem 23472

Solve the initial value problem: $\frac{d y}{d t}=1+\frac{8}{t}$ , with $y=2$ when $t=1$ , for $t>0$ .

See Solution

Problem 23473

Solve the initial value problem: $\frac{d P}{d t}=2 e^{3 t}$ , $t \geq 0$ , $P(0)=8$ .

See Solution

Problem 23474

Evaluate the limit: $\lim _{x \rightarrow \infty}\left(1-\frac{3}{x}\right)^{x}$ .

See Solution

Problem 23475

Evaluate the integral: $\int y^{2}(9-y^{3})^{2/3} dy$ (use $C$ for the constant of integration).

See Solution

Problem 23476

Evaluate the integral: $\int\left(x-\frac{1}{x^{2}}\right)\left(x^{2}+\frac{2}{x}\right)^{9} dx + C$

See Solution

Problem 23477

Evaluate the integral: $\int \frac{9+5 x}{1+x^{2}} d x$ (use $C$ for the constant).

See Solution

Problem 23478

Evaluate the integral: $\int_{0}^{13} \frac{d x}{\sqrt[3]{(1+2 x)^{2}}}$

See Solution

Problem 23479

A 48 cm wire is cut into two squares.
(a) Find the area function $A(x)$ for side length $x$ .
(b) What $x$ minimizes the area?
(c) What is the minimum area?

See Solution

Problem 23480

Given $f(x)=\frac{x^{2}+1}{x^{2}-4}$ , determine: (a) intervals of increase/decrease, (b) local max/min, (c) concavity intervals, (d) inflection points.

See Solution

Problem 23481

Find $\frac{d y}{d x}$ at the points given, then find the tangent and normal line equations. a) $y^{2}=\frac{x^{2}-4}{x^{2}+4}$ at $(2,0)$ b) $(x+y)^{3}=x^{3}+y^{3}$ at $(-1,1)$

See Solution

Problem 23482

Estimate $2(1.97)^{2}(1.03)-1.97$ using linear approximation and find the percentage error with a calculator.

See Solution

Problem 23483

Evaluate the integral $\int x^{2} \sin(x^{3}) \, dx$ .

See Solution

Problem 23484

A manufacturer wants to minimize aluminum cost for a soda can with volume $V=412 \mathrm{~cm}^{3}$ . Find $S(r)$ as a function of $r$ .

See Solution

Problem 23485

Lashonda has 360 m of fencing for a rectangular field.
(a) Find the area function $A(x)$ in terms of side length $x$ . (b) What side length $x$ maximizes the area? (c) What is the maximum area?

See Solution

Problem 23486

Approximate the integral of $f(x) = \frac{1}{2}x + 3$ over $[-2, 4]$ using the trapezoidal rule with 6 trapezoids.

See Solution

Problem 23487

Determine if $x=-5$ is a relative max, min, or neither given that $g'(x)$ is positive for $x<-5$ and $-5<x<-1$ , and negative for $x>-1$ .

See Solution

Problem 23488

Find the general indefinite integral: $\int\left(\frac{n}{\sqrt{1-x^{2}}}-\frac{1}{x}\right) d x$ . Show your work.

See Solution

Problem 23489

Fish interactions can be modeled by the Morse potential $V(r)=e^{-r}-A e^{-a r}$ . For $a=\frac{1}{2}$ and $A=1$ , find $\lim_{r \to 0} V(r)$ . What is the limit?

See Solution

Problem 23490

Analyze the Leonard-Jones 6-3 potential $V(r)=\frac{1}{r^{6}}-\frac{A}{r^{3}}$ as $r \rightarrow 0$ . What happens to $V(r)$ ?

See Solution

Problem 23491

Find the limit: $\lim _{x \rightarrow 4} \frac{x-4}{\sqrt{x}-2}$ .

See Solution

Problem 23492

Find the slope of the tangent line to $2 e^{2 y}-2 x=\ln x$ at $(1,0)$ . Options: a) $\frac{3}{2}$ b) $\frac{3}{4}$ c) $-\frac{1}{4}$ d) $-\frac{1}{2}$

See Solution

Problem 23493

Calculate $\frac{d H}{d t}$ for $H=-29.95+1.971 t^{2}-0.9971 t^{2} \log t$ . Compare growth rates at $t=8$ and $t=36$ weeks.

See Solution

Problem 23494

Find the nitrogen level $N$ that maximizes the crop yield $Y=\frac{k N}{25+N^{2}}$ . What is $N$ ?

See Solution

Problem 23495

Calculate the integral $\int_{-3}^{5}|x+2| d x$ .

See Solution

Problem 23496

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{3^{h+1}-3}{h}$ .

See Solution

Problem 23497

Evaluate the integral: $\int \frac{x^{2}+1}{(x-1)^{2}(x+3)} \, dx$

See Solution

Problem 23498

Find the general antiderivative of $f(t)=\frac{2t-4+6\sqrt{t}}{\sqrt{t}}$ . What is $F(t)$ ?

See Solution

Problem 23499

If $f(x)=\tan \left(x^{2}\right)$ , find the equivalent expression for $f^{\prime \prime}(x)$ . Options: a, b, c, d.

See Solution

Problem 23500

Given $f^{\prime \prime}(x)=36 x$ , find $f^{\prime}(x)$ and $f(x)$ using constants $C$ and $D$ .

See Solution

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Calculus

Problem 23401

Calculate the integral ∫06(3−∣3−x∣) dx\int_{0}^{6}(3-|3-x|) \, dx∫06​(3−∣3−x∣)dx.

Problem 23402

Find the velocity and acceleration at t=1 st=1 \mathrm{~s}t=1 s for s(t)=t2−3ts(t)=t^{2}-3ts(t)=t2−3t. v(1)=□ v(1)=\square v(1)=□

Problem 23403

Calculate the integral from 0 to 6 of the function 3−(3−x)3 - (3 - x)3−(3−x).

Problem 23404

Find the slope of the tangent line to the curve y=x2+6xy=x^{2}+6xy=x2+6x at the point where x=−3x=-3x=−3 using the limit definition.

Problem 23405

Tree growth rate GGG relates to mass MMM with G=CM0.7G=C M^{0.7}G=CM0.7. Differentiate to find dGdt\frac{d G}{d t}dtdG​ in terms of dMdt\frac{d M}{d t}dtdM​. dGdt=□dMdt\frac{d G}{d t}=\square \frac{d M}{d t}dtdG​=□dtdM​

Problem 23406

Find the velocity v(1)v(1)v(1) and acceleration a(1)a(1)a(1) for the position function s(t)=t4−3ts(t)=t^{4}-3ts(t)=t4−3t at t=1t=1t=1 s.

Problem 23407

Three carts of mass 1250 kg1250 \mathrm{~kg}1250 kg move at 15.0 m/s15.0 \mathrm{~m/s}15.0 m/s at D. Find speeds at B ([16.6 m/s][16.6 \mathrm{~m/s}][16.6 m/s]) and C ([10.4 m/s][10.4 \mathrm{~m/s}][10.4 m/s]). Heights: A: 16m, B: 0m, C: 8.5m, D: 2.5m.

Problem 23408

For the reaction 2 A→B2 \mathrm{~A} \rightarrow \mathrm{B}2 A→B with a rate constant of 0.853 s−10.853 \mathrm{~s}^{-1}0.853 s−1, how long until 60%60\%60% of A remains?

Problem 23409

Find the tangent line equation for the curve y=x4−2x3+5y=x^{4}-2x^{3}+5y=x4−2x3+5 at the point where x=2x=2x=2.

Problem 23410

Problem 23411

Evaluate ∫01xex2cos⁡(ex2) dx\int_{0}^{1} x e^{x^{2}} \cos(e^{x^{2}}) \, \mathrm{d} x∫01​xex2cos(ex2)dx using substitution.

Problem 23412

Calculate the integral ∫069−(x−3)2 dx\int_{0}^{6} \sqrt{9-(x-3)^{2}} \, dx∫06​9−(x−3)2​dx.

Problem 23413

Chewing frequency ccc is related to body mass MMM by c=kM−0.128c=kM^{-0.128}c=kM−0.128. (a) Given M(t)=1+2tM(t)=1+2\sqrt{t}M(t)=1+2t​, find dcdt\frac{dc}{dt}dtdc​.(b) With jaw length L=rM0.312L=rM^{0.312}L=rM0.312, find dcdL\frac{dc}{dL}dLdc​.

Problem 23414

Find points on the curve y=13x3+9xy=\frac{1}{3} x^{3}+\frac{9}{x}y=31​x3+x9​ where the tangent is parallel to y=8x+3y=8 x+3y=8x+3.

Problem 23415

Find the volume of the solid formed by rotating the region bounded by f(x)=−3x2+12x+36f(x)=-3 x^{2}+12 x+36f(x)=−3x2+12x+36, x=0x=0x=0, and y=0y=0y=0 about the yyy-axis.

Problem 23416

Substitute u=x2+1u=x^{2}+1u=x2+1 in the integral ∫022xln⁡(x2+1)dx\int_{0}^{2} 2 x \ln \left(x^{2}+1\right) d x∫02​2xln(x2+1)dx. Which integral results?

Problem 23417

Find the tangent line equation to the curve y=(x+3)(x−8)xy=\frac{(x+3)(x-8)}{x}y=x(x+3)(x−8)​ at x=2x=2x=2 where dydx=1+24x2\frac{dy}{dx}=1+\frac{24}{x^{2}}dxdy​=1+x224​.

Problem 23418

Find the antiderivative of f(x)=e−x−cos⁡(x)+cos⁡(e)f(x)=e^{-x}-\cos (x)+\cos (e)f(x)=e−x−cos(x)+cos(e). Choose the correct option.

Problem 23419

Find F′(x)F^{\prime}(x)F′(x) for F(x)=∫0xet2dtF(x)=\int_{0}^{x} e^{t^{2}} d tF(x)=∫0x​et2dt. Which option is correct?

Problem 23420

Given that g(x)g(x)g(x) is continuous with critical points at x=−5x=-5x=−5 and x=−1x=-1x=−1, determine if x=−5x=-5x=−5 is a max, min, or neither.

Problem 23421

Is the function f(x)={e−x,x≤0cos⁡x,0<x≤1x2−1,1<xf(x)=\left\{\begin{array}{ll}e^{-x}, & x \leq 0 \\ \cos x, & 0<x \leq 1 \\ x^{2}-1, & 1<x\end{array}\right.f(x)=⎩⎨⎧​e−x,cosx,x2−1,​x≤00<x≤11<x​ continuous on [−1,1][-1,1][−1,1]?

Problem 23422

Calculate the integral from 0 to 1 of the function 2x−342x - \frac{3}{4}2x−43​. Provide your answer as a decimal.

Problem 23423

Find the derivative of 1x2−x+10\frac{1}{x^{2}-x+10}x2−x+101​. What is ddx1x2−x+10\frac{d}{dx} \frac{1}{x^{2}-x+10}dxd​x2−x+101​?

Problem 23424

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=log⁡5(2x)y=\log _{5}(2 x)y=log5​(2x).

Problem 23425

Find the derivative of (x3+1)2(x^{3}+1)^{2}(x3+1)2. What is ddx(x3+1)2\frac{d}{d x}\left(x^{3}+1\right)^{2}dxd​(x3+1)2?

Problem 23426

Find the limit as xxx approaches infinity for 3x+5x−4\frac{3x + 5}{x - 4}x−43x+5​. Options: ∞\infty∞, 0, 3, 4.

Problem 23427

Differentiate xy=cot⁡(xy)xy=\cot(xy)xy=cot(xy) implicitly to find dydx\frac{dy}{dx}dxdy​. Choose from: xy\frac{x}{y}yx​, yx\frac{y}{x}xy​, −xy-\frac{x}{y}−yx​, −yx-\frac{y}{x}−xy​.

Problem 23428

Find the integral of 17x\frac{1}{7 x}7x1​ with respect to xxx. What is the result?

Problem 23429

Calculate the integral from 0 to π/2 of sin(x) dx. Options: 1, 0, -1, π/4.

Problem 23430

Find the derivative of the equation 8x2y−ey−4=08x^{2}y - e^{y} - 4 = 08x2y−ey−4=0 and solve for dydx\frac{dy}{dx}dxdy​.

Problem 23431

Find the integral of e−xe^{-x}e−x and provide the result.

Problem 23432

Find the Riemann sum of fff from [1,7][1,7][1,7] using 3 equal intervals and left endpoints, given f(1)=2f(1)=2f(1)=2, f(2)=3f(2)=3f(2)=3, f(3)=−1f(3)=-1f(3)=−1, f(4)=4f(4)=4f(4)=4, f(5)=1f(5)=1f(5)=1, f(6)=3f(6)=3f(6)=3, f(7)=5f(7)=5f(7)=5. Options: (a) 0 (b) 4 (c) 10 (d) 14 (e) 20.

Problem 23433

Substitute to turn the integrand into a rational function and find the integral: ∫5x3x2+13dx \int \frac{5 x^{3}}{\sqrt[3]{x^{2}+1}} d x ∫3x2+1​5x3​dx

Problem 23434

Find lim⁡x→1ln⁡xx−1\lim _{x \rightarrow 1} \frac{\ln x}{x-1}limx→1​x−1lnx​ using L'Hospital's Rule. Options: 0, does not exist, 1, −1-1−1.

Problem 23435

Evaluate the limit: lim⁡x→0+x\lim _{x \rightarrow 0^{+}} xlimx→0+​x. Choices: 1, limit does not exist, -1, 0.

Problem 23436

Calculate the balance from a \$20000 investment at 7.5\% interest compounded continuously for 20 years.

Problem 23437

An 80.0 kg80.0 \mathrm{~kg}80.0 kg student running at 3.5 m/s3.5 \mathrm{~m/s}3.5 m/s grabs a rope. How high will they swing?

Problem 23438

True or False: The general antiderivative of f(x)=x−2f(x)=x^{-2}f(x)=x−2 is F(x)=−1x+CF(x)=-\frac{1}{x}+CF(x)=−x1​+C.

Problem 23439

Calculate the average rate of change of h(x)=−2x2+6h(x)=-2x^{2}+6h(x)=−2x2+6 between x=5x=5x=5 and x=9x=9x=9.

Problem 23440

Evaluate: lim⁡x→0+(x−sin⁡x)x\lim _{x \rightarrow 0^{+}}(x-\sin x)^{x}limx→0+​(x−sinx)x. Options: 0, 1, e, or limit does not exist.

Calculate the integral $\int_{0}^{6}(3-|3-x|) \, dx$ .

Find the velocity and acceleration at $t=1 \mathrm{~s}$ for $s(t)=t^{2}-3t$ .
$v(1)=\square$

Calculate the integral from 0 to 6 of the function $3 - (3 - x)$ .

Find the slope of the tangent line to the curve $y=x^{2}+6x$ at the point where $x=-3$ using the limit definition.

Tree growth rate $G$ relates to mass $M$ with $G=C M^{0.7}$ . Differentiate to find $\frac{d G}{d t}$ in terms of $\frac{d M}{d t}$ . $\frac{d G}{d t}=\square \frac{d M}{d t}$

Find the velocity $v(1)$ and acceleration $a(1)$ for the position function $s(t)=t^{4}-3t$ at $t=1$ s.

Three carts of mass $1250 \mathrm{~kg}$ move at $15.0 \mathrm{~m/s}$ at D. Find speeds at B ( $[16.6 \mathrm{~m/s}]$ ) and C ( $[10.4 \mathrm{~m/s}]$ ). Heights: A: 16m, B: 0m, C: 8.5m, D: 2.5m.

For the reaction $2 \mathrm{~A} \rightarrow \mathrm{B}$ with a rate constant of $0.853 \mathrm{~s}^{-1}$ , how long until $60\%$ of A remains?

Find the tangent line equation for the curve $y=x^{4}-2x^{3}+5$ at the point where $x=2$ .

Evaluate $\int_{0}^{1} x e^{x^{2}} \cos(e^{x^{2}}) \, \mathrm{d} x$ using substitution.

Calculate the integral $\int_{0}^{6} \sqrt{9-(x-3)^{2}} \, dx$ .

Chewing frequency $c$ is related to body mass $M$ by $c=kM^{-0.128}$ .
(a) Given $M(t)=1+2\sqrt{t}$ , find $\frac{dc}{dt}$ .
(b) With jaw length $L=rM^{0.312}$ , find $\frac{dc}{dL}$ .

Find points on the curve $y=\frac{1}{3} x^{3}+\frac{9}{x}$ where the tangent is parallel to $y=8 x+3$ .

Find the volume of the solid formed by rotating the region bounded by $f(x)=-3 x^{2}+12 x+36$ , $x=0$ , and $y=0$ about the $y$ -axis.

Substitute $u=x^{2}+1$ in the integral $\int_{0}^{2} 2 x \ln \left(x^{2}+1\right) d x$ . Which integral results?

Find the tangent line equation to the curve $y=\frac{(x+3)(x-8)}{x}$ at $x=2$ where $\frac{dy}{dx}=1+\frac{24}{x^{2}}$ .

Find the antiderivative of $f(x)=e^{-x}-\cos (x)+\cos (e)$ . Choose the correct option.

Find $F^{\prime}(x)$ for $F(x)=\int_{0}^{x} e^{t^{2}} d t$ . Which option is correct?

Given that $g(x)$ is continuous with critical points at $x=-5$ and $x=-1$ , determine if $x=-5$ is a max, min, or neither.

Is the function $f(x)=\left\{\begin{array}{ll}e^{-x}, & x \leq 0 \\ \cos x, & 0<x \leq 1 \\ x^{2}-1, & 1<x\end{array}\right.$ continuous on $[-1,1]$ ?

Calculate the integral from 0 to 1 of the function $2x - \frac{3}{4}$ . Provide your answer as a decimal.

Find the derivative of $\frac{1}{x^{2}-x+10}$ . What is $\frac{d}{dx} \frac{1}{x^{2}-x+10}$ ?

Find the derivative $\frac{d y}{d x}$ for the function $y=\log _{5}(2 x)$ .

Find the derivative of $(x^{3}+1)^{2}$ . What is $\frac{d}{d x}\left(x^{3}+1\right)^{2}$ ?

Find the limit as $x$ approaches infinity for $\frac{3x + 5}{x - 4}$ . Options: $\infty$ , 0, 3, 4.

Differentiate $xy=\cot(xy)$ implicitly to find $\frac{dy}{dx}$ . Choose from: $\frac{x}{y}$ , $\frac{y}{x}$ , $-\frac{x}{y}$ , $-\frac{y}{x}$ .

Find the integral of $\frac{1}{7 x}$ with respect to $x$ . What is the result?

Find the derivative of the equation $8x^{2}y - e^{y} - 4 = 0$ and solve for $\frac{dy}{dx}$ .

Find the integral of $e^{-x}$ and provide the result.

Find the Riemann sum of $f$ from $[1,7]$ using 3 equal intervals and left endpoints, given $f(1)=2$ , $f(2)=3$ , $f(3)=-1$ , $f(4)=4$ , $f(5)=1$ , $f(6)=3$ , $f(7)=5$ . Options: (a) 0 (b) 4 (c) 10 (d) 14 (e) 20.

Substitute to turn the integrand into a rational function and find the integral:
$\int \frac{5 x^{3}}{\sqrt[3]{x^{2}+1}} d x$

Find $\lim _{x \rightarrow 1} \frac{\ln x}{x-1}$ using L'Hospital's Rule. Options: 0, does not exist, 1, $-1$ .

Evaluate the limit: $\lim _{x \rightarrow 0^{+}} x$ . Choices: 1, limit does not exist, -1, 0.

An $80.0 \mathrm{~kg}$ student running at $3.5 \mathrm{~m/s}$ grabs a rope. How high will they swing?

True or False: The general antiderivative of $f(x)=x^{-2}$ is $F(x)=-\frac{1}{x}+C$ .

Calculate the average rate of change of $h(x)=-2x^{2}+6$ between $x=5$ and $x=9$ .

Evaluate: $\lim _{x \rightarrow 0^{+}}(x-\sin x)^{x}$ . Options: 0, 1, e, or limit does not exist.

Evaluate if $\lim _{x \rightarrow 0} e^{x}=1$ is True or False.

Two cars approach an intersection: one at $25 \mathrm{~m/s}$ east and the other at $\frac{50}{3} \mathrm{~m/s}$ south. Find their approach rate when one is $200 \mathrm{~m}$ and the other $150 \mathrm{~m}$ from the intersection.

Calculate $L_{4}, R_{4}$ , and their average for the integral $\int_{1}^{5}(x^{2}+5) \, dx$ .

The average velocity of a particle over $[a, b]$ is the slope of the secant line on the position vs. time graph. True or False?

The derivative of $f(x)=x e^{2 x+1}$ uses the product rule. True or False?

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{5}{x-9}$ , where $h \neq 0$ . Simplify your answer.

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{x}{x+3}$ and simplify. $\frac{f(x+h)-f(x)}{h}=\square$

Calculate the half-life of a radioactive element with decay model $A(t)=A_{0} e^{-0.0313 t}$ . Round to the nearest tenth.

Show that $f''(0) < 0$ for Hill's equation when $m = 1$ and rewrite $f(P)$ with $m = 1$ . Simplify your answer.

An employee invests \ $60,000 in two pension plans: one at$ 6\% $semiannually and another at$ 5.25\%$ continuously. Which earns more in 5 years?

Find the correct limit formula for the instantaneous rate of change of $f(x)=x^{2/3}$ at $x=8$ .

Find the slope of the secant line for $f(x)=x^{3}-4x^{2}+5x$ between $x=-1$ and $x=2$ .

Identify $f(x)$ , $a$ , and $L$ in the limit: $\lim _{x \rightarrow 3}(-x+2)=-1$ .

Differentiate $L(x)=L_{\infty}-\left(L_{\infty}-L_{0}\right) e^{-k x}$ and show $\frac{d}{d x} L(x)=k\left(L_{\infty}-L(x)\right)$ .

Find the minimum number of zeros for the continuous function $f(x)$ on $[-1,3]$ using the Intermediate Value Theorem.

Calculate the doubling time for an investment with continuous compounding at a $3\%$ interest rate.

Calculate the four second-order partial derivatives of the function $f(x, y)=5 x^{7} y^{4}+8 x^{8} y^{5}$ .

Find the derivative of the function $h(x)=\int_{1}^{\sqrt{x}} \frac{z^{2}}{z^{4}+9} dz$ . What is $h'(x)$ ?

Find the derivative of the function $h(x)=\int_{1}^{\sqrt{x}} \frac{z^{2}}{z^{4}+9} dz$ . What is $h'(x)$ ?

Find the derivative of the function $\frac{4 x^{2}}{e^{x}}$ with respect to $x$ .

Evaluate the integral $\int_{0}^{1} \frac{4 x^{3}}{\sqrt{1-x^{8}}} \mathrm{dx}$ .

Calculate the integral from 1 to 64 of $x^{-5/6}$ with respect to $x$ .

Find values of $k$ for which $\lim_{x \to \infty} \frac{e^{-kx}-7}{e^{-5x}+3}$ exists. What are the limits?

Calculate the Riemann integral $\int_{0}^{8} x^{2} dx$ using $n=250$ subintervals.

Show there exists an interval around 0 where the distance from $f(x)=\sin x$ to $x$ is less than $0.1|x|$ .

Evaluate the integral: $\int_{1 / 2}^{1 / \sqrt{2}} \frac{3}{\sqrt{1-x^{2}}} d x$

Show that $\int_{0}^{2 \sqrt{e \pi}} x^{2} \cos \left(x^{3}\right) \leq 4 e \pi$ .

Let $N=f(t)$ be the population in millions since 2010. Explain: (a) $f^{\prime}(5)=35$ (b) $f^{-1}(695)=15$ (c) $(f^{-1})^{\prime}(695)=\frac{7}{365}$ .

Find $F(v)$ given $f(v)=\frac{3}{7} \sec v \tan v$ and $F(0)=3$ , where $-\frac{\pi}{2}<v<\frac{\pi}{2}$ .

Solve the initial value problem: $\frac{d y}{d t}=1+\frac{8}{t}$ , with $y=2$ when $t=1$ , for $t>0$ .

Solve the initial value problem: $\frac{d P}{d t}=2 e^{3 t}$ , $t \geq 0$ , $P(0)=8$ .

Evaluate the limit: $\lim _{x \rightarrow \infty}\left(1-\frac{3}{x}\right)^{x}$ .