Calculus

Problem 1401

Choose $u$ for integration by parts in $\int x^{6} \ln (x) \, dx$ . Recall $\int u \, dv = uv - \int v \, du$ .

See Solution

Problem 1402

Calculate the indefinite integral of $\ln \left(x^{2}\right)$ with respect to $x$ : $\int \ln \left(x^{2}\right) d x=$

See Solution

Problem 1403

Calculate the volume of the solid formed by rotating the area between $x=0$ , $x=1$ , $y=0$ , and $y=9+x^{4}$ around the $x$ -axis. $V=$

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Problem 1404

Find the volume of the solid formed by rotating the area between $y=2 x^{2}$ , $x=1$ , and $y=0$ around the $x$ -axis. $V=$

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Problem 1405

Calculate the volume of the solid formed by rotating the area between $y=e^{1 x}+5$ , $y=0$ , $x=0$ , and $x=0.6$ around the $x$ -axis.

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Problem 1406

Find the volume of the solid formed by rotating the area between $y=\sqrt{x-1}$ , $y=0$ , $x=2$ , and $x=3$ around the $x$ -axis. $\text{Volume} =$

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Problem 1407

Find the volume of the solid formed by rotating the area under $y=x \sqrt{4-x^{2}}$ from $x=0$ to $x=2$ around the $x$ -axis. $\text { Volume }=$

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Problem 1408

Find the volume of the solid formed by rotating the area between $y=x^{2}+x-2$ and $y=0$ around the $x$ -axis.

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Problem 1409

Find the volume of the solid formed by rotating the area under $y=x \sqrt{4-x^{2}}$ from $x=0$ to $x=2$ about the $y$ -axis using $\int_{0}^{2} 2 \pi \sqrt{4-y^{2}} d y$ . Evaluate the integral for the volume.

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Problem 1410

Calculate the volume of the solid formed by rotating the area between $y=x^{2}$ , $x=2$ , and $y=0$ around the $x$ -axis. $\text { Volume }=$

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Problem 1411

Find the limit as $x$ approaches 3 for the expression $5x^{2} + 4x + 2$ .

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Problem 1412

Solve the initial value problem: $y'' + 18x = 0$ , with $y(0) = 2$ and $y'(0) = 2$ . Find $y = \quad: \cdots \quad:$

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Problem 1413

Evaluate the limit: $\lim _{x \rightarrow 196} \frac{\sqrt{x}-14}{x-196}$ to three decimal places or state DNE.

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Problem 1414

Solve the initial value problem: $y'' + 18x = 0$ , with $y(0) = 2$ and $y'(0) = 2$ . Find $y =$ .

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Problem 1415

Solve the initial value problem: $y^{\prime \prime}+18 x=0$ , with $y(0)=2$ , $y^{\prime}(0)=2$ . Find $y=...$

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Problem 1416

Find all values of $r$ for which $y=r x^{2}$ satisfies the equation $y^{\prime}=9 x$ . Enter answers as a list. $r=$

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Problem 1417

Evaluate the limit or state if it doesn't exist: $\lim _{x \rightarrow 0} \frac{\sin (15 x)}{x}=$ (3 decimal places or DNE)

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Problem 1418

Find the growth constant $k$ for a trout population growing from 2500 to 6250 in 1 year, with a capacity of 25000. When will it reach 12900? $k=$ $\therefore \mathrm{yr}^{-1}$ Time to 12900: $\approx$ - years.

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Problem 1419

Find the average rate of change of $f(t)=t^{2}-2t$ from $t=2$ to $t=5$ . Show your work.

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Problem 1420

Evaluate the integral using substitution: $\int -x(5-8x)^{5} \, dx = C$ .

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Problem 1421

Find the limit: $\lim_{x \rightarrow -6} \frac{x+6}{x^{2}+x-30}$ and give your answer to three decimal places.

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Problem 1422

Find the one-sided limit: $\lim _{x \rightarrow 0^{-}} \frac{2 \sin (x)}{3|x|}$ to three decimal places.

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Problem 1423

Find the one-sided limit: $\lim _{x \rightarrow 7^{+}} \frac{x+9}{x-7}=$ (Enter DNE if it doesn't exist.)

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Problem 1424

Plot the function and estimate the limit value: $\lim _{x \rightarrow 0} \frac{\sin (2 x)}{\sin (4 x)}=$ (two decimal places).

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Problem 1425

Find the left and right limits of $\frac{7 x}{x-2}$ as $x$ approaches 2: $\lim _{x \rightarrow 2^{+}}$ and $\lim _{x \rightarrow 2^{-}}$ .

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Problem 1426

A rocket starts from rest with mass $m_0$ and burns fuel at rate $k$ . Find $v(t)$ from $m \frac{d v}{d t}=c k-m g$ .
(a) $v(t)=\quad \theta_{-} \mathrm{m} / \mathrm{sec}$
(b) If fuel is 80% of $m_0$ and lasts 110 s, find $v(110)$ with $g=9.8 \mathrm{~m/s}^2$ and $c=2500 \mathrm{~m/s}$ . Round to nearest whole number.

See Solution

Problem 1427

Find and simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the functions: 71. $f(x)=4x$ , 72. $f(x)=7x$ .

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Problem 1428

Simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x) = x^2 - 4x + 3$ , with $h \neq 0$ .

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Problem 1429

Find the limit: $\lim _{x \rightarrow 8^{+}} \frac{1}{x-8}$ .

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Problem 1430

Find the limit: $\lim _{x \rightarrow 2^{+}} \frac{x-6}{(x-2)^{2}}$ .

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Problem 1431

The area between $y=\sin (x)$ and the $x$ -axis from $x=-\pi$ to $x=\pi$ is given by $\int_{-\pi}^{\pi} \sin (x) dx$ . True or False?

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Problem 1432

False. The solutions to the differential equation $\frac{d y}{d x}=y^{2}-4 y$ include other values.

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Problem 1433

Verify if $\int \frac{1}{1+x^{2}} \, dx = \tan^{-1}(x) + C$ and show the proof step by step.

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Problem 1434

Find the limits for the piecewise function $f(x)=\left\{\begin{array}{ll}x^{2}+16, & x<-16 \\ \sqrt{x+16}, & x \geq-16\end{array}\right.$ : a. $\lim _{x \rightarrow-16^{-}} f(x)$ b. $\lim _{x \rightarrow-16^{+}} f(x)$ c. $\lim _{x \rightarrow-16} f(x)$

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Problem 1435

Is it true or false that $\int_{1}^{x^{2}} \ln(1+t^{2}) dt = \int_{1}^{x} 2t \ln(1+t^{4}) dt$ ?

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Problem 1436

Find the value of $K$ in the solution $y(t)=\frac{K}{1+A e^{-r t}}$ for the IVP $\frac{d y}{d t}=3 y(1-\frac{y}{12})$ , $y(0)=2$ .

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Problem 1437

Find the limits of the function $g(x)$ defined as:
$g(x)=\begin{cases} 0 & \text{if } x \leq -6 \\ \sqrt{36-x^{2}} & \text{if } -6 < x < 6 \\ x & \text{if } x \geq 6 \end{cases}$
for $x \to -6^{-}, -6^{+}, 6^{-}, 6^{+}, 6$ .

See Solution

Problem 1438

Solve the initial value problem: $\frac{d y}{d t}=3 y\left(1-\frac{y}{12}\right)$ , $y(0)=2$ . Find $y(t)$ .

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Problem 1439

Calculate the work done by the force $F(x)=2 x^{-2}+4 x$ from $x=1$ to $x=2$ : evaluate $\int_{1}^{2}\left(2 x^{-2}+4 x\right) d x$ . Provide the answer to three decimal places.

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Problem 1440

Find the inflection point of $g(x)=\int_{1}^{x^{2}} e^{2 \sqrt{t}} d t$ and give the exact $x$ -coordinate.

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Problem 1441

Find the value of $f(\pi)$ where $f(T)=\int_{0}^{x} x \cos (x) d x$ . Provide the answer to three decimal places.

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Problem 1442

Find the value of $f(\pi)$ where $f(T)=\int_{0}^{T} x \cos (x) d x$ . Round to three decimal places.

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Problem 1443

Does the integral $\int_{0}^{3} \frac{1}{x} dx$ converge or diverge?

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Problem 1444

Determine if the integral $\int_{-1}^{4} \frac{1}{x^{2 / 3}} d x$ converges or diverges.

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Problem 1445

Does the integral $\int_{-\infty}^{\infty} e^{-3 x} dx$ converge or diverge?

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Problem 1446

Find the limit: $\lim _{t \rightarrow 4^{+}} \frac{|7 t-28|}{t^{2}-16}$ .

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Problem 1447

Find the tangent line equation for the curve at the specified parameter value for each case.
1. $x=t^{4}+1, \; y=t^{3}+t; \; t=-1$
2. $x=t-t^{-1}, \; y=1+t^{2}; \; t=1$
3. $x=e^{\sqrt{t}}, \; y=t-\ln t^{2}; \; t=1$
4. $x=\cos \theta+\sin 2 \theta, \; y=\sin \theta+\cos 2 \theta; \; \theta=0$

See Solution

Problem 1448

Find the value of $y(1)$ for the initial value problem $\frac{d y}{d x}=\frac{\ln (x)}{x}$ with $y(e)=3$ .

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Problem 1449

Find the value of the integral $\int_{4}^{4} f(x) d x$ given that $f$ and $g$ are continuous functions.

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Problem 1450

Find the value of $\int_{2}^{5} f(x) d x$ given that $\int_{0}^{2} f(x) dx=-1$ and $\int_{0}^{5} f(x) dx=3$ .

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Problem 1451

Find the value of the integral $\int_{2}^{0} g(x) d x$ given $\int_{0}^{2} g(x) d x=8$ . Enter as a decimal or DNE.

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Problem 1452

Calculate the integral $\int_{0}^{2}(5 f(x)+g(x)) d x$ given that $\int_{0}^{2} f(x) d x=-1$ and $\int_{0}^{2} g(x) d x=8$ .

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Problem 1453

Given that $f(x)$ is said to be nonnegative but has negative values in the table, clarify this. Find $y(4)$ from $\frac{d y}{d x}=y^{2} f^{\prime}(x), \quad y(3)=1$ .

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Problem 1454

Find the value of $a$ so that the function $f(x)=\frac{30+x-x^{2}}{x-6}$ is continuous at $x=6$ .

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Problem 1455

Check if the function $f(x)=\left\{\begin{array}{ll} \frac{x^{2}-1}{x-1} & \text{if } x \neq 1 \\ 4 & \text{if } x=1 \end{array}\right.$ is continuous at $a=1$ .

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Problem 1456

Prove there’s a solution to $\cos (x)=7-2 x$ using the Intermediate Value Theorem. Find intervals for $a$ and $b$ .

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Problem 1457

Find the limit: $\lim _{x \rightarrow 4^{+}} \frac{3-x}{x^{2}-2 x-8}$ .

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Problem 1458

Find the derivatives of these functions: 1) $y=\sin 2x + (2x-5)^{2}$ 2) $y=\cot x + \sec^{2} x + 5$

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Problem 1459

Find the derivative of these functions:
1. $y=\cot ^{4} x$
2. $y=\sec ^{3} x$
3. $y=\cos x^{2}$
4. $y=\sqrt{\frac{\sec x+\tan x}{\sec x-\tan x}}$
5. $y=\sin 2 x+(2 x-5)^{2}$
6. $y=\cot x+\sec ^{2} x+5$
7. $y=\frac{\sec -1}{\sec x+1}$
8. $y=\operatorname{cosec} 3 x^{3}$
9. $y=\frac{\sin 3 x}{x+1}$
10. $y=\frac{\sin 2 x}{2 x+5}$
11. $y=\cot 2 x$
12. $y=\frac{(3 x+1) \cos 2 x}{e^{x}}$
13. $y=\sec x$

See Solution

Problem 1460

Find the derivatives of these functions: 1. $y=\frac{\sin 3 x}{x+1}$ , 2. $y=\frac{\sin 2 x}{2 x+5}$ , 3. $y=\cot 2 x$ , 4. $y=\frac{(3 x+1) \cos 2 x}{e^{x}}$ , 5. $y=\sec x$ .

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Problem 1461

Find $c \in (3, 5)$ such that $\cos(c) = 7 - 2c$ using the Intermediate Value Theorem.

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Problem 1462

Is the statement true or false? The derivative of $f(x)$ is the instantaneous rate of change of $y=f(x)$ with respect to $x$ . Choose A, B, C, or D.

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Problem 1463

Find the average velocity from $t=2$ to $t=3$ and $t=2$ to $t=4$ , and the instantaneous velocity at $t=2$ .

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Problem 1464

Find the average velocity of a particle given $s(t)=t^{2}+4t-3$ over intervals [2,3] and [2,4].

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Problem 1465

Is it true or false that a derivative can exist where a function does not? Explain your reasoning.

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Problem 1466

Find the derivatives of these functions:
1. $y=\cot x+\sec ^{2} x+5$
2. $y=\frac{1}{\sec x+1}$ or $y=\frac{\sec (x-1)}{\sec x+1}$ .

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Problem 1467

Find the derivative of $f(x)=-x^{2}+9x-6$ using the limit definition: $\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ .

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Problem 1468

Find the average velocity of a ball rolling down a ramp for functions $s(t)=10 t^{2}$ and $s(t)=12 t^{2}$ over specified intervals.

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Problem 1469

Find the derivative of the implicit function: $y^2 + \sin y + x + y + y \sin x + y \sin y + \cos(y^2 + 1) + \sin y + x^2 + 4y = \cos x$ and $3xy^2 + \cos y^2 = 2x^3 + 5$ and $\tan(5y) - y \sin x + 3xy^2 = 9$ .

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Problem 1470

Find the derivative $f^{\prime}(x)$ of the function $f(x)=9x-2$ .

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Problem 1471

Find the derivative $f'(x)$ of the function $f(x) = x^2 - 5$ .

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Problem 1472

Find the derivative $f^{\prime}(x)$ for the function $f(x)=7 x^{2}+x$ .

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Problem 1473

Find the derivative $R'(t)$ for $R(t) = -0.1 t^{2}$ and calculate $R'(1)$ .

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Problem 1474

Find the derivative of $R(t)=-0.1 t^{2}$ and evaluate it at $t=1$ . What is $R'(1)$ ?

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Problem 1475

Find the equation of the tangent line to the curve $y=\frac{4}{x}$ at the point $\left(3, \frac{4}{3}\right)$ .

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Problem 1476

Estimate the growth rate of Whooping cranes in 2007 using $r$ and given values. Choose all correct answers. $d N / d t=r^{} N_{t} \\ r=\ln (\text { lambda }) \\ d N / d t=0.086^{} 258 \\ r=\ln (1.09)$

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Problem 1477

Find the derivative $U'(t)$ of $U(t)=5.1 t^{2}-1.2 t$ and evaluate it at $t=8$ .

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Problem 1478

Find the tangent line equation to the curve $y=-x^{2}+5x-4$ at $x=1$ .

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Problem 1479

Find the limit as $x$ approaches 4 for the expression $x^{3}+x$ .

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Problem 1480

Evaluate these limits: 1. $\lim_{x \to 4}(x^3 + x)$ 2. $\lim_{x \to 4}(x^2 + 1)$ 3. $\lim_{x \to 2} \frac{x^2 - x - 2}{x^2 - 2x}$ 4. $\lim_{x \to 1} \frac{x^2 - 2x + 1}{x^3 - x}$ 5. $\lim_{x \to \infty} \frac{\sqrt{x^2 + 1}}{x^2}$ 6. $\lim_{x \to 4}(x^2 + 3x - 5)$ 7. $\lim_{y \to \infty}(y^3 - 2y + 7)$ 8. $\lim_{t \to 0} \frac{2t^2 + 1}{t^3 + 3t - 4}$ 9. $\lim_{x \to 1} \frac{(s + 1)^2}{2x^2 + 3}$ 10. $\lim_{w \to 2} \frac{3w^2 - 4w + 2}{w^3 - 5}$ 11. $\lim_{w \to -1} \frac{3w^2 - 2w + 7}{w^2 + 1}$ 12. $\lim_{x \to 2} \frac{\sqrt{x - 2}}{\sqrt{x^2 - 4}}$ 13. $\lim_{x \to 2} \frac{(1 - x^2)^{1}}{(1 - x^2)^2}$ 14. $\lim_{x \to 3} \frac{x - 3}{\sqrt{x^2 - 9}}$ 15. $\lim_{x \to \infty} \frac{4}{2x^2 - 3}$ 16. $\lim_{x \to \infty} \frac{2x^2}{3x^2 + 5}$ 17. $\lim_{x \to \infty} \frac{x^2}{3x^2 - 4x + 1}$ 18. $\lim_{x \to \infty} 2^{\frac{3}{x}}$ 19. $\lim_{x \to 0^+} 2^{\frac{1}{x}}$ 20. $\lim_{x \to 0^+} \frac{1}{1 + 2^{\frac{1}{x}}}$

See Solution

Problem 1481

Find the average rate of change of $y=2 \times 3^{x}$ from $x=0$ to $x=4$ . Options: A 40.5 B 162 C 158 D 40 E 4

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Problem 1482

Find $\frac{d y}{d x}$ for $y=\frac{2 x^{4}+9 x^{2}}{4 x}$ . Choices include options A to E.

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Problem 1483

Find $\frac{d y}{d x}$ for $y=\frac{2 x^{4}+9 x^{2}}{4 x}$ . Options: A, B, C, D, E.

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Problem 1484

Find $\frac{d y}{d x}$ for $y=\frac{2 x^{4}+9 x^{2}}{4 x}$ . Choices are A, B, C, D, E.

See Solution

Problem 1485

Find the derivative of $f(x)=\frac{3x+5}{4-x}$ .

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Problem 1486

Find the derivative of $f(x)=(2x-x^{3})\sqrt{2-x^{2}}$ .

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Problem 1487

Find the derivative of $f(x)=\frac{2-5 x}{\cos 10 x}$ .

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Problem 1488

Find the slope of the function $f(x)=\frac{-x-1}{7 \sqrt{x}}$ at the point (8, -0.45).

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Problem 1489

Find the slope of $f(x)=-5x-1+3\sqrt{7x}$ at the point $(2,0.22)$ using a graphing tool and round to the nearest hundredth.

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Problem 1490

Find the slope of $f(x)=\frac{2x-1}{3\sqrt{x}}$ at the point (2, 0.71).

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Problem 1491

Find the slope of the function $f(x)=4x^{3}+5x^{2}+3x+8$ at the point (3, 170).

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Problem 1492

Find the slope of the function $f(x)=-3 x^{3}-4 x^{2}+5 x+3$ at the point (4, -233).

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Problem 1493

Calculate the limit $\frac{f(x+h)-f(x)}{h}$ for $h \neq 0$ where $f(x)=x^{2}+5x-1$ and $f(x)=\frac{1}{x+3}$ .

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Problem 1494

Find the meaning of $f^{\prime}(6)=-6$ for the function $f(x)=-x^{3}+8 x^{2}+6 x+5$ .

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Problem 1495

Find the meaning of $f^{\prime}(5)=-767$ for the function $f(x)=-9 x^{3}-10 x^{2}+8 x+3$ .

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Problem 1496

Find the limit of $f(x)$ as $x$ approaches 1, where $f(x)=\frac{(x-1)^{2}(x+1)}{|x-1|}$ for $x \neq 1$ and $f(1)=2$ .

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Problem 1497

Find $\lim _{x \rightarrow \frac{\pi}{4}} g(x)$ for $g(x)=\frac{2 \cos ^{2} x-1}{\cos x-\sin x}$ .

See Solution

Problem 1498

Find $\lim _{x \rightarrow \frac{\pi}{4}} g(x)$ for $g(x)=\frac{2 \cos ^{2} x-1}{\cos x-\sin x}$ . Which option is equivalent?

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Problem 1499

Find $\lim_{x \rightarrow 1} f(x)$ if $g(x) \leq f(x) \leq h(x)$ where $g(x)=\sin \left(\frac{\pi}{2} x\right)+4$ and $h(x)=-\frac{1}{4} x^{3}+\frac{3}{4} x+\frac{9}{2}$ .

See Solution

Problem 1500

Calculate the average rate of change of $g(x)=-5 x^{3}+4$ between $x=-4$ and $x=4$ .

See Solution

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Calculus

Problem 1401

Choose uuu for integration by parts in ∫x6ln⁡(x) dx\int x^{6} \ln (x) \, dx∫x6ln(x)dx. Recall ∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du∫udv=uv−∫vdu.

Problem 1402

Calculate the indefinite integral of ln⁡(x2)\ln \left(x^{2}\right)ln(x2) with respect to xxx: ∫ln⁡(x2)dx=\int \ln \left(x^{2}\right) d x=∫ln(x2)dx=

Problem 1403

Calculate the volume of the solid formed by rotating the area between x=0x=0x=0, x=1x=1x=1, y=0y=0y=0, and y=9+x4y=9+x^{4}y=9+x4 around the xxx-axis. V=V=V=

Problem 1404

Find the volume of the solid formed by rotating the area between y=2x2y=2 x^{2}y=2x2, x=1x=1x=1, and y=0y=0y=0 around the xxx-axis. V= V= V=

Problem 1405

Calculate the volume of the solid formed by rotating the area between y=e1x+5y=e^{1 x}+5y=e1x+5, y=0y=0y=0, x=0x=0x=0, and x=0.6x=0.6x=0.6 around the xxx-axis.

Problem 1406

Find the volume of the solid formed by rotating the area between y=x−1y=\sqrt{x-1}y=x−1​, y=0y=0y=0, x=2x=2x=2, and x=3x=3x=3 around the xxx-axis. Volume= \text{Volume} = Volume=

Problem 1407

Find the volume of the solid formed by rotating the area under y=x4−x2y=x \sqrt{4-x^{2}}y=x4−x2​ from x=0x=0x=0 to x=2x=2x=2 around the xxx-axis. Volume = \text { Volume }= Volume =

Problem 1408

Find the volume of the solid formed by rotating the area between y=x2+x−2y=x^{2}+x-2y=x2+x−2 and y=0y=0y=0 around the xxx-axis.

Problem 1409

Find the volume of the solid formed by rotating the area under y=x4−x2y=x \sqrt{4-x^{2}}y=x4−x2​ from x=0x=0x=0 to x=2x=2x=2 about the yyy-axis using ∫022π4−y2dy\int_{0}^{2} 2 \pi \sqrt{4-y^{2}} d y∫02​2π4−y2​dy. Evaluate the integral for the volume.

Problem 1410

Calculate the volume of the solid formed by rotating the area between y=x2y=x^{2}y=x2, x=2x=2x=2, and y=0y=0y=0 around the xxx-axis. Volume =\text { Volume }= Volume =

Problem 1411

Find the limit as xxx approaches 3 for the expression 5x2+4x+25x^{2} + 4x + 25x2+4x+2.

Problem 1412

Solve the initial value problem: y′′+18x=0y'' + 18x = 0y′′+18x=0, with y(0)=2y(0) = 2y(0)=2 and y′(0)=2y'(0) = 2y′(0)=2. Find y=:⋯:y = \quad: \cdots \quad:y=:⋯:

Problem 1413

Evaluate the limit: lim⁡x→196x−14x−196\lim _{x \rightarrow 196} \frac{\sqrt{x}-14}{x-196}x→196lim​x−196x​−14​ to three decimal places or state DNE.

Problem 1414

Solve the initial value problem: y′′+18x=0y'' + 18x = 0y′′+18x=0, with y(0)=2y(0) = 2y(0)=2 and y′(0)=2y'(0) = 2y′(0)=2. Find y=y =y=.

Problem 1415

Solve the initial value problem: y′′+18x=0y^{\prime \prime}+18 x=0y′′+18x=0, with y(0)=2y(0)=2y(0)=2, y′(0)=2y^{\prime}(0)=2y′(0)=2. Find y=...y=...y=...

Problem 1416

Find all values of rrr for which y=rx2y=r x^{2}y=rx2 satisfies the equation y′=9xy^{\prime}=9 xy′=9x. Enter answers as a list. r= r= r=

Problem 1417

Evaluate the limit or state if it doesn't exist: lim⁡x→0sin⁡(15x)x=\lim _{x \rightarrow 0} \frac{\sin (15 x)}{x}=x→0lim​xsin(15x)​= (3 decimal places or DNE)

Problem 1418

Find the growth constant kkk for a trout population growing from 2500 to 6250 in 1 year, with a capacity of 25000. When will it reach 12900? k= k= k= ∴yr−1\therefore \mathrm{yr}^{-1}∴yr−1 Time to 12900: ≈\approx≈ - years.

Problem 1419

Find the average rate of change of f(t)=t2−2tf(t)=t^{2}-2tf(t)=t2−2t from t=2t=2t=2 to t=5t=5t=5. Show your work.

Problem 1420

Evaluate the integral using substitution: ∫−x(5−8x)5 dx=C\int -x(5-8x)^{5} \, dx = C∫−x(5−8x)5dx=C.

Problem 1421

Find the limit: lim⁡x→−6x+6x2+x−30\lim_{x \rightarrow -6} \frac{x+6}{x^{2}+x-30}limx→−6​x2+x−30x+6​ and give your answer to three decimal places.

Problem 1422

Find the one-sided limit: lim⁡x→0−2sin⁡(x)3∣x∣\lim _{x \rightarrow 0^{-}} \frac{2 \sin (x)}{3|x|}x→0−lim​3∣x∣2sin(x)​ to three decimal places.

Problem 1423

Find the one-sided limit: lim⁡x→7+x+9x−7=\lim _{x \rightarrow 7^{+}} \frac{x+9}{x-7}=x→7+lim​x−7x+9​= (Enter DNE if it doesn't exist.)

Problem 1424

Plot the function and estimate the limit value: lim⁡x→0sin⁡(2x)sin⁡(4x)=\lim _{x \rightarrow 0} \frac{\sin (2 x)}{\sin (4 x)}=x→0lim​sin(4x)sin(2x)​= (two decimal places).

Problem 1425

Find the left and right limits of 7xx−2\frac{7 x}{x-2}x−27x​ as xxx approaches 2: lim⁡x→2+\lim _{x \rightarrow 2^{+}}limx→2+​ and lim⁡x→2−\lim _{x \rightarrow 2^{-}}limx→2−​.

Problem 1426

Problem 1427

Find and simplify the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for the functions: 71. f(x)=4xf(x)=4xf(x)=4x, 72. f(x)=7xf(x)=7xf(x)=7x.

Problem 1428

Simplify the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=x2−4x+3f(x) = x^2 - 4x + 3f(x)=x2−4x+3, with h≠0h \neq 0h=0.

Problem 1429

Find the limit: lim⁡x→8+1x−8\lim _{x \rightarrow 8^{+}} \frac{1}{x-8}limx→8+​x−81​.

Problem 1430

Find the limit: lim⁡x→2+x−6(x−2)2\lim _{x \rightarrow 2^{+}} \frac{x-6}{(x-2)^{2}}limx→2+​(x−2)2x−6​.

Problem 1431

The area between y=sin⁡(x)y=\sin (x)y=sin(x) and the xxx-axis from x=−πx=-\pix=−π to x=πx=\pix=π is given by ∫−ππsin⁡(x)dx\int_{-\pi}^{\pi} \sin (x) dx∫−ππ​sin(x)dx. True or False?

Problem 1432

False. The solutions to the differential equation dydx=y2−4y\frac{d y}{d x}=y^{2}-4 ydxdy​=y2−4y include other values.

Problem 1433

Verify if ∫11+x2 dx=tan⁡−1(x)+C\int \frac{1}{1+x^{2}} \, dx = \tan^{-1}(x) + C∫1+x21​dx=tan−1(x)+C and show the proof step by step.

Problem 1434

Problem 1435

Is it true or false that ∫1x2ln⁡(1+t2)dt=∫1x2tln⁡(1+t4)dt\int_{1}^{x^{2}} \ln(1+t^{2}) dt = \int_{1}^{x} 2t \ln(1+t^{4}) dt∫1x2​ln(1+t2)dt=∫1x​2tln(1+t4)dt?

Problem 1436

Find the value of KKK in the solution y(t)=K1+Ae−rty(t)=\frac{K}{1+A e^{-r t}}y(t)=1+Ae−rtK​ for the IVP dydt=3y(1−y12)\frac{d y}{d t}=3 y(1-\frac{y}{12})dtdy​=3y(1−12y​), y(0)=2y(0)=2y(0)=2.

Problem 1437

Problem 1438

Solve the initial value problem: dydt=3y(1−y12)\frac{d y}{d t}=3 y\left(1-\frac{y}{12}\right)dtdy​=3y(1−12y​), y(0)=2y(0)=2y(0)=2. Find y(t)y(t)y(t).

Problem 1439

Calculate the work done by the force F(x)=2x−2+4xF(x)=2 x^{-2}+4 xF(x)=2x−2+4x from x=1x=1x=1 to x=2x=2x=2: evaluate ∫12(2x−2+4x)dx\int_{1}^{2}\left(2 x^{-2}+4 x\right) d x∫12​(2x−2+4x)dx. Provide the answer to three decimal places.

Problem 1440

Find the inflection point of g(x)=∫1x2e2tdtg(x)=\int_{1}^{x^{2}} e^{2 \sqrt{t}} d tg(x)=∫1x2​e2t​dt and give the exact xxx-coordinate.

Problem 1441

Find the value of f(π)f(\pi)f(π) where f(T)=∫0xxcos⁡(x)dxf(T)=\int_{0}^{x} x \cos (x) d xf(T)=∫0x​xcos(x)dx. Provide the answer to three decimal places.

Choose $u$ for integration by parts in $\int x^{6} \ln (x) \, dx$ . Recall $\int u \, dv = uv - \int v \, du$ .

Calculate the indefinite integral of $\ln \left(x^{2}\right)$ with respect to $x$ : $\int \ln \left(x^{2}\right) d x=$

Calculate the volume of the solid formed by rotating the area between $x=0$ , $x=1$ , $y=0$ , and $y=9+x^{4}$ around the $x$ -axis. $V=$

Find the volume of the solid formed by rotating the area between $y=2 x^{2}$ , $x=1$ , and $y=0$ around the $x$ -axis. $V=$

Calculate the volume of the solid formed by rotating the area between $y=e^{1 x}+5$ , $y=0$ , $x=0$ , and $x=0.6$ around the $x$ -axis.

Find the volume of the solid formed by rotating the area between $y=\sqrt{x-1}$ , $y=0$ , $x=2$ , and $x=3$ around the $x$ -axis. $\text{Volume} =$

Find the volume of the solid formed by rotating the area under $y=x \sqrt{4-x^{2}}$ from $x=0$ to $x=2$ around the $x$ -axis. $\text { Volume }=$

Find the volume of the solid formed by rotating the area between $y=x^{2}+x-2$ and $y=0$ around the $x$ -axis.

Find the volume of the solid formed by rotating the area under $y=x \sqrt{4-x^{2}}$ from $x=0$ to $x=2$ about the $y$ -axis using $\int_{0}^{2} 2 \pi \sqrt{4-y^{2}} d y$ . Evaluate the integral for the volume.

Calculate the volume of the solid formed by rotating the area between $y=x^{2}$ , $x=2$ , and $y=0$ around the $x$ -axis. $\text { Volume }=$

Find the limit as $x$ approaches 3 for the expression $5x^{2} + 4x + 2$ .

Solve the initial value problem: $y'' + 18x = 0$ , with $y(0) = 2$ and $y'(0) = 2$ . Find $y = \quad: \cdots \quad:$

Evaluate the limit: $\lim _{x \rightarrow 196} \frac{\sqrt{x}-14}{x-196}$ to three decimal places or state DNE.

Solve the initial value problem: $y'' + 18x = 0$ , with $y(0) = 2$ and $y'(0) = 2$ . Find $y =$ .

Solve the initial value problem: $y^{\prime \prime}+18 x=0$ , with $y(0)=2$ , $y^{\prime}(0)=2$ . Find $y=...$

Find all values of $r$ for which $y=r x^{2}$ satisfies the equation $y^{\prime}=9 x$ . Enter answers as a list. $r=$

Evaluate the limit or state if it doesn't exist: $\lim _{x \rightarrow 0} \frac{\sin (15 x)}{x}=$ (3 decimal places or DNE)

Find the growth constant $k$ for a trout population growing from 2500 to 6250 in 1 year, with a capacity of 25000. When will it reach 12900? $k=$ $\therefore \mathrm{yr}^{-1}$ Time to 12900: $\approx$ - years.

Find the average rate of change of $f(t)=t^{2}-2t$ from $t=2$ to $t=5$ . Show your work.

Evaluate the integral using substitution: $\int -x(5-8x)^{5} \, dx = C$ .

Find the limit: $\lim_{x \rightarrow -6} \frac{x+6}{x^{2}+x-30}$ and give your answer to three decimal places.

Find the one-sided limit: $\lim _{x \rightarrow 0^{-}} \frac{2 \sin (x)}{3|x|}$ to three decimal places.

Find the one-sided limit: $\lim _{x \rightarrow 7^{+}} \frac{x+9}{x-7}=$ (Enter DNE if it doesn't exist.)

Plot the function and estimate the limit value: $\lim _{x \rightarrow 0} \frac{\sin (2 x)}{\sin (4 x)}=$ (two decimal places).

Find the left and right limits of $\frac{7 x}{x-2}$ as $x$ approaches 2: $\lim _{x \rightarrow 2^{+}}$ and $\lim _{x \rightarrow 2^{-}}$ .

Find and simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the functions: 71. $f(x)=4x$ , 72. $f(x)=7x$ .

Simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x) = x^2 - 4x + 3$ , with $h \neq 0$ .

Find the limit: $\lim _{x \rightarrow 8^{+}} \frac{1}{x-8}$ .

Find the limit: $\lim _{x \rightarrow 2^{+}} \frac{x-6}{(x-2)^{2}}$ .

The area between $y=\sin (x)$ and the $x$ -axis from $x=-\pi$ to $x=\pi$ is given by $\int_{-\pi}^{\pi} \sin (x) dx$ . True or False?

False. The solutions to the differential equation $\frac{d y}{d x}=y^{2}-4 y$ include other values.

Verify if $\int \frac{1}{1+x^{2}} \, dx = \tan^{-1}(x) + C$ and show the proof step by step.

Is it true or false that $\int_{1}^{x^{2}} \ln(1+t^{2}) dt = \int_{1}^{x} 2t \ln(1+t^{4}) dt$ ?

Find the value of $K$ in the solution $y(t)=\frac{K}{1+A e^{-r t}}$ for the IVP $\frac{d y}{d t}=3 y(1-\frac{y}{12})$ , $y(0)=2$ .

Solve the initial value problem: $\frac{d y}{d t}=3 y\left(1-\frac{y}{12}\right)$ , $y(0)=2$ . Find $y(t)$ .

Calculate the work done by the force $F(x)=2 x^{-2}+4 x$ from $x=1$ to $x=2$ : evaluate $\int_{1}^{2}\left(2 x^{-2}+4 x\right) d x$ . Provide the answer to three decimal places.

Find the inflection point of $g(x)=\int_{1}^{x^{2}} e^{2 \sqrt{t}} d t$ and give the exact $x$ -coordinate.

Find the value of $f(\pi)$ where $f(T)=\int_{0}^{x} x \cos (x) d x$ . Provide the answer to three decimal places.

Find the value of $f(\pi)$ where $f(T)=\int_{0}^{T} x \cos (x) d x$ . Round to three decimal places.

Does the integral $\int_{0}^{3} \frac{1}{x} dx$ converge or diverge?

Determine if the integral $\int_{-1}^{4} \frac{1}{x^{2 / 3}} d x$ converges or diverges.

Does the integral $\int_{-\infty}^{\infty} e^{-3 x} dx$ converge or diverge?

Find the limit: $\lim _{t \rightarrow 4^{+}} \frac{|7 t-28|}{t^{2}-16}$ .

Find the value of $y(1)$ for the initial value problem $\frac{d y}{d x}=\frac{\ln (x)}{x}$ with $y(e)=3$ .

Find the value of the integral $\int_{4}^{4} f(x) d x$ given that $f$ and $g$ are continuous functions.

Find the value of $\int_{2}^{5} f(x) d x$ given that $\int_{0}^{2} f(x) dx=-1$ and $\int_{0}^{5} f(x) dx=3$ .

Find the value of the integral $\int_{2}^{0} g(x) d x$ given $\int_{0}^{2} g(x) d x=8$ . Enter as a decimal or DNE.

Calculate the integral $\int_{0}^{2}(5 f(x)+g(x)) d x$ given that $\int_{0}^{2} f(x) d x=-1$ and $\int_{0}^{2} g(x) d x=8$ .

Given that $f(x)$ is said to be nonnegative but has negative values in the table, clarify this. Find $y(4)$ from $\frac{d y}{d x}=y^{2} f^{\prime}(x), \quad y(3)=1$ .

Find the value of $a$ so that the function $f(x)=\frac{30+x-x^{2}}{x-6}$ is continuous at $x=6$ .

Check if the function $f(x)=\left\{\begin{array}{ll} \frac{x^{2}-1}{x-1} & \text{if } x \neq 1 \\ 4 & \text{if } x=1 \end{array}\right.$ is continuous at $a=1$ .

Prove there’s a solution to $\cos (x)=7-2 x$ using the Intermediate Value Theorem. Find intervals for $a$ and $b$ .

Find the limit: $\lim _{x \rightarrow 4^{+}} \frac{3-x}{x^{2}-2 x-8}$ .

Find the derivatives of these functions: 1) $y=\sin 2x + (2x-5)^{2}$ 2) $y=\cot x + \sec^{2} x + 5$

Find the derivatives of these functions: 1. $y=\frac{\sin 3 x}{x+1}$ , 2. $y=\frac{\sin 2 x}{2 x+5}$ , 3. $y=\cot 2 x$ , 4. $y=\frac{(3 x+1) \cos 2 x}{e^{x}}$ , 5. $y=\sec x$ .

Find $c \in (3, 5)$ such that $\cos(c) = 7 - 2c$ using the Intermediate Value Theorem.

Is the statement true or false? The derivative of $f(x)$ is the instantaneous rate of change of $y=f(x)$ with respect to $x$ . Choose A, B, C, or D.

Find the average velocity from $t=2$ to $t=3$ and $t=2$ to $t=4$ , and the instantaneous velocity at $t=2$ .

Find the average velocity of a particle given $s(t)=t^{2}+4t-3$ over intervals [2,3] and [2,4].

Find the derivatives of these functions:
1. $y=\cot x+\sec ^{2} x+5$
2. $y=\frac{1}{\sec x+1}$ or $y=\frac{\sec (x-1)}{\sec x+1}$ .

Find the derivative of $f(x)=-x^{2}+9x-6$ using the limit definition: $\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ .

Find the average velocity of a ball rolling down a ramp for functions $s(t)=10 t^{2}$ and $s(t)=12 t^{2}$ over specified intervals.

Find the derivative $f^{\prime}(x)$ of the function $f(x)=9x-2$ .

Find the derivative $f'(x)$ of the function $f(x) = x^2 - 5$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=7 x^{2}+x$ .

Find the derivative $R'(t)$ for $R(t) = -0.1 t^{2}$ and calculate $R'(1)$ .

Find the derivative of $R(t)=-0.1 t^{2}$ and evaluate it at $t=1$ . What is $R'(1)$ ?