Calculus

Problem 31501

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{(-8+h)^{2}-64}{h}$

See Solution

Problem 31502

Find $\delta$ values that satisfy $\lim _{x \rightarrow -1} 5x + 3 = -2$ with $\varepsilon = 0.4$ . Choices: $\delta=0.0064$ , $\delta=0.02667$ , $\delta=0.16$ , $\delta=0.08$ .

See Solution

Problem 31503

Evaluate the limits of $f(x) = \frac{1}{(x+1)^{2}}$ as $x$ approaches -1 from both sides and directly.
a) $\lim_{x \rightarrow -1^{-}} f(x) =$
b) $\lim_{x \rightarrow -1^{+}} f(x) =$
c) $\lim_{x \rightarrow -1} f(x) =$

See Solution

Problem 31504

Estimate the instantaneous rate of change of daily receipts 3 weeks after the opening day. Use data:
Weeks: 0, 3, 6, 9, 12, 15, 18, 21, 24, 27 Receipts: 77.025, 50.306, 31.28, 16.24, 6.526, 0.98, 0.92, 0.335, 0.076, 0.044
Round to four decimal places.

See Solution

Problem 31505

Evaluate left, right, and midpoint Riemann sums for $f(x)=\cos(2x)$ from $0$ to $\frac{\pi}{4}$ with $n=60$ . Conjecture the area.

See Solution

Problem 31506

Find the limit as $x$ approaches 0 for $\frac{x-9}{x^{2}(x-2)}$ .

See Solution

Problem 31507

Find the limits of $f(x) = x \sqrt{7 + x^{-2}}$ as $x$ approaches 0 from the left, right, and directly at 0.

See Solution

Problem 31508

Find the $y$ -coordinate on the ellipse $4 x^{2}+25 y^{2}=100$ at $x=1$ , the tangent line equation, triangle area, and $c$ for min area.

See Solution

Problem 31509

Given the graph of $f(x)$ , find:
a. $g(1)$ ; b. $g'(-2)$ ; c. $g''(1)$ ; d. The interval where $g(x)$ is decreasing.
Points: $f(-3)=2$ , $f(-2)=1$ , $f(-1)=0$ , $f(1)=-2$ , $f(3)=0$ . Ignore $f(1)=1$ .

See Solution

Problem 31510

Find the limit as $x$ approaches 0 for $\frac{x-9}{x^{2}(x-2)}$ . Limit =

See Solution

Problem 31511

Find the horizontal limits of the function $f(x)=\frac{3 x^{3}-6 x^{2}-10 x}{7-11 x-3 x^{3}}$ . Calculate $\lim _{x \rightarrow-\infty} f(x)$ and $\lim _{x \rightarrow \infty} f(x)$ .

See Solution

Problem 31512

Find the horizontal asymptotes by calculating these limits:
1. $\lim_{x \to \infty} \frac{-14x}{15 + 2x}$
2. $\lim_{x \to -\infty} \frac{7x - 10}{x^3 + 11x - 14}$
3. $\lim_{x \to \infty} \frac{x^2 - 6x - 12}{2 - 6x^2}$
4. $\lim_{x \to \infty} \frac{\sqrt{x^2 + 4x}}{5 - 13x}$
5. $\lim_{x \to -\infty} \frac{\sqrt{x^2 + 4x}}{5 - 13x}$

See Solution

Problem 31513

Find the value of $a$ for the function $f(x)=\frac{6 x^{2}-2 x}{2 x}$ (for $x \neq 0$ ) to be continuous at $x=0$ .

See Solution

Problem 31514

Find the general solution of $(x+\sin y) dx+(x \cos y-2 y) dy=0$ . If homogeneous, use $y=v x$ .

See Solution

Problem 31515

Let $f(x)=\frac{6}{1+x^{2}}$ . Find the area of region $R$ , volume when rotated about $y=7$ , and volume of semicircular solid. Bounds: $-1$ to $1$ .

See Solution

Problem 31516

Find the limit: $\lim _{x \rightarrow 11} \frac{x-11}{x^{2}-11 x}=\square$ (integer or simplified fraction) or state if it doesn't exist.

See Solution

Problem 31517

Find the limit: $\lim _{x \rightarrow 0} \frac{x-11}{x^{2}-11 x}=\square$ or state if it doesn't exist.

See Solution

Problem 31518

Find the limit: $\lim _{x \rightarrow 121} \frac{x-11}{x^{2}-11 x}=\square$ or state if it does not exist.

See Solution

Problem 31519

Let $f(x)=\frac{6}{1+x^{2}}$ .
(a) Find the area of the region bounded by $f$ and $y=3$ . (b) Find the volume when this region is rotated about $y=7$ . (c) Find the volume of the solid with semicircular cross sections perpendicular to the $x$ -axis.

See Solution

Problem 31520

Find the limit: $\lim _{x \rightarrow 8} 5 x$ . A. $\lim _{x \rightarrow 8} 5 x=\square$ B. Limit does not exist.

See Solution

Problem 31521

Find the limits for the function $f(x)=-\frac{6 x-6}{(x-6)^{2}}$ as $x \rightarrow 6^{-}$ and $x \rightarrow 6^{+}$ .

See Solution

Problem 31522

Calculate $2 \int_{0}^{6}|x-4| dx$ . Options: (F) 6, (G) 10, (H) 22, (1) 42.

See Solution

Problem 31523

Calculate the integral $\int_{0}^{6}|x-4| d x=$ with options: (F) 6, (C) 10, (16) 22, (1) 42.

See Solution

Problem 31524

Two people start at the same point. One walks east at $3 \mathrm{mi/h}$ , the other northeast at $4 \mathrm{mi/h}$ . Find the distance change rate after 15 minutes.

See Solution

Problem 31525

Calculate the integral $\int_{0}^{6}|x-4| d x$ .

See Solution

Problem 31526

A trough is 8 ft long with triangular ends 2 ft wide and 1 ft high. Water fills at 10 ft³/min. Find the rise rate when water is 6 in deep.

See Solution

Problem 31527

Find the general solution of the homogeneous equation using $y=v x$ : $(x+\sin y) dx+(x \cos y-2 x) dy=0$ .

See Solution

Problem 31528

Find the object's acceleration at $t=5$ given the position function $s(t)=\frac{1}{15} t^{5}-\frac{1}{2} t^{2}+\frac{5}{t}$ .

See Solution

Problem 31529

Find the limit: $\lim _{x \rightarrow 4} \frac{x^{2}-5 x+4}{2 x-8}$

See Solution

Problem 31530

Determine the horizontal asymptotes of the function $f(x)=\frac{3 x}{\sqrt{x^{2}+4}}$ .

See Solution

Problem 31531

Is the function $f(x)=\frac{x}{1+x^{2}}$ increasing on the interval $(-6,6)$ ?

See Solution

Problem 31532

Find the interval of convergence for the series from $f(x)=\frac{7}{9+x^{2}}$ . Provide the answer in interval notation.

See Solution

Problem 31533

Approximate $\int_{0}^{10} f(x) dx$ using a trapezoidal sum with values $f(0)=4$ , $f(1)=5$ , $f(4)=10$ , $f(8)=12$ , $f(10)=8$ .

See Solution

Problem 31534

A ball falls under gravity with a retarding force $F=bv$ . Find the ball's acceleration at any time.

See Solution

Problem 31535

Find the general solution for the equation: $(x^2 - xy + y^2) dx - xy dy = 0$ . Use $y = vx$ if homogeneous.

See Solution

Problem 31536

Solve the differential equation $\frac{d y}{d x}=\frac{1}{2} \sin \left(\frac{\pi}{2} x\right) \sqrt{y+7}$ with $f(1)=2$ . Find the tangent line at $x=1$ to estimate $f(0.8)$ .

See Solution

Problem 31537

Use cylindrical shells to find the volume $V$ generated by $y=x^{2}$ and $x=y^{2}$ around $y=-5$ .

See Solution

Problem 31538

Find the limit: $\lim _{x \rightarrow 2} \frac{7-f(x)}{x+g(x)}$ given $\lim _{x \rightarrow 2} f(x)=-2$ and $\lim _{x \rightarrow 2} g(x)=7$ .

See Solution

Problem 31539

Determine if the function $f(x)=\ln (x-2)$ on $[3,11]$ satisfies $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$ .

See Solution

Problem 31540

Estimate waste accumulation $Q(t)=10^{4}(t^{2}+15t+70)$ for $0 \leq t \leq 10$ . Find: a) current waste, b) avg. change in 3 years, c) present change, d) when change hits $3 \times 10^{5}$ .

See Solution

Problem 31541

Find limits for the piecewise function $f(x)=\left\{\begin{array}{ll}x^{2}, & x<2 \\ 3x, & x>2\end{array}\right.$ at $x=2$ .

See Solution

Problem 31542

Find the limits for the piecewise function $f(x)=\left\{\begin{array}{ll}x^{2}, & x<2 \\ 3x, & x>2\end{array}\right.$ at $x=2$ .

See Solution

Problem 31543

Find the limits and value of the function $f(x)=\frac{x-6}{|x-6|}$ at $x=6$ : a) $\lim _{x \rightarrow 6^{+}} f(x)$ , b) $\lim _{x \rightarrow 6^{-}} f(x)$ , c) $\lim _{x \rightarrow 6} f(x)$ , d) $f(6)$ .

See Solution

Problem 31544

Find the limit: $f(x)=\begin{cases}x^{2}, & x<2 \\ 3x, & x>2\end{cases}$ . What is $\lim _{x \rightarrow 2^{+}} f(x)$ ?

See Solution

Problem 31545

Find limits for $f(x)=\left\{\begin{array}{ll}x^{2}, & x<2 \\ 3x, & x>2\end{array}\right.$ at $x=2$ . Complete parts (A)-(D).

See Solution

Problem 31546

Find the limits of $f(x)=\frac{x-2}{x^{2}-2 x}$ as $x$ approaches 0, 2, and 4.

See Solution

Problem 31547

Differentiate the function $f(t)=\frac{t^{2}-1}{t^{2}+1}$ .

See Solution

Problem 31548

Find the antiderivative of $f(x)=(x-1)(2x+3)$ . Choose all correct options from the list provided.

See Solution

Problem 31549

Explain how to find a limit and describe right-sided, left-sided, and two-sided limits.

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Problem 31550

Find all antiderivatives of $f(x)=(x-1)(2x+3)$ . Incorrect answers lose credit.

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Problem 31551

Find the limits: a) $\lim _{x \rightarrow 8} f(x)$ , b) $\lim _{x \rightarrow 0} f(x)$ , c) $\lim _{x \rightarrow -4} f(x)$ for $f(x)=\frac{x^{2}-4 x-32}{x-8}$ .

See Solution

Problem 31552

Find the general antiderivative $F(x)$ of $f(x)=\frac{3}{x}+2x+1$ for $x<0$ , including constant $C$ .

See Solution

Problem 31553

Differentiate the function $g(x)=\ln \left(x^{-4}+x^{4}\right)$ .

See Solution

Problem 31554

Find the exact value of $F(2)$ if $F$ is the antiderivative of $f(x)=2x+7$ and $F(1)=9$ .

See Solution

Problem 31555

Differentiate the function $R(w)=4^{w}-5 \log _{9} w$ .

See Solution

Problem 31556

Find the limit as $x$ approaches $\infty$ for $\frac{3x+7}{8x-6}$ . Is it a number, $-\infty$ , or $\infty$ ?

See Solution

Problem 31557

Find the limits for $f(x)=\frac{x^{2}-7 x-18}{x-9}$ as $x$ approaches 9 from the left, right, and directly.

See Solution

Problem 31558

Find the limits: a) $\lim_{x \rightarrow 8} f(x)$ , b) $\lim_{x \rightarrow 0} f(x)$ , c) $\lim_{x \rightarrow -2} f(x)$ for $f(x)=\frac{x^{2}-6x-16}{x-8}$ .

See Solution

Problem 31559

Find the derivative $\frac{d y}{d x}$ for $y=\frac{1}{x^{8}}$ . What is $\frac{d y}{d x}=$ ?

See Solution

Problem 31560

Find the derivative of $y=x^{-5}$ . What is $\frac{d y}{d x}=\square$ ?

See Solution

Problem 31561

Find the derivative $f^{\prime}(x)$ of $f(x)=2 x^{2}+3 x-2$ , then calculate $f^{\prime}(1)$ , $f^{\prime}(4)$ , and $f^{\prime}(9)$ .

See Solution

Problem 31562

Find the derivative $f^{\prime}(x)$ of $f(x)=2x^{2}+3x-2$ , then calculate $f^{\prime}(1)$ , $f^{\prime}(4)$ , $f^{\prime}(9)$ .

See Solution

Problem 31563

Differentiate $\frac{5 x^{2}}{3}-\frac{9}{4 x^{2}}$ with respect to $x$ .

See Solution

Problem 31564

Find the derivative $h^{\prime}(t)$ for the function $h(t)=7.9-8.5 t+0.2 t^{2}$ .

See Solution

Problem 31565

Find $f'(x)$ for $f(x)=x^{4}-4x^{3}+3$ , the slope at $x=-1$ , the tangent line equation at $x=-1$ , and where it's horizontal.

See Solution

Problem 31566

Find the marginal cost function for $C(x)=190+4.4x-0.01x^{2}$ . What is $C^{\prime}(x)$ ?

See Solution

Problem 31567

Find the derivative of $\frac{17 x+39}{x}$ . What is $\frac{d}{d x} \frac{17 x+39}{x}=\square$ ?

See Solution

Problem 31568

Sales function $S(t)=0.04 t^{3}+0.5 t^{2}+6 t+3$ . Find $S^{\prime}(t)$ , $S(7)$ , $S^{\prime}(7)$ , and interpret $S(11)$ and $S^{\prime}(11)$ .

See Solution

Problem 31569

Find the marginal revenue function for $R(x)=x(19-0.04 x)$ . What is $R^{\prime}(x)=?$

See Solution

Problem 31570

Find $f_{xz}$ for $f(x, y, z)=e^{xyz}$ at the point $(1,-1,-1)$ . Options: a. None, b. $e$ , c. $-e$ , d. 0.

See Solution

Problem 31571

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

See Solution

Problem 31572

Find $\lim _{t \rightarrow 0} \left( t \mathbf{i}+\sin t \mathbf{j}-2 \frac{\sin t}{t} \mathbf{k} \right)$ .

See Solution

Problem 31573

Find the derivative of $f(x)=x^{2}-5x+5$ and the slope of the tangent line at $x=\frac{3}{2}$ and $x=2$ .

See Solution

Problem 31574

The cost function for $x$ food processors is $C(x)=2300+70x-0.1x^{2}$ .
(A) Find the exact cost of the 71st processor. (B) Estimate the cost using marginal cost.
(A) Exact cost: \$\square. (B) Approximate cost: \$\square.

See Solution

Problem 31575

Find the average cost per unit for 400 frames, marginal average cost at 400 units, and estimate for 401 frames.

See Solution

Problem 31576

Find the limit: $\lim _{x \rightarrow 2}(9 x-6)$ . Choose A or B for your answer.

See Solution

Problem 31577

Find the limit: $\lim _{x \rightarrow 3} \frac{x^{2}-4}{x-2}$ . Is it A. = \square or B. does not exist?

See Solution

Problem 31578

Find the limit: $\lim _{x \rightarrow 7} 1$ . Choose A or B and fill in the answer.

See Solution

Problem 31579

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

See Solution

Problem 31580

Find the volume of the solid formed by revolving the area bounded by $x=y^{3/2}$ , $x=0$ , $y=2$ around the $y$ -axis.

See Solution

Problem 31581

1. Calculate the rate of change in the first 30 minutes after takeoff: $\frac{\Delta y}{\Delta x}=\frac{r t}{\text{hours}}$ .
2. Determine the rate of change from 1.5 hours to 3 hours.
3. Find the rate of change from 4.5 hours to landing altitude.
4. At 6:00 a.m. ( $58^{\circ}$ ) and 2:00 p.m. ( $76^{\circ}$ ), find the rate of change in degrees per hour.

See Solution

Problem 31582

Find the derivative of $y=\frac{x^{5}}{x^{3}}$ using the Quotient Rule. Choose the correct answer below.

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Problem 31583

Find the derivative of $y=\frac{x^{7}}{x^{5}}$ using the Quotient Rule. Choose the correct answer format.

See Solution

Problem 31584

Find $\delta$ so that $|f(x)-0.25|<0.1$ for $0<|x-4|<\delta$ , where $f(x)=\frac{1}{x}$ .

See Solution

Problem 31585

Find the derivative of $y=\frac{x^{8}}{x^{4}}$ using the Quotient Rule. Choose the correct answer from options A-D.

See Solution

Problem 31586

Find the derivative of $y=\frac{x^{8}}{x^{4}}$ using the Quotient Rule and select the correct answer from options A-D.

See Solution

Problem 31587

Find all times $t$ in $0 < t < 90$ when Stephan changes direction, given his velocity $v(t)=2.38 e^{-0.02 t} \sin \left(\frac{\pi}{56} t\right)$ .

See Solution

Problem 31588

How long to grow \$8,000 to \$13,900 at 4.50\% interest, compounded continuously? (Round to two decimal places.)

See Solution

Problem 31589

Find the derivative of $f(x)=\frac{1}{(x^{2}-9)^{\frac{1}{2}}$ using the chain rule.

See Solution

Problem 31590

Find the limits for $f(x)=\frac{x^{2}-x-56}{x-8}$ as $x$ approaches 8 from the left, right, and directly.

See Solution

Problem 31591

Find the limits: a) $\lim_{x \rightarrow 8} f(x)$ , b) $\lim_{x \rightarrow 0} f(x)$ , c) $\lim_{x \rightarrow -6} f(x)$ for $f(x)=\frac{x^{2}-2x-48}{x-8}$ .

See Solution

Problem 31592

Find horizontal and vertical asymptotes for $f(x)=\frac{5 x+4}{6 x^{2}+1}$ . Identify and fill in the answers.

See Solution

Problem 31593

Find the derivative of $y=\frac{x^{9}}{x^{5}}$ using the Quotient Rule and by dividing first.

See Solution

Problem 31594

Find the horizontal and vertical asymptotes of $f(x)=\frac{x^{2}+36}{x^{2}-36}$ .

See Solution

Problem 31595

Find the limit as $x$ approaches $\infty$ for $\frac{8x+7}{3x-8}$ . Is it a number or does it not exist?

See Solution

Problem 31596

Calculate the limit: $\lim _{x \rightarrow-\infty} \frac{50 x^{2}+600 x+5}{7 x^{3}+9}$ . Does it exist?

See Solution

Problem 31597

Find $f^{\prime}(x)$ for $f(x)=2 x^{2}-5 x+3$ and calculate $f^{\prime}(0)$ , $f^{\prime}(1)$ , $f^{\prime}(5)$ .

See Solution

Problem 31598

Find the derivative of $y = x^{-8}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 31599

Find the limit: $\lim _{x \rightarrow+\infty} \frac{5}{7-5.7 e^{-3 x}}$ . State if it exists or not.

See Solution

Problem 31600

Find the derivative of the function $f(w)=w^{\frac{3}{4}}+8w$ .

See Solution

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Calculus

Problem 31501

Evaluate the limit: lim⁡h→0(−8+h)2−64h\lim _{h \rightarrow 0} \frac{(-8+h)^{2}-64}{h}limh→0​h(−8+h)2−64​

Problem 31502

Find δ\deltaδ values that satisfy lim⁡x→−15x+3=−2\lim _{x \rightarrow -1} 5x + 3 = -2limx→−1​5x+3=−2 with ε=0.4\varepsilon = 0.4ε=0.4. Choices: δ=0.0064\delta=0.0064δ=0.0064, δ=0.02667\delta=0.02667δ=0.02667, δ=0.16\delta=0.16δ=0.16, δ=0.08\delta=0.08δ=0.08.

Problem 31503

Problem 31504

Estimate the instantaneous rate of change of daily receipts 3 weeks after the opening day. Use data: Weeks: 0, 3, 6, 9, 12, 15, 18, 21, 24, 27 Receipts: 77.025, 50.306, 31.28, 16.24, 6.526, 0.98, 0.92, 0.335, 0.076, 0.044 Round to four decimal places.

Problem 31505

Evaluate left, right, and midpoint Riemann sums for f(x)=cos⁡(2x)f(x)=\cos(2x)f(x)=cos(2x) from 000 to π4\frac{\pi}{4}4π​ with n=60n=60n=60. Conjecture the area.

Problem 31506

Find the limit as xxx approaches 0 for x−9x2(x−2)\frac{x-9}{x^{2}(x-2)}x2(x−2)x−9​.

Problem 31507

Find the limits of f(x)=x7+x−2 f(x) = x \sqrt{7 + x^{-2}} f(x)=x7+x−2​ as x x x approaches 0 from the left, right, and directly at 0.

Problem 31508

Find the yyy-coordinate on the ellipse 4x2+25y2=1004 x^{2}+25 y^{2}=1004x2+25y2=100 at x=1x=1x=1, the tangent line equation, triangle area, and ccc for min area.

Problem 31509

Problem 31510

Find the limit as xxx approaches 0 for x−9x2(x−2)\frac{x-9}{x^{2}(x-2)}x2(x−2)x−9​. Limit =

Problem 31511

Problem 31512

Problem 31513

Find the value of aaa for the function f(x)=6x2−2x2xf(x)=\frac{6 x^{2}-2 x}{2 x}f(x)=2x6x2−2x​ (for x≠0x \neq 0x=0) to be continuous at x=0x=0x=0.

Problem 31514

Find the general solution of (x+sin⁡y)dx+(xcos⁡y−2y)dy=0(x+\sin y) dx+(x \cos y-2 y) dy=0(x+siny)dx+(xcosy−2y)dy=0. If homogeneous, use y=vxy=v xy=vx.

Problem 31515

Let f(x)=61+x2f(x)=\frac{6}{1+x^{2}}f(x)=1+x26​. Find the area of region RRR, volume when rotated about y=7y=7y=7, and volume of semicircular solid. Bounds: −1-1−1 to 111.

Problem 31516

Find the limit: lim⁡x→11x−11x2−11x=□\lim _{x \rightarrow 11} \frac{x-11}{x^{2}-11 x}=\squarelimx→11​x2−11xx−11​=□ (integer or simplified fraction) or state if it doesn't exist.

Problem 31517

Find the limit: lim⁡x→0x−11x2−11x=□\lim _{x \rightarrow 0} \frac{x-11}{x^{2}-11 x}=\squarelimx→0​x2−11xx−11​=□ or state if it doesn't exist.

Problem 31518

Find the limit: lim⁡x→121x−11x2−11x=□\lim _{x \rightarrow 121} \frac{x-11}{x^{2}-11 x}=\squarelimx→121​x2−11xx−11​=□ or state if it does not exist.

Problem 31519

Let f(x)=61+x2f(x)=\frac{6}{1+x^{2}}f(x)=1+x26​. (a) Find the area of the region bounded by fff and y=3y=3y=3. (b) Find the volume when this region is rotated about y=7y=7y=7. (c) Find the volume of the solid with semicircular cross sections perpendicular to the xxx-axis.

Problem 31520

Find the limit: lim⁡x→85x\lim _{x \rightarrow 8} 5 xlimx→8​5x. A. lim⁡x→85x=□\lim _{x \rightarrow 8} 5 x=\squarelimx→8​5x=□ B. Limit does not exist.

Problem 31521

Find the limits for the function f(x)=−6x−6(x−6)2f(x)=-\frac{6 x-6}{(x-6)^{2}}f(x)=−(x−6)26x−6​ as x→6−x \rightarrow 6^{-}x→6− and x→6+x \rightarrow 6^{+}x→6+.

Problem 31522

Calculate 2∫06∣x−4∣dx2 \int_{0}^{6}|x-4| dx2∫06​∣x−4∣dx. Options: (F) 6, (G) 10, (H) 22, (1) 42.

Problem 31523

Calculate the integral ∫06∣x−4∣dx=\int_{0}^{6}|x-4| d x=∫06​∣x−4∣dx= with options: (F) 6, (C) 10, (16) 22, (1) 42.

Problem 31524

Two people start at the same point. One walks east at 3mi/h3 \mathrm{mi/h}3mi/h, the other northeast at 4mi/h4 \mathrm{mi/h}4mi/h. Find the distance change rate after 15 minutes.

Problem 31525

Calculate the integral ∫06∣x−4∣dx\int_{0}^{6}|x-4| d x∫06​∣x−4∣dx.

Problem 31526

A trough is 8 ft long with triangular ends 2 ft wide and 1 ft high. Water fills at 10 ft³/min. Find the rise rate when water is 6 in deep.

Problem 31527

Find the general solution of the homogeneous equation using y=vxy=v xy=vx: (x+sin⁡y)dx+(xcos⁡y−2x)dy=0(x+\sin y) dx+(x \cos y-2 x) dy=0(x+siny)dx+(xcosy−2x)dy=0.

Problem 31528

Find the object's acceleration at t=5t=5t=5 given the position function s(t)=115t5−12t2+5ts(t)=\frac{1}{15} t^{5}-\frac{1}{2} t^{2}+\frac{5}{t}s(t)=151​t5−21​t2+t5​.

Problem 31529

Find the limit: lim⁡x→4x2−5x+42x−8\lim _{x \rightarrow 4} \frac{x^{2}-5 x+4}{2 x-8}limx→4​2x−8x2−5x+4​

Problem 31530

Determine the horizontal asymptotes of the function f(x)=3xx2+4f(x)=\frac{3 x}{\sqrt{x^{2}+4}}f(x)=x2+4​3x​.

Problem 31531

Is the function f(x)=x1+x2f(x)=\frac{x}{1+x^{2}}f(x)=1+x2x​ increasing on the interval (−6,6)(-6,6)(−6,6)?

Problem 31532

Find the interval of convergence for the series from f(x)=79+x2f(x)=\frac{7}{9+x^{2}}f(x)=9+x27​. Provide the answer in interval notation.

Problem 31533

Approximate ∫010f(x)dx\int_{0}^{10} f(x) dx∫010​f(x)dx using a trapezoidal sum with values f(0)=4f(0)=4f(0)=4, f(1)=5f(1)=5f(1)=5, f(4)=10f(4)=10f(4)=10, f(8)=12f(8)=12f(8)=12, f(10)=8f(10)=8f(10)=8.

Problem 31534

A ball falls under gravity with a retarding force F=bvF=bvF=bv. Find the ball's acceleration at any time.

Problem 31535

Find the general solution for the equation: (x2−xy+y2)dx−xydy=0(x^2 - xy + y^2) dx - xy dy = 0(x2−xy+y2)dx−xydy=0. Use y=vxy = vxy=vx if homogeneous.

Problem 31536

Solve the differential equation dydx=12sin⁡(π2x)y+7\frac{d y}{d x}=\frac{1}{2} \sin \left(\frac{\pi}{2} x\right) \sqrt{y+7}dxdy​=21​sin(2π​x)y+7​ with f(1)=2f(1)=2f(1)=2. Find the tangent line at x=1x=1x=1 to estimate f(0.8)f(0.8)f(0.8).

Problem 31537

Use cylindrical shells to find the volume VVV generated by y=x2y=x^{2}y=x2 and x=y2x=y^{2}x=y2 around y=−5y=-5y=−5.

Problem 31538

Find the limit: lim⁡x→27−f(x)x+g(x)\lim _{x \rightarrow 2} \frac{7-f(x)}{x+g(x)}limx→2​x+g(x)7−f(x)​ given lim⁡x→2f(x)=−2\lim _{x \rightarrow 2} f(x)=-2limx→2​f(x)=−2 and lim⁡x→2g(x)=7\lim _{x \rightarrow 2} g(x)=7limx→2​g(x)=7.

Problem 31539

Determine if the function f(x)=ln⁡(x−2)f(x)=\ln (x-2)f(x)=ln(x−2) on [3,11][3,11][3,11] satisfies f′(c)=f(b)−f(a)b−af^{\prime}(c)=\frac{f(b)-f(a)}{b-a}f′(c)=b−af(b)−f(a)​.

Problem 31540

Estimate waste accumulation Q(t)=104(t2+15t+70)Q(t)=10^{4}(t^{2}+15t+70)Q(t)=104(t2+15t+70) for 0≤t≤100 \leq t \leq 100≤t≤10. Find: a) current waste, b) avg. change in 3 years, c) present change, d) when change hits 3×1053 \times 10^{5}3×105.

Problem 31541

Find limits for the piecewise function f(x)={x2,x<23x,x>2f(x)=\left\{\begin{array}{ll}x^{2}, & x<2 \\ 3x, & x>2\end{array}\right.f(x)={x2,3x,​x<2x>2​ at x=2x=2x=2.

Problem 31542

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{(-8+h)^{2}-64}{h}$

Find $\delta$ values that satisfy $\lim _{x \rightarrow -1} 5x + 3 = -2$ with $\varepsilon = 0.4$ . Choices: $\delta=0.0064$ , $\delta=0.02667$ , $\delta=0.16$ , $\delta=0.08$ .

Estimate the instantaneous rate of change of daily receipts 3 weeks after the opening day. Use data:
Weeks: 0, 3, 6, 9, 12, 15, 18, 21, 24, 27 Receipts: 77.025, 50.306, 31.28, 16.24, 6.526, 0.98, 0.92, 0.335, 0.076, 0.044
Round to four decimal places.

Evaluate left, right, and midpoint Riemann sums for $f(x)=\cos(2x)$ from $0$ to $\frac{\pi}{4}$ with $n=60$ . Conjecture the area.

Find the limit as $x$ approaches 0 for $\frac{x-9}{x^{2}(x-2)}$ .

Find the limits of $f(x) = x \sqrt{7 + x^{-2}}$ as $x$ approaches 0 from the left, right, and directly at 0.

Find the $y$ -coordinate on the ellipse $4 x^{2}+25 y^{2}=100$ at $x=1$ , the tangent line equation, triangle area, and $c$ for min area.

Find the limit as $x$ approaches 0 for $\frac{x-9}{x^{2}(x-2)}$ . Limit =

Find the value of $a$ for the function $f(x)=\frac{6 x^{2}-2 x}{2 x}$ (for $x \neq 0$ ) to be continuous at $x=0$ .

Find the general solution of $(x+\sin y) dx+(x \cos y-2 y) dy=0$ . If homogeneous, use $y=v x$ .

Let $f(x)=\frac{6}{1+x^{2}}$ . Find the area of region $R$ , volume when rotated about $y=7$ , and volume of semicircular solid. Bounds: $-1$ to $1$ .

Find the limit: $\lim _{x \rightarrow 11} \frac{x-11}{x^{2}-11 x}=\square$ (integer or simplified fraction) or state if it doesn't exist.

Find the limit: $\lim _{x \rightarrow 0} \frac{x-11}{x^{2}-11 x}=\square$ or state if it doesn't exist.

Find the limit: $\lim _{x \rightarrow 121} \frac{x-11}{x^{2}-11 x}=\square$ or state if it does not exist.

Let $f(x)=\frac{6}{1+x^{2}}$ .
(a) Find the area of the region bounded by $f$ and $y=3$ . (b) Find the volume when this region is rotated about $y=7$ . (c) Find the volume of the solid with semicircular cross sections perpendicular to the $x$ -axis.

Find the limit: $\lim _{x \rightarrow 8} 5 x$ . A. $\lim _{x \rightarrow 8} 5 x=\square$ B. Limit does not exist.

Find the limits for the function $f(x)=-\frac{6 x-6}{(x-6)^{2}}$ as $x \rightarrow 6^{-}$ and $x \rightarrow 6^{+}$ .

Calculate $2 \int_{0}^{6}|x-4| dx$ . Options: (F) 6, (G) 10, (H) 22, (1) 42.

Calculate the integral $\int_{0}^{6}|x-4| d x=$ with options: (F) 6, (C) 10, (16) 22, (1) 42.

Two people start at the same point. One walks east at $3 \mathrm{mi/h}$ , the other northeast at $4 \mathrm{mi/h}$ . Find the distance change rate after 15 minutes.

Calculate the integral $\int_{0}^{6}|x-4| d x$ .

Find the general solution of the homogeneous equation using $y=v x$ : $(x+\sin y) dx+(x \cos y-2 x) dy=0$ .

Find the object's acceleration at $t=5$ given the position function $s(t)=\frac{1}{15} t^{5}-\frac{1}{2} t^{2}+\frac{5}{t}$ .

Find the limit: $\lim _{x \rightarrow 4} \frac{x^{2}-5 x+4}{2 x-8}$

Determine the horizontal asymptotes of the function $f(x)=\frac{3 x}{\sqrt{x^{2}+4}}$ .

Is the function $f(x)=\frac{x}{1+x^{2}}$ increasing on the interval $(-6,6)$ ?

Find the interval of convergence for the series from $f(x)=\frac{7}{9+x^{2}}$ . Provide the answer in interval notation.

Approximate $\int_{0}^{10} f(x) dx$ using a trapezoidal sum with values $f(0)=4$ , $f(1)=5$ , $f(4)=10$ , $f(8)=12$ , $f(10)=8$ .

A ball falls under gravity with a retarding force $F=bv$ . Find the ball's acceleration at any time.

Find the general solution for the equation: $(x^2 - xy + y^2) dx - xy dy = 0$ . Use $y = vx$ if homogeneous.

Solve the differential equation $\frac{d y}{d x}=\frac{1}{2} \sin \left(\frac{\pi}{2} x\right) \sqrt{y+7}$ with $f(1)=2$ . Find the tangent line at $x=1$ to estimate $f(0.8)$ .

Use cylindrical shells to find the volume $V$ generated by $y=x^{2}$ and $x=y^{2}$ around $y=-5$ .

Find the limit: $\lim _{x \rightarrow 2} \frac{7-f(x)}{x+g(x)}$ given $\lim _{x \rightarrow 2} f(x)=-2$ and $\lim _{x \rightarrow 2} g(x)=7$ .

Determine if the function $f(x)=\ln (x-2)$ on $[3,11]$ satisfies $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$ .

Estimate waste accumulation $Q(t)=10^{4}(t^{2}+15t+70)$ for $0 \leq t \leq 10$ . Find: a) current waste, b) avg. change in 3 years, c) present change, d) when change hits $3 \times 10^{5}$ .

Find limits for the piecewise function $f(x)=\left\{\begin{array}{ll}x^{2}, & x<2 \\ 3x, & x>2\end{array}\right.$ at $x=2$ .

Find the limits for the piecewise function $f(x)=\left\{\begin{array}{ll}x^{2}, & x<2 \\ 3x, & x>2\end{array}\right.$ at $x=2$ .

Find the limit: $f(x)=\begin{cases}x^{2}, & x<2 \\ 3x, & x>2\end{cases}$ . What is $\lim _{x \rightarrow 2^{+}} f(x)$ ?

Find limits for $f(x)=\left\{\begin{array}{ll}x^{2}, & x<2 \\ 3x, & x>2\end{array}\right.$ at $x=2$ . Complete parts (A)-(D).

Find the limits of $f(x)=\frac{x-2}{x^{2}-2 x}$ as $x$ approaches 0, 2, and 4.

Differentiate the function $f(t)=\frac{t^{2}-1}{t^{2}+1}$ .

Find the antiderivative of $f(x)=(x-1)(2x+3)$ . Choose all correct options from the list provided.

Find all antiderivatives of $f(x)=(x-1)(2x+3)$ . Incorrect answers lose credit.

Find the limits: a) $\lim _{x \rightarrow 8} f(x)$ , b) $\lim _{x \rightarrow 0} f(x)$ , c) $\lim _{x \rightarrow -4} f(x)$ for $f(x)=\frac{x^{2}-4 x-32}{x-8}$ .

Find the general antiderivative $F(x)$ of $f(x)=\frac{3}{x}+2x+1$ for $x<0$ , including constant $C$ .

Differentiate the function $g(x)=\ln \left(x^{-4}+x^{4}\right)$ .

Find the exact value of $F(2)$ if $F$ is the antiderivative of $f(x)=2x+7$ and $F(1)=9$ .

Differentiate the function $R(w)=4^{w}-5 \log _{9} w$ .

Find the limit as $x$ approaches $\infty$ for $\frac{3x+7}{8x-6}$ . Is it a number, $-\infty$ , or $\infty$ ?

Find the limits for $f(x)=\frac{x^{2}-7 x-18}{x-9}$ as $x$ approaches 9 from the left, right, and directly.

Find the limits: a) $\lim_{x \rightarrow 8} f(x)$ , b) $\lim_{x \rightarrow 0} f(x)$ , c) $\lim_{x \rightarrow -2} f(x)$ for $f(x)=\frac{x^{2}-6x-16}{x-8}$ .

Find the derivative $\frac{d y}{d x}$ for $y=\frac{1}{x^{8}}$ . What is $\frac{d y}{d x}=$ ?

Find the derivative of $y=x^{-5}$ . What is $\frac{d y}{d x}=\square$ ?

Find the derivative $f^{\prime}(x)$ of $f(x)=2 x^{2}+3 x-2$ , then calculate $f^{\prime}(1)$ , $f^{\prime}(4)$ , and $f^{\prime}(9)$ .

Find the derivative $f^{\prime}(x)$ of $f(x)=2x^{2}+3x-2$ , then calculate $f^{\prime}(1)$ , $f^{\prime}(4)$ , $f^{\prime}(9)$ .

Differentiate $\frac{5 x^{2}}{3}-\frac{9}{4 x^{2}}$ with respect to $x$ .

Find the derivative $h^{\prime}(t)$ for the function $h(t)=7.9-8.5 t+0.2 t^{2}$ .

Find $f'(x)$ for $f(x)=x^{4}-4x^{3}+3$ , the slope at $x=-1$ , the tangent line equation at $x=-1$ , and where it's horizontal.

Find the marginal cost function for $C(x)=190+4.4x-0.01x^{2}$ . What is $C^{\prime}(x)$ ?

Find the derivative of $\frac{17 x+39}{x}$ . What is $\frac{d}{d x} \frac{17 x+39}{x}=\square$ ?

Sales function $S(t)=0.04 t^{3}+0.5 t^{2}+6 t+3$ . Find $S^{\prime}(t)$ , $S(7)$ , $S^{\prime}(7)$ , and interpret $S(11)$ and $S^{\prime}(11)$ .

Find the marginal revenue function for $R(x)=x(19-0.04 x)$ . What is $R^{\prime}(x)=?$

Find $f_{xz}$ for $f(x, y, z)=e^{xyz}$ at the point $(1,-1,-1)$ . Options: a. None, b. $e$ , c. $-e$ , d. 0.

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

Find $\lim _{t \rightarrow 0} \left( t \mathbf{i}+\sin t \mathbf{j}-2 \frac{\sin t}{t} \mathbf{k} \right)$ .

Find the derivative of $f(x)=x^{2}-5x+5$ and the slope of the tangent line at $x=\frac{3}{2}$ and $x=2$ .

The cost function for $x$ food processors is $C(x)=2300+70x-0.1x^{2}$ .
(A) Find the exact cost of the 71st processor. (B) Estimate the cost using marginal cost.
(A) Exact cost: \$\square. (B) Approximate cost: \$\square.