Calculus

Problem 9401

Change the order of integration and evaluate $\int_{0}^{\sqrt{8}} \int_{x^{2}}^{8} \frac{e^{y}}{\sqrt{y}} d y d x$ .

See Solution

Problem 9402

Find where the function $f(x)=\frac{x^{4}}{4}+x^{3}-2 x^{2}+9$ has a horizontal tangent line.

See Solution

Problem 9403

Given the function $f(t)=\frac{2}{t}$ , find the net change and average rate of change from $t=a$ to $t=a+h$ .

See Solution

Problem 9404

Find the derivative of the function $f(x) = a^{2} x$ . What is $f'(x)$ ?

See Solution

Problem 9405

A 17 ft ladder leans against a wall. If the top slips down at $4 \mathrm{ft/s}$ , how fast does the base move when the top is 16 ft high?

See Solution

Problem 9406

Calculate the integral from 2 to 4 of $\sqrt{2x+4}$ .

See Solution

Problem 9407

Find the absolute minimum and maximum of $f(x)=2 x^{3}+3 x^{2}-120 x+8$ for $-5 \leq x \leq 5$ .

See Solution

Problem 9408

Gravel is dumped at $10 \mathrm{ft}^{3} / \mathrm{min}$ . How fast is the height increasing when it's $15 \mathrm{ft}$ high?

See Solution

Problem 9409

Calculate the integral: $\int_{-1}^{0.5} 6 \cdot e^{4x-1} \cdot e^{2x} \, dx$

See Solution

Problem 9410

Gegeben ist die Funktion $y=f_{1}(x)=\left(x+\frac{1}{t}\right) \cdot e^{-t x}$ .
a) Untersuchen Sie Nullstellen, Extrempunkte und Wendepunkte. b) Bestimmen Sie die Gleichung für Wendepunkte. c) Zeichnen Sie $f_{-1}$ und $f_{0,5}$ . d) Berechnen Sie den Flächeninhalt $A(k)$ für $k>0$ und $\lim_{k \rightarrow \infty} A(k)$ .

See Solution

Problem 9411

Evaluate the integral: $\int_{4}^{5}\left(\frac{2}{t^{2}}+\frac{2}{t}\right) d t$

See Solution

Problem 9412

Find the extreme values of $f(x)=-9 e^{x} \cos x$ on $[0, \pi]$ . State the $x$ -values or DNE if none exist.

See Solution

Problem 9413

Determine the vertical, horizontal, and oblique asymptotes of the function $T(x)=\frac{x^{2}}{x^{4}-16}$ .

See Solution

Problem 9414

Determine the vertical, horizontal, and oblique asymptotes for the function $T(x)=\frac{x^{2}}{x^{4}-1}$ .

See Solution

Problem 9415

Estimate the limit as x approaches 4 for $\sqrt{x^{2}+14 x+49}$ .

See Solution

Problem 9416

Check if the Mean Value Theorem applies to $f(x)=\sqrt{4-x^{2}}$ on $[0,2]$ . If yes, find $c$ in $[0,2]$ . If no, enter DNE.

See Solution

Problem 9417

Find the limit as $s$ approaches 4 for $P(s)=\frac{97+41s}{s+5}$ . A. 65 B. 29 C. Does not exist D. 52

See Solution

Problem 9418

Evaluate the limit: $\lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}$ . Does it exist?

See Solution

Problem 9419

Find the average slope of the function $f(x)=2 x^{3}-9 x^{2}-108 x+2$ on the interval $[-5,7]$ .

See Solution

Problem 9420

Find the limit as $x$ approaches 5 from the left: $\lim _{x \rightarrow 5^{-}} \frac{x^{2}+25}{x+5}$

See Solution

Problem 9421

Find the limit: $\lim _{x \rightarrow 5^{-}} \frac{x^{2}+25}{x+5}$ . Options: A. 10 B. 5 C. 0 D. Does not exist.

See Solution

Problem 9422

Find the limit: $\lim _{x \rightarrow 4^{+}} \sqrt{x^{2}+6 x+9}$ .

See Solution

Problem 9423

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x^{2}-8 x+6}{x^{3}-9 x^{2}+11}$ . Options: A. 1 B. 0 C. $\infty$ D. $\frac{6}{11}$

See Solution

Problem 9424

Find the limit: $\lim _{x \rightarrow 4^{+}} \sqrt{x^{2}+6 x+9}$ . Options: A. \pm 7 B. 49 C. 7 D. Does not exist.

See Solution

Problem 9425

Calculate the work done in moving an object from $x=1$ ft to $x=14$ ft with a force of $3 x^{-2}$ pounds.

See Solution

Problem 9426

Find the limit as $s$ approaches $\infty$ for $P(s)=\frac{84+46 s}{s+5}$ . What is $\lim_{s \to \infty} P(s)$ ? A. 84 B. 22 C. 46 D. Does not exist

See Solution

Problem 9427

Find the tangent line equation for $g(x)=\sqrt{x}$ at $x=16$ , using $g^{\prime}(x)=\frac{1}{2\sqrt{x}}$ . $y=$

See Solution

Problem 9428

Find the derivative $f^{\prime}(2)$ for the function $f(x)=(\ln (x))^{4}$ .

See Solution

Problem 9429

Find lower and upper estimates for $\int_{10}^{30} f(x) \, dx$ using the table values and five equal subintervals.

See Solution

Problem 9430

Evaluate the integral: $\int \frac{y}{(y+2)(5 y-1)} d y$ (include absolute values and use $C$ for the constant).

See Solution

Problem 9431

A spring requires $5 \mathrm{~J}$ to stretch from $34 \mathrm{~cm}$ to $43 \mathrm{~cm}$ . Find $k$ in $\mathrm{N}/\mathrm{m}$ .
Then, (a) how much work to stretch from $39 \mathrm{~cm}$ to $41 \mathrm{~cm}$ ? (b) How far can a $25 \mathrm{~N}$ force stretch it?

See Solution

Problem 9432

Evaluate the integral: $\int \frac{1}{x(x-a)} d x$ (Use absolute values and $C$ for the constant.)

See Solution

Problem 9433

Find the derivative of the function $w(r)=\sqrt{r^{6}+7}$ .

See Solution

Problem 9434

Differentiate $y=e^{x}-x^{2} \arctan x$ . What is $y'$ ?

See Solution

Problem 9435

Find the derivative of $w(r)=\sqrt{r^{6}+7}$ . What is $\frac{d w}{d r}$ ?

See Solution

Problem 9436

Find the limit of the function $f$ as $x$ approaches 2 from the right, given points at $(2,2)$ and $(2,4)$ . Calculate: $\lim _{x \rightarrow 2^{+}} f(x)$

See Solution

Problem 9437

Differentiate $y=\frac{x}{\cos x}$ .

See Solution

Problem 9438

Find the derivative of $y = \tan^{-1}(\sqrt{3x^2 - 1})$ with respect to $x$ .

See Solution

Problem 9439

Calculate the average rate of change for $f(x)=\sqrt{3x-2}$ from $x=1$ to $x=2$ . What expression gives this?

See Solution

Problem 9440

Find critical numbers of $f(x)=-5 \sin (x) \cos (x)$ on $(-\pi, \pi)$ . Where is $f$ increasing and decreasing?

See Solution

Problem 9441

You have a \ $150,000 mortgage. Interpret$ P(2)=554.43 $and$ P'(2)=75.01$. What do these values mean?

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Problem 9442

Minimize the surface area of a cylindrical can with volume $60 \mathrm{~cm}^{3}$ , height $\geq 1 \mathrm{~cm}$ , diameter $\geq 4 \mathrm{~cm}$ . Find optimal dimensions.

See Solution

Problem 9443

Sketch the graph of a function $g$ where $g^{\prime}(x)>0$ for $x<3$ and $g^{\prime}(x)<0$ for $x>3$ .

See Solution

Problem 9444

What does $s^{\prime}(7)=-224$ mean?
A. 7 seconds after drop, velocity is $-224 \mathrm{ft} / \mathrm{s}$ . B. Dropped from 224 feet, velocity is $7 \mathrm{ft} / \mathrm{s}$ . C. 7 seconds after drop, 224 feet above ocean. D. Dropped from 7 feet, velocity is $-224 \mathrm{ft} / \mathrm{s}$ at ocean.

See Solution

Problem 9445

Find $\lim _{x \rightarrow 4} 2 f(x) g(x)$ given $\lim _{x \rightarrow 4} f(x)=5$ and $\lim _{x \rightarrow 4} g(x)=-3$ .

See Solution

Problem 9446

Find the average rate of change of the function $y=3^{x}$ from $x=0$ to $x=1$ .

See Solution

Problem 9447

Find the limits: (a) $\lim _{x \rightarrow 4} 2 f(x) g(x)$ and (b) $\lim _{x \rightarrow 4} \frac{x^{2}}{f(x)}$ given $\lim _{x \rightarrow 4} f(x)=5$ and $\lim _{x \rightarrow 4} g(x)=-3$ .

See Solution

Problem 9448

Find the average velocity of a TV dropped from a building, where $h(t)=-16 t^{2}+48 t+340$ , over $[1,3.2]$ .

See Solution

Problem 9449

Find the wind speed $v$ (0 ≤ $v$ ≤ 45) that minimizes the wind chill factor $W(v)=-0.17 v^{3}+18.4 v^{2}-584 v-239$ .

See Solution

Problem 9450

Find the limits: (a) $\lim _{x \rightarrow 4} 2 f(x) g(x)$ , (b) $\lim _{x \rightarrow 4} \frac{x^{2}}{f(x)}$ , (c) $\lim _{x \rightarrow 4^{-}} g(x)^{2}-f(x)$ given $\lim _{x \rightarrow 4} f(x)=5$ and $\lim _{x \rightarrow 4} g(x)=-3$ .

See Solution

Problem 9451

Evaluate if the following statements about a continuous function $f$ on $[a, b]$ are true or false, and explain why:
a) $f$ has an absolute max $f(c)$ and min $f(d)$ in $[a, b]$ b) $f^{\prime}(c)$ exists for all $c$ in $[a, b]$ c) $f$ takes every value between $f(a)$ and $f(b)$

See Solution

Problem 9452

Find the fourth derivative of $f(x)=\frac{12x}{1-x}$ , denoted as $f^{(4)}(x)$ .

See Solution

Problem 9453

Given $\lim _{x \rightarrow 4} f(x)=5$ and $\lim _{x \rightarrow 4} g(x)=-3$ , find these limits: (a) $\lim _{x \rightarrow 4} 2 f(x) g(x)$ , (b) $\lim _{x \rightarrow 4} \frac{x^{2}}{f(x)}$ , (c) $\lim _{x \rightarrow 4^{-}} g(x)^{2}-f(x)$ , (d) $\lim _{x \rightarrow 4^{+}} \sqrt{-g(x)}$ , (e) $\lim _{x \rightarrow 4} \frac{f(x)}{f(x)+12 g(x)}$ .

See Solution

Problem 9454

Find the limit of $f(x)=\frac{3x}{x-2}$ as $x$ approaches 2.

See Solution

Problem 9455

Find values of $x$ in $[0, 2\pi]$ where the graph of $y=\frac{\cos x}{2+\sin x}$ has a horizontal tangent. $x=$ $\| x=$

See Solution

Problem 9456

Determine if the following statements are true or false, and explain your reasoning: a) If $f$ and $g$ are increasing on $I$ , then $f+g$ is increasing on $I$ . b) If $f$ and $g$ are positive, increasing on $I$ , then $f g$ is increasing on $I$ . c) If $f^{\prime \prime}(2)=0$ , then $f$ has an inflection point at $x=2$ .

See Solution

Problem 9457

Given $f(1)=6$ and $f^{\prime}(x) \geq 2$ for $1 \leq x \leq 3$ , prove that $f(3) \geq 10$ .

See Solution

Problem 9458

Check if the Mean Value Theorem applies to $f(x)=\frac{x+4}{x^{2}-1}$ on $[-4,0]$ and find values of $c$ .

See Solution

Problem 9459

Find the tangent line equation for $y(x) = 7 \cot^{6}(x)$ at $x = \frac{\pi}{4}$ .

See Solution

Problem 9460

Find the derivative of the function $f(x)=\tan^{2}(x^{4})$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 9461

Compute the limits using conjugates: (a) $\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{x+3}-2}$ , (b) $\lim _{x \rightarrow 4} \frac{4-x}{5-\sqrt{x^{2}+9}}$ .

See Solution

Problem 9462

The graph of the derivative $f^{\prime}$ of a function $f$ crosses the x-axis at -2, 0, 2, 4.
a) Where is $f$ increasing? b) Where is $f$ decreasing? c) Find critical values of $f$ . d) Classify critical values as min, max, or neither.

See Solution

Problem 9463

Find the derivative $f'(x)$ for $f(x)=\frac{2 \sin x}{6 \sin x+4 \cos x}$ . Then, determine the tangent line at $a=0$ : $y=mx+b$ with $m=$ and $b=$ .

See Solution

Problem 9464

Find the curvature of the curve $\vec{r}(t)=\langle 5 \cos (3 t), 5 \sin (3 t), t\rangle$ at $t=0$ . Round to two decimal places.

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Problem 9465

A company's revenue is $R(q)=-q^{3}+250 q^{2}$ and cost is $C(q)=300+16 q$ .
A) Find the marginal profit function $M P(q)$ .
B) Determine the quantity (in hundreds) to maximize profits. Round to two decimal places.

See Solution

Problem 9466

A company's revenue is $R(q)=-q^{3}+370 q^{2}$ and cost is $C(q)=400+20 q$ . Find the marginal profit $M P(q)$ and max profit quantity.

See Solution

Problem 9467

Find the derivative of $f(x)=x^{10}(x^{2}+7)^{3}$ without simplifying.

See Solution

Problem 9468

Find the derivative of $f(x) = 9 \sqrt{x}$ .

See Solution

Problem 9469

Find the derivative of $f(x)=\sqrt[3]{5 x^{2}-2 x}$ .

See Solution

Problem 9470

Find the derivative of $f(x)=e^{-x}(\ln x)$ .

See Solution

Problem 9471

Find the derivative of $g(t)=\frac{\ln t}{t^{2}}$ using the quotient rule: $(g * f' - f * g') / g²$ . Here, $f(t) = \ln(t)$ and $g(t) = t²$ .

See Solution

Problem 9472

Find the limit using I'Hospital's Rule: $\lim _{x \rightarrow-\infty} x^{2} e^{4 x}$ .

See Solution

Problem 9473

Find the derivative of $y=\ln \sqrt{\frac{x+1}{x-1}}$ .

See Solution

Problem 9474

Find the rate of change of the function $y=\frac{3}{2}x+4$ .

See Solution

Problem 9475

Find the $x$ in $[-2,1]$ for $y=3x^{2}+2x+1$ where the derivative equals the secant slope.

See Solution

Problem 9476

Prove that the derivative of $\csc(x)$ is $-\csc(x) \cot(x)$ .

See Solution

Problem 9477

Find the derivative of $f(x)=\sec (x)-x$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 9478

Find the tangent line equation for $y=6 x \sin (x)$ at $\left(\frac{\pi}{2}, 3 \pi\right)$ . $y=$

See Solution

Problem 9479

Find the values of $x$ where the graph of $f(x) = x - 2 \sin(x)$ has a horizontal tangent. List answers as comma-separated values.

See Solution

Problem 9480

Differentiate the function $y=10^{6-x^{2}}$ . Find $y^{\prime}$ .

See Solution

Problem 9481

Find the limit as $\theta$ approaches 0: $\lim _{\theta \rightarrow 0} \frac{\sin (\theta)}{5 \theta+\tan (\theta)}$

See Solution

Problem 9482

Find the one-sided limit $\lim _{x \rightarrow 5^{+}} f(x)$ for the piecewise function $f(x)=\{2x-3 \text{ if } x \leq 5; x^2+1 \text{ if } x>5\}$ .

See Solution

Problem 9483

Calculate the limit: $\lim _{x \rightarrow 0} \frac{\sin (3 x)}{5 x}$ .

See Solution

Problem 9484

Find the derivative $f^{\prime}(x)$ for $f(x)=\frac{14 \sec (x)}{5(1+3 \tan (x))}$ .

See Solution

Problem 9485

Estimate the root of $x^{3}-2 x-5=0$ using Newton's method with initial guess $x_{0}=2$ for one iteration.

See Solution

Problem 9486

Find the derivative of the function $f(x)=8 x^{3} \arctan(9 x^{3})$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 9487

Find the exact value of $\lim_{x \rightarrow 0} g(x)$ given $6x^{2}+3x+4 < g(x) < 10x^{2}-3x+4$ for all $x \neq 0$ .

See Solution

Problem 9488

Differentiate $R(t)=(3t+e^{t})(3-\sqrt{t})$ . Find $R^{\prime}(t)$ .

See Solution

Problem 9489

Find the value of $\lim _{x \rightarrow 3}(f g)(x)$ given that $\lim _{x \rightarrow 3} f(x)=3$ and $\lim _{x \rightarrow 3} g(x)=5$ .

See Solution

Problem 9490

Find the next approximation $x_{1}$ using Newton's method for $f(x)=0$ with initial guess $x_{0}=4$ where $y=5x-1$ is tangent at $(4,19)$ .

See Solution

Problem 9491

True or false: The graph of $y=f(x)$ has a vertical asymptote at $x=a$ if $\lim _{x \rightarrow a^{+}} f(x)=-\infty$ .

See Solution

Problem 9492

Find $\left(f^{-1}\right)^{\prime}(-5)$ given $f^{\prime}(-4)=-8$ , $f^{\prime}(-5)=7$ , $f^{\prime}(6)=-10$ .

See Solution

Problem 9493

True or false: If $\lim_{x \rightarrow a^{+}} f(x)=L$ and $\lim_{x \rightarrow a^{-}} f(x)=L$ , is $f$ continuous at $x=a$ ?

See Solution

Problem 9494

Find the limit as $x$ approaches $-\infty$ : $\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}+x}}{-6 x}$ .

See Solution

Problem 9495

Find the limit as $x$ approaches infinity: $\lim_{x \rightarrow \infty}\left(5+\frac{1}{x}\right)$ .

See Solution

Problem 9496

Identify which integrals are improper and explain why for each: (a) $\int_{-1}^{0} \frac{8 x}{4 x^{2}-1} d x$ (b) $\int_{-2}^{-1} \frac{8 x}{4 x^{2}-1} d x$ (c) $\int_{1/2}^{1} \frac{\sin(x)}{x} d x$

See Solution

Problem 9497

Find the value of $\lim_{x \rightarrow 0} g(x)$ given that $10 x^{2}+10 x+4<g(x)<9 x^{2}-2 x+4$ for $x \neq 0$ .

See Solution

Problem 9498

Find the value of $\lim _{x \rightarrow 2} \sqrt{x+f(x)}$ given that $\lim _{x \rightarrow 2} f(x)=5$ . Round to three decimal places.

See Solution

Problem 9499

Sketch a function $f$ with these features: $f(0)=f(6)=0$ , $f'(3)=f'(5)=0$ , $f'$ positive for $x<3$ and $3<x<5$ , negative for $x>5$ , $f''<0$ for $x<3$ or $x>4$ , and $f''>0$ for $3<x<4$ .

See Solution

Problem 9500

Find critical numbers of $f(x)=x^{2}-8x+3$ , intervals of increase/decrease, and relative extrema using the First Derivative Test.

See Solution

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Calculus

Problem 9401

Change the order of integration and evaluate ∫08∫x28eyydydx\int_{0}^{\sqrt{8}} \int_{x^{2}}^{8} \frac{e^{y}}{\sqrt{y}} d y d x∫08​​∫x28​y​ey​dydx.

Problem 9402

Find where the function f(x)=x44+x3−2x2+9f(x)=\frac{x^{4}}{4}+x^{3}-2 x^{2}+9f(x)=4x4​+x3−2x2+9 has a horizontal tangent line.

Problem 9403

Given the function f(t)=2tf(t)=\frac{2}{t}f(t)=t2​, find the net change and average rate of change from t=at=at=a to t=a+ht=a+ht=a+h.

Problem 9404

Find the derivative of the function f(x)=a2xf(x) = a^{2} xf(x)=a2x. What is f′(x)f'(x)f′(x)?

Problem 9405

A 17 ft ladder leans against a wall. If the top slips down at 4ft/s4 \mathrm{ft/s}4ft/s, how fast does the base move when the top is 16 ft high?

Problem 9406

Calculate the integral from 2 to 4 of 2x+4\sqrt{2x+4}2x+4​.

Problem 9407

Find the absolute minimum and maximum of f(x)=2x3+3x2−120x+8f(x)=2 x^{3}+3 x^{2}-120 x+8f(x)=2x3+3x2−120x+8 for −5≤x≤5-5 \leq x \leq 5−5≤x≤5.

Problem 9408

Gravel is dumped at 10ft3/min10 \mathrm{ft}^{3} / \mathrm{min}10ft3/min. How fast is the height increasing when it's 15ft15 \mathrm{ft}15ft high?

Problem 9409

Calculate the integral: ∫−10.56⋅e4x−1⋅e2x dx\int_{-1}^{0.5} 6 \cdot e^{4x-1} \cdot e^{2x} \, dx∫−10.5​6⋅e4x−1⋅e2xdx

Problem 9410

Problem 9411

Evaluate the integral: ∫45(2t2+2t)dt\int_{4}^{5}\left(\frac{2}{t^{2}}+\frac{2}{t}\right) d t∫45​(t22​+t2​)dt

Problem 9412

Find the extreme values of f(x)=−9excos⁡xf(x)=-9 e^{x} \cos xf(x)=−9excosx on [0,π][0, \pi][0,π]. State the xxx-values or DNE if none exist.

Problem 9413

Determine the vertical, horizontal, and oblique asymptotes of the function T(x)=x2x4−16T(x)=\frac{x^{2}}{x^{4}-16}T(x)=x4−16x2​.

Problem 9414

Determine the vertical, horizontal, and oblique asymptotes for the function T(x)=x2x4−1T(x)=\frac{x^{2}}{x^{4}-1}T(x)=x4−1x2​.

Problem 9415

Estimate the limit as x approaches 4 for x2+14x+49\sqrt{x^{2}+14 x+49}x2+14x+49​.

Problem 9416

Check if the Mean Value Theorem applies to f(x)=4−x2f(x)=\sqrt{4-x^{2}}f(x)=4−x2​ on [0,2][0,2][0,2]. If yes, find ccc in [0,2][0,2][0,2]. If no, enter DNE.

Problem 9417

Find the limit as sss approaches 4 for P(s)=97+41ss+5P(s)=\frac{97+41s}{s+5}P(s)=s+597+41s​. A. 65 B. 29 C. Does not exist D. 52

Problem 9418

Evaluate the limit: lim⁡x→1x4−1x−1\lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}x→1lim​x−1x4−1​. Does it exist?

Problem 9419

Find the average slope of the function f(x)=2x3−9x2−108x+2f(x)=2 x^{3}-9 x^{2}-108 x+2f(x)=2x3−9x2−108x+2 on the interval [−5,7][-5,7][−5,7].

Problem 9420

Find the limit as xxx approaches 5 from the left: lim⁡x→5−x2+25x+5\lim _{x \rightarrow 5^{-}} \frac{x^{2}+25}{x+5}x→5−lim​x+5x2+25​

Problem 9421

Find the limit: lim⁡x→5−x2+25x+5\lim _{x \rightarrow 5^{-}} \frac{x^{2}+25}{x+5}limx→5−​x+5x2+25​. Options: A. 10 B. 5 C. 0 D. Does not exist.

Problem 9422

Find the limit: lim⁡x→4+x2+6x+9\lim _{x \rightarrow 4^{+}} \sqrt{x^{2}+6 x+9}x→4+lim​x2+6x+9​.

Problem 9423

Find the limit: lim⁡x→−∞x2−8x+6x3−9x2+11\lim _{x \rightarrow-\infty} \frac{x^{2}-8 x+6}{x^{3}-9 x^{2}+11}x→−∞lim​x3−9x2+11x2−8x+6​. Options: A. 1 B. 0 C. ∞\infty∞ D. 611\frac{6}{11}116​

Problem 9424

Find the limit: lim⁡x→4+x2+6x+9\lim _{x \rightarrow 4^{+}} \sqrt{x^{2}+6 x+9}x→4+lim​x2+6x+9​. Options: A. \pm 7 B. 49 C. 7 D. Does not exist.

Problem 9425

Calculate the work done in moving an object from x=1x=1x=1 ft to x=14x=14x=14 ft with a force of 3x−23 x^{-2}3x−2 pounds.

Problem 9426

Find the limit as sss approaches ∞\infty∞ for P(s)=84+46ss+5P(s)=\frac{84+46 s}{s+5}P(s)=s+584+46s​. What is lim⁡s→∞P(s)\lim_{s \to \infty} P(s)lims→∞​P(s)? A. 84 B. 22 C. 46 D. Does not exist

Problem 9427

Find the tangent line equation for g(x)=xg(x)=\sqrt{x}g(x)=x​ at x=16x=16x=16, using g′(x)=12xg^{\prime}(x)=\frac{1}{2\sqrt{x}}g′(x)=2x​1​. y=y=y=

Problem 9428

Find the derivative f′(2)f^{\prime}(2)f′(2) for the function f(x)=(ln⁡(x))4f(x)=(\ln (x))^{4}f(x)=(ln(x))4.

Problem 9429

Find lower and upper estimates for ∫1030f(x) dx\int_{10}^{30} f(x) \, dx∫1030​f(x)dx using the table values and five equal subintervals.

Problem 9430

Evaluate the integral: ∫y(y+2)(5y−1)dy\int \frac{y}{(y+2)(5 y-1)} d y∫(y+2)(5y−1)y​dy (include absolute values and use CCC for the constant).

Problem 9431

Problem 9432

Evaluate the integral: ∫1x(x−a)dx\int \frac{1}{x(x-a)} d x∫x(x−a)1​dx (Use absolute values and CCC for the constant.)

Problem 9433

Find the derivative of the function w(r)=r6+7w(r)=\sqrt{r^{6}+7}w(r)=r6+7​.

Problem 9434

Differentiate y=ex−x2arctan⁡xy=e^{x}-x^{2} \arctan xy=ex−x2arctanx. What is y′y'y′?

Problem 9435

Find the derivative of w(r)=r6+7w(r)=\sqrt{r^{6}+7}w(r)=r6+7​. What is dwdr\frac{d w}{d r}drdw​?

Problem 9436

Find the limit of the function fff as xxx approaches 2 from the right, given points at (2,2)(2,2)(2,2) and (2,4)(2,4)(2,4). Calculate: lim⁡x→2+f(x) \lim _{x \rightarrow 2^{+}} f(x) x→2+lim​f(x)

Problem 9437

Differentiate y=xcos⁡xy=\frac{x}{\cos x}y=cosxx​.

Problem 9438

Find the derivative of y=tan⁡−1(3x2−1)y = \tan^{-1}(\sqrt{3x^2 - 1})y=tan−1(3x2−1​) with respect to xxx.

Problem 9439

Calculate the average rate of change for f(x)=3x−2f(x)=\sqrt{3x-2}f(x)=3x−2​ from x=1x=1x=1 to x=2x=2x=2. What expression gives this?

Problem 9440

Find critical numbers of f(x)=−5sin⁡(x)cos⁡(x)f(x)=-5 \sin (x) \cos (x)f(x)=−5sin(x)cos(x) on (−π,π)(-\pi, \pi)(−π,π). Where is fff increasing and decreasing?

Problem 9441

Change the order of integration and evaluate $\int_{0}^{\sqrt{8}} \int_{x^{2}}^{8} \frac{e^{y}}{\sqrt{y}} d y d x$ .

Find where the function $f(x)=\frac{x^{4}}{4}+x^{3}-2 x^{2}+9$ has a horizontal tangent line.

Given the function $f(t)=\frac{2}{t}$ , find the net change and average rate of change from $t=a$ to $t=a+h$ .

Find the derivative of the function $f(x) = a^{2} x$ . What is $f'(x)$ ?

A 17 ft ladder leans against a wall. If the top slips down at $4 \mathrm{ft/s}$ , how fast does the base move when the top is 16 ft high?

Calculate the integral from 2 to 4 of $\sqrt{2x+4}$ .

Find the absolute minimum and maximum of $f(x)=2 x^{3}+3 x^{2}-120 x+8$ for $-5 \leq x \leq 5$ .

Gravel is dumped at $10 \mathrm{ft}^{3} / \mathrm{min}$ . How fast is the height increasing when it's $15 \mathrm{ft}$ high?

Calculate the integral: $\int_{-1}^{0.5} 6 \cdot e^{4x-1} \cdot e^{2x} \, dx$

Evaluate the integral: $\int_{4}^{5}\left(\frac{2}{t^{2}}+\frac{2}{t}\right) d t$

Find the extreme values of $f(x)=-9 e^{x} \cos x$ on $[0, \pi]$ . State the $x$ -values or DNE if none exist.

Determine the vertical, horizontal, and oblique asymptotes of the function $T(x)=\frac{x^{2}}{x^{4}-16}$ .

Determine the vertical, horizontal, and oblique asymptotes for the function $T(x)=\frac{x^{2}}{x^{4}-1}$ .

Estimate the limit as x approaches 4 for $\sqrt{x^{2}+14 x+49}$ .

Check if the Mean Value Theorem applies to $f(x)=\sqrt{4-x^{2}}$ on $[0,2]$ . If yes, find $c$ in $[0,2]$ . If no, enter DNE.

Find the limit as $s$ approaches 4 for $P(s)=\frac{97+41s}{s+5}$ . A. 65 B. 29 C. Does not exist D. 52

Evaluate the limit: $\lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}$ . Does it exist?

Find the average slope of the function $f(x)=2 x^{3}-9 x^{2}-108 x+2$ on the interval $[-5,7]$ .

Find the limit as $x$ approaches 5 from the left: $\lim _{x \rightarrow 5^{-}} \frac{x^{2}+25}{x+5}$

Find the limit: $\lim _{x \rightarrow 5^{-}} \frac{x^{2}+25}{x+5}$ . Options: A. 10 B. 5 C. 0 D. Does not exist.

Find the limit: $\lim _{x \rightarrow 4^{+}} \sqrt{x^{2}+6 x+9}$ .

Find the limit: $\lim _{x \rightarrow-\infty} \frac{x^{2}-8 x+6}{x^{3}-9 x^{2}+11}$ . Options: A. 1 B. 0 C. $\infty$ D. $\frac{6}{11}$

Find the limit: $\lim _{x \rightarrow 4^{+}} \sqrt{x^{2}+6 x+9}$ . Options: A. \pm 7 B. 49 C. 7 D. Does not exist.

Calculate the work done in moving an object from $x=1$ ft to $x=14$ ft with a force of $3 x^{-2}$ pounds.

Find the limit as $s$ approaches $\infty$ for $P(s)=\frac{84+46 s}{s+5}$ . What is $\lim_{s \to \infty} P(s)$ ? A. 84 B. 22 C. 46 D. Does not exist

Find the tangent line equation for $g(x)=\sqrt{x}$ at $x=16$ , using $g^{\prime}(x)=\frac{1}{2\sqrt{x}}$ . $y=$

Find the derivative $f^{\prime}(2)$ for the function $f(x)=(\ln (x))^{4}$ .

Find lower and upper estimates for $\int_{10}^{30} f(x) \, dx$ using the table values and five equal subintervals.

Evaluate the integral: $\int \frac{y}{(y+2)(5 y-1)} d y$ (include absolute values and use $C$ for the constant).

Evaluate the integral: $\int \frac{1}{x(x-a)} d x$ (Use absolute values and $C$ for the constant.)

Find the derivative of the function $w(r)=\sqrt{r^{6}+7}$ .

Differentiate $y=e^{x}-x^{2} \arctan x$ . What is $y'$ ?

Find the derivative of $w(r)=\sqrt{r^{6}+7}$ . What is $\frac{d w}{d r}$ ?

Find the limit of the function $f$ as $x$ approaches 2 from the right, given points at $(2,2)$ and $(2,4)$ . Calculate: $\lim _{x \rightarrow 2^{+}} f(x)$

Differentiate $y=\frac{x}{\cos x}$ .

Find the derivative of $y = \tan^{-1}(\sqrt{3x^2 - 1})$ with respect to $x$ .

Calculate the average rate of change for $f(x)=\sqrt{3x-2}$ from $x=1$ to $x=2$ . What expression gives this?

Find critical numbers of $f(x)=-5 \sin (x) \cos (x)$ on $(-\pi, \pi)$ . Where is $f$ increasing and decreasing?

You have a \ $150,000 mortgage. Interpret$ P(2)=554.43 $and$ P'(2)=75.01$. What do these values mean?

Minimize the surface area of a cylindrical can with volume $60 \mathrm{~cm}^{3}$ , height $\geq 1 \mathrm{~cm}$ , diameter $\geq 4 \mathrm{~cm}$ . Find optimal dimensions.

Sketch the graph of a function $g$ where $g^{\prime}(x)>0$ for $x<3$ and $g^{\prime}(x)<0$ for $x>3$ .

Find $\lim _{x \rightarrow 4} 2 f(x) g(x)$ given $\lim _{x \rightarrow 4} f(x)=5$ and $\lim _{x \rightarrow 4} g(x)=-3$ .

Find the average rate of change of the function $y=3^{x}$ from $x=0$ to $x=1$ .

Find the average velocity of a TV dropped from a building, where $h(t)=-16 t^{2}+48 t+340$ , over $[1,3.2]$ .

Find the wind speed $v$ (0 ≤ $v$ ≤ 45) that minimizes the wind chill factor $W(v)=-0.17 v^{3}+18.4 v^{2}-584 v-239$ .

Find the fourth derivative of $f(x)=\frac{12x}{1-x}$ , denoted as $f^{(4)}(x)$ .

Find the limit of $f(x)=\frac{3x}{x-2}$ as $x$ approaches 2.

Find values of $x$ in $[0, 2\pi]$ where the graph of $y=\frac{\cos x}{2+\sin x}$ has a horizontal tangent. $x=$ $\| x=$

Given $f(1)=6$ and $f^{\prime}(x) \geq 2$ for $1 \leq x \leq 3$ , prove that $f(3) \geq 10$ .

Check if the Mean Value Theorem applies to $f(x)=\frac{x+4}{x^{2}-1}$ on $[-4,0]$ and find values of $c$ .

Find the tangent line equation for $y(x) = 7 \cot^{6}(x)$ at $x = \frac{\pi}{4}$ .

Find the derivative of the function $f(x)=\tan^{2}(x^{4})$ . What is $f^{\prime}(x)$ ?

Compute the limits using conjugates: (a) $\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{x+3}-2}$ , (b) $\lim _{x \rightarrow 4} \frac{4-x}{5-\sqrt{x^{2}+9}}$ .

The graph of the derivative $f^{\prime}$ of a function $f$ crosses the x-axis at -2, 0, 2, 4.
a) Where is $f$ increasing? b) Where is $f$ decreasing? c) Find critical values of $f$ . d) Classify critical values as min, max, or neither.

Find the derivative $f'(x)$ for $f(x)=\frac{2 \sin x}{6 \sin x+4 \cos x}$ . Then, determine the tangent line at $a=0$ : $y=mx+b$ with $m=$ and $b=$ .

Find the curvature of the curve $\vec{r}(t)=\langle 5 \cos (3 t), 5 \sin (3 t), t\rangle$ at $t=0$ . Round to two decimal places.

A company's revenue is $R(q)=-q^{3}+250 q^{2}$ and cost is $C(q)=300+16 q$ .
A) Find the marginal profit function $M P(q)$ .
B) Determine the quantity (in hundreds) to maximize profits. Round to two decimal places.

A company's revenue is $R(q)=-q^{3}+370 q^{2}$ and cost is $C(q)=400+20 q$ . Find the marginal profit $M P(q)$ and max profit quantity.

Find the derivative of $f(x)=x^{10}(x^{2}+7)^{3}$ without simplifying.

Find the derivative of $f(x) = 9 \sqrt{x}$ .

Find the derivative of $f(x)=\sqrt[3]{5 x^{2}-2 x}$ .

Find the derivative of $f(x)=e^{-x}(\ln x)$ .

Find the derivative of $g(t)=\frac{\ln t}{t^{2}}$ using the quotient rule: $(g * f' - f * g') / g²$ . Here, $f(t) = \ln(t)$ and $g(t) = t²$ .

Find the limit using I'Hospital's Rule: $\lim _{x \rightarrow-\infty} x^{2} e^{4 x}$ .

Find the derivative of $y=\ln \sqrt{\frac{x+1}{x-1}}$ .

Find the rate of change of the function $y=\frac{3}{2}x+4$ .