Calculus

Problem 26501

Differentiate the function: $y=\frac{x}{10}-\frac{10}{x}$ .

See Solution

Problem 26502

Find the left and right limits of $f(x)$ as $x$ approaches 1. State the limit if it exists.
$f(x)=\left\{\begin{array}{ll} |x|-1, & \text { if } x \neq 1 \\ x^{3}, & \text { if } x=1 \end{array}\right.$
If no limit exists, write DNE.
$\lim _{x \rightarrow 1^{-}} f(x)= \\ \lim _{x \rightarrow 1^{+}} f(x)= \\ \lim _{x \rightarrow 1} f(x)=$

See Solution

Problem 26503

Use a graphing tool to find the left and right limits of $f(x)$ as $x$ approaches 1. State the limit or DNE.
$f(x)=\left\{\begin{array}{ll} |x|-1, & \text { if } x \neq 1 \\ x^{3}, & \text { if } x=1 \end{array}\right.$
$\lim _{x \rightarrow 1^{-}} f(x)= \\ \lim _{x \rightarrow 1^{+}} f(x)= \\ \lim _{x \rightarrow 1} f(x)=$

See Solution

Problem 26504

Determine if the limit of $f(x)=\frac{x^{2}-6x-7}{x^{2}-7x}$ exists at $x=7$ . Round to two decimal places or enter DNE.

See Solution

Problem 26505

Berechne den mittleren Benzinverbrauch über eine Strecke von 50 km mit der Funktion $v(s) = -0,0000006 s^{3} + 0,00006 s^{2} - 0,0017 s + 0,06$ .

See Solution

Problem 26506

Find the limit as $x$ approaches 5 for the expression $\frac{x^{2}-25}{x-5}$ .

See Solution

Problem 26507

A function $h$ is differentiable. Let $k$ be its inverse.
Part A: Find $k'$ using the inverse function rule.
Part B: Use the values to find $k'(1)$ .

See Solution

Problem 26508

A 240 g ball moves at 20.0 m/s with a GPE of 70.0 J. What speed will it hit the ground considering air resistance?

See Solution

Problem 26509

1. Find vertical tangent lines for $y^{2}-2y=6x$ .
2. Given $g(x)=f^{-1}(x)$ , find $g^{\prime}(7)$ with $f(7)=4$ , $f^{\prime}(7)=-3$ .
3. If $f(x)=g(x)\sqrt{h(x)}$ , find $f^{\prime}(2)$ .
4. Find $\frac{d}{dx} g^{-1}(4)$ .

See Solution

Problem 26510

Ableitungsregeln anwenden: Bestimmen Sie $\mathrm{f}^{\prime}$ für die Funktionen a) $f(x)=\frac{1}{4} x^{4}-2 x^{2}$ bis f) $f(x)=\frac{4}{x}+1$ . Steigungen an $x_{0}$ : a) $f(x)=\frac{1}{2} x^{2}-2, x_{0}=2$ bis f) $f(x)=x+\frac{1}{x}, x_{0}=1$ .

See Solution

Problem 26511

Evaluate the limit: $\lim _{x \rightarrow 6^{-}}\left(\frac{|x-6|}{6-x}\right)=$ . If DNE, enter DNE.

See Solution

Problem 26512

Evaluate the limit: $\lim _{h \rightarrow 0}\left(\frac{\sqrt{13-h}-\sqrt{13}}{h}\right)=$

See Solution

Problem 26513

Find the left and right limits of $f(x)$ as $x$ approaches -5, where $f(x)=\frac{|x+5|}{x+5}$ . State the limit or DNE.

See Solution

Problem 26514

Is the function $f(x)=6 \cdot 3^{x+4}$ continuous everywhere? If yes, state the domain; if no, list points of discontinuity.

See Solution

Problem 26515

Find the limit: $\lim _{x \rightarrow 5^{-}} \frac{x^{2}}{x^{2}-25}$ . Enter DNE if it doesn't exist.

See Solution

Problem 26516

Evaluate the limit: $\lim _{x \rightarrow 4} \frac{\sqrt{x-3}-1}{x^{2}+x-20}=$ Enter DNE if it doesn't exist.

See Solution

Problem 26517

Bestimme die Ableitungsfunktion von $f(x)=x+2\sqrt{x}$ .

See Solution

Problem 26518

A person throws ball A up at $10 \mathrm{~m/s}$ and ball B down from $5 \mathrm{~m}$ . How long after B hits the ground does A?

See Solution

Problem 26519

Use the integral test to find how many terms of the series $\sum_{n=1}^{\infty} \frac{6}{n^{4}}$ ensure an error < 0.002.

See Solution

Problem 26520

Calculate the average rate of change of $f(x)=\sqrt{2x+9}$ on the interval $[-4,8]$ .

See Solution

Problem 26521

Find the interval of convergence for the series: $\sum_{n=1}^{\infty} \frac{(-1)^{n} 4^{n}(x-2)^{n}}{n}$ .

See Solution

Problem 26522

Find the area of the region enclosed by $y=x^{2}+2$ , $y=2x+2$ , and $y=4$ . Sketch and find intersections.

See Solution

Problem 26523

A ball is thrown down from a 12.0 m building at 3.5 m/s. What is its speed just before hitting the ground?

See Solution

Problem 26524

Find the domain of $f(x)=\frac{x^{2}+3 x-4}{-x^{3}+27}$ , check its continuity, and identify asymptotes.

See Solution

Problem 26525

Find the average value of $y = x^2 - 1$ on $[0, \sqrt{3}]$ and the point where it equals this average.

See Solution

Problem 26526

Find the derivative of $h(t) = \frac{10 \log_{4} t}{t}$ .

See Solution

Problem 26527

Find the derivative $f^{\prime}(x)$ , the partition numbers for $f^{\prime}$ , and the critical numbers of $f$ for $f(x)=\frac{1}{x+8}$ .

See Solution

Problem 26528

Find the derivative $f^{\prime}(x)$ , the partition numbers for it, and the critical numbers of $f(x)=\frac{2}{x+1}$ .

See Solution

Problem 26529

Find (A) $f^{\prime}(x)$ , (B) partition numbers for $f^{\prime}$ , (C) critical numbers of $f$ for $f(x)=\frac{2}{x+1}$ .

See Solution

Problem 26530

Determine where $f(x)$ is increasing, decreasing, and find local extrema for $f(x)=(x-3)e^{-9x}$ .

See Solution

Problem 26531

Determine where the function $f(x)$ is increasing, decreasing, and find local extrema for $f(x)=x^{3}+6 x-27$ .

See Solution

Problem 26532

Find where $f(x)=x^{3}+6x-27$ is increasing, decreasing, and its local extrema. Use interval notation for answers.

See Solution

Problem 26533

Find the limit as $x$ approaches 2 for $\frac{3 f(x)-15}{x-2}$ given $f(2)=5$ and $f'(2)=-5$ .

See Solution

Problem 26534

Bestimme die Tangentengleichung an $f(x)=x^{3}$ im Punkt $P(-1, f(-1))$ .

See Solution

Problem 26535

Find the limit: $\lim _{x \rightarrow-\infty}\left(-x^{4}+x^{2}+x+4\right)$ .

See Solution

Problem 26536

Find the limit: $\lim _{x \rightarrow-\infty}\left(x^{5}-4 x^{3}+4 x\right)$ .

See Solution

Problem 26537

Find the limit: $\lim _{x \rightarrow \infty} \frac{3 x}{x+1}$ .

See Solution

Problem 26538

Find the average rate of change of the function $y = 2x^2 + 1$ over the interval $[-1, 1]$ .

See Solution

Problem 26539

Bestimme die Tangentengleichung der Funktion $f(x) = 2x^2 - x + 1$ im Punkt $P(2, f(2))$ .

See Solution

Problem 26540

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

See Solution

Problem 26541

Find the volume of the solid formed by rotating the area between $y = x^{2} + 2$ and $y = 4 - x^{2}$ around $y = -3$ .

See Solution

Problem 26542

Find the limit as $x$ approaches 2 for $\frac{3 f(x)-15}{x-2}$ given $f(0)=3$ , $f(1)=4$ , $f(2)=5$ .

See Solution

Problem 26543

Find the average rate of change of $y=\frac{1}{x-1}$ over the interval $[-1,-\frac{1}{2}]$ .

See Solution

Problem 26544

Find the derivative of the function $y = \sqrt{2x + 4}$ .

See Solution

Problem 26545

Find the tangent line equation for the curve $y = -x^3 + x^2 + 3$ at the point $(1, 3)$ .

See Solution

Problem 26546

Find the sum $\sum_{n=1}^{\infty}(-1)^{n} \frac{n !}{n^{n} b_{1} b_{2} \ldots b_{n}}$ where $b_n \to \frac{1}{2}$ .

See Solution

Problem 26547

Find the rate of change of $y$ with respect to $x$ from the equation $x+\frac{1}{2} y=5$ .

See Solution

Problem 26548

Find $c$ in the interval $(1, 3)$ such that $h'(c) = \frac{h(3) - h(1)}{2}$ . Given $h(x)=x^{3}-x$ .

See Solution

Problem 26549

Find the slope of the tangent line to $f(x)=\sqrt{x}$ when $x=25$ .

See Solution

Problem 26550

Find the slope of the tangent line to $g(x)=x^{2}$ when $x=5$ .

See Solution

Problem 26551

Find the slope of the tangent line to $g(x)=\frac{1}{x}$ at $x=6$ using limits.

See Solution

Problem 26552

Find the height $x$ (in cm) that maximizes the cross-sectional area $A = x(28 - 2x)$ . Round to the nearest cm.

See Solution

Problem 26553

Determine the slope of the tangent line to $k(x)=x^{3}$ when $x=3$ .

See Solution

Problem 26554

Find the slope of the tangent line to $g(x)=\frac{1}{x}$ at the point where $x=6$ .

See Solution

Problem 26555

A 65 kg bike rider starts at rest 200 m high. What is his velocity at 150 m high?

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Problem 26556

Bestimme die Normalengleichung der Funktion $f(x)=x^{2}-4$ am Punkt $P(1|-3)$ .

See Solution

Problem 26557

Find the limit as $x$ approaches 8 from the left of $\frac{3 p(x)-2 x}{15-3 x}$ , given $p(8)=3$ .

See Solution

Problem 26558

Given the curve $x y^{2}-2 x^{3}=2$ for $y \geq 0$ :
(a) Show $\frac{d y}{d x}=\frac{6 x^{2}-y^{2}}{2 x y}$ .
(b) Find the tangent line equation at $(1,2)$ .
(c) Determine the $x$ -coordinate where the tangent line is horizontal.
(d) Calculate $\frac{d^{2} y}{d x^{2}}$ at $(1,2)$ .

See Solution

Problem 26559

Determine the half-life of a radioactive element with decay function $A(t)=A_{0} e^{-0.0375 t}$ .

See Solution

Problem 26560

Find the relative minima and maxima of the polynomial function $y = x^{4} - x^{2} + 4$ .

See Solution

Problem 26561

A bullet is shot up at $+742 \mathrm{~m/s}$ . Find its velocity at $40 \mathrm{~s}$ , $80 \mathrm{~s}$ , and $120 \mathrm{~s}$ .

See Solution

Problem 26562

The temperature function $f(t)$ is defined in two parts. Answer these questions:
(a) What is the average rate of change of temperature from $t=0$ to $t=12$ hours? (b) Approximate $f^{\prime}(10)$ using the table data. (c) Is $f$ continuous for $0 \leq t \leq 24$ ? Justify. (d) Find $f^{\prime}(20)$ .

See Solution

Problem 26563

A sculptor forms a cylinder with volume $V=\pi r^{2} h$ . Given $h=8$ in, $r=3$ in, and $\frac{dr}{dt}=-\frac{1}{2}$ in/min, find:
(a) Rate of area decrease of the circular cross-section. (b) Rate of height increase. (c) Expression for $\frac{dr}{dh}$ in terms of $h$ and $r$ .

See Solution

Problem 26564

Given values for a function $h$ and its derivative $h^{\prime}$ , find $g^{\prime \prime}(e^{0.7})$ , $f^{\prime}(1)$ , tangent line for $h^{-1}$ at $x=-2$ , and $\frac{d y}{d x}$ at $(2,1)$ .

See Solution

Problem 26565

Find the volume of the solid formed by rotating the area between $y=\sin \left(x^{2}\right)$ , $[0, \sqrt{\pi}]$ , and the $x$ -axis around the $y$ -axis.

See Solution

Problem 26566

Verify if $y=(x+1)-\frac{1}{3} e^{x}$ satisfies $\frac{dy}{dx}=y-x$ .

See Solution

Problem 26567

Calculate the average rate of change of $k(x)=2 x^{2}-3$ from $x=1$ to $x=6$ .

See Solution

Problem 26568

Approximate the volume of a bird's egg shell with inner radius 5 mm and outer radius 5.2 mm using $V(r)=\frac{4}{3} \pi r^{3}$ . Volume is approximately $\square \mathrm{mm}^{3}$ . (Round to nearest hundredth.)

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Problem 26569

Solve the initial value problem: $\frac{d y}{d x}=y^{2}(1-x^{2})$ , with $y(0)=1$ .

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Problem 26570

Approximate the volume of a bird's egg shell with inner radius 7 mm and outer radius 7.3 mm using $V(r)=\frac{4}{3} \pi r^{3}$ . Volume is approximately $\square \mathrm{mm}^{3}$ .

See Solution

Problem 26571

Find the derivatives: 1) $y=-3 \tan(5x^3)$ , 2) $y=\sec^{-1}(x^5)$ , 3) $y=\ln(\sin(4x))-x^4$ , 4) $\cos(x-y)=5x$ , 5) $y=\cos^2(\sqrt{3x})$ . Also, find the slope of the tangent line for $f(x)=\ln(1-x)$ at $x=-1$ .

See Solution

Problem 26572

Find $\frac{d y}{d x}$ for the equation $3 x - \tan y = 4$ in terms of $y$ . Options: (A) $3 \sin^2 y$ , (B) $3 \cos^2 y$ , (C) $3 \cos y \cot y$ , (D) $\frac{3}{1 + 9 y^2}$ .

See Solution

Problem 26573

Approximate the shell volume of an egg with inner radius 8 mm and outer radius 8.4 mm using $V(r)=\frac{4}{3} \pi r^{3}$ . The volume is approximately $\square \mathrm{mm}^{3}$ .

See Solution

Problem 26574

Approximate the volume of a bird's egg shell with inner radius 8 mm and outer radius 8.2 mm using $V(r)=\frac{4}{3} \pi r^{3}$ . Volume: $\square \mathrm{mm}^{3}$ .

See Solution

Problem 26575

Calculate the average rate of change of $g(x)=-x^{3}+10 x-20$ from $x=-4$ to $x=4$ . Round your answer to the nearest tenth.

See Solution

Problem 26576

Estimate the local max/min values and their $x$ locations from the graph of $f$ . Also find intervals of increase and decrease.

See Solution

Problem 26577

Does the function $f(x)=\frac{x^{3}-5}{x^{2}+5}$ have a slant asymptote?

See Solution

Problem 26578

Find the slope of the tangent line for these functions:
15. $g(x)=\sin ^{2}(3 x)$ at $x=\frac{\pi}{12}$ .
16. $2 y^{5}+3 x-2=x^{3}-\ln y$ at $(0,1)$ .
17. Find $\frac{d^{2} y}{d x^{2}}$ for $x y^{3}=4$ .

See Solution

Problem 26579

Determine the half-life of a radioactive element with decay function $A(t)=A_{0} e^{-0.0289 t}$ .

See Solution

Problem 26580

Find the normal line equation for $f(x)=x^{4}+3x^{2}-1$ where $f^{\prime}(x)=2$ . Choose from options a) to e).

See Solution

Problem 26581

Find the normal line equation for $f(x)=x^{4}+3 x^{2}-1$ at the point where $f^{\prime}(x)=2$ .

See Solution

Problem 26582

Find the slope of the tangent line for these functions at the specified points:
1. $g(x)=\sin^{2}(3x)$ at $x=\frac{\pi}{12}$
2. $2y^{5}+3x-2=x^{3}-\ln y$ at $(0,1)$
3. Find $\frac{d^{2}y}{dx^{2}}$ for $xy^{3}=4$ and evaluate it.
4. $y=2\sin x$ at $x=\frac{3\pi}{2}$
5. $e^{y}+4x=8$ at $(0,2)$

See Solution

Problem 26583

Find the work to pump oil from a half-full spherical tank (radius 8 ft, oil weight 50 lb/ft³) through the top.

See Solution

Problem 26584

Find the half-life of a radioactive element with decay function $A(t)=A_{0} e^{-0.033 t}$ .

See Solution

Problem 26585

20. Find $\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sin ^{2}(x)-\sin ^{2}\left(\frac{\pi}{3}\right)}{x-\frac{\pi}{3}}$ .
21. Which function has a vertical asymptote at $x=4$ ? (A) $\frac{x+5}{x^{2}-4}$ (B) $\frac{x^{2}-16}{x-4}$ (C) $\frac{4 x}{x+1}$ (D) $\frac{x+6}{x^{2}-7 x+12}$ (E) None.
22. Find $\lim _{x \rightarrow-\infty} \frac{2 x+3}{\sqrt{x^{2}+x+1}}$ .

See Solution

Problem 26586

Find the derivative of $f(x)=\arccos \left(x^{3}\right)$ , i.e., determine $f^{\prime}(x)$ .

See Solution

Problem 26587

Find the value of $f^{\prime}\left(\frac{3}{4}\right)$ if $f(x)=\tan^{-1}(x)$ .

See Solution

Problem 26588

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\arctan \left(\frac{\sqrt{7}}{4}+h\right)-\arctan \left(\frac{\sqrt{7}}{4}\right)}{h}$ .

See Solution

Problem 26589

Find the derivative of the function $f(x)=\arccos \left(x^{2}\right)$ , i.e., compute $f^{\prime}(x)$ .

See Solution

Problem 26590

Evaluate the limit: $\lim _{h \rightarrow 0} \frac{\arcsin \left(\frac{4}{5}+h\right)-\arcsin \left(\frac{4}{5}\right)}{h}$ .

See Solution

Problem 26591

1. Approximate $W^{\prime}(20)$ using the temperature data every 5 days over 30 days. Show calculations and units.
2. Find $f^{\prime}(4)$ for the apple removal rate $f(t)=8+(0.7 t) \cos \left(\frac{t^{3}}{50}\right)$ and explain its meaning.
3. Determine the snow volume change rate at 9 A.M. given $s(t)=2 t e^{\sin t}$ and Javier's removal rate $r(t)$ .
4. Interpret $r(3)=0.988$ and $r^{\prime}(1)=0.025$ for the storm's impact on the distance between the road and water.

See Solution

Problem 26592

1. Relate the change in surface area $A=2 \pi r^{2}+2 \pi r h$ to the change in height $h$ (radius constant).
2. Relate the volume change $V=\frac{4}{3} \pi r^{3}$ to the radius change $r$ of an expanding balloon.
3. Relate the runner's speed to the changing distance from home plate while running from second to third base.
4. Relate the rates of change of two planes flying at right angles as they converge at a point.
5. Relate the change in angle of elevation to the height of a rocket rising vertically, 5 miles from the observer.

See Solution

Problem 26593

Determine where the function $f(x) = x^{4} - 4x^{3} - 19$ is increasing.

See Solution

Problem 26594

Tentukan fungsi penerimaan marginal dari fungsi permintaan $P=4-10 Q$ .

See Solution

Problem 26595

Find the derivative of the function $f(x) = a \exp\left(-\frac{(x-b)^2}{2c^2}\right)$ with respect to $x$ .

See Solution

Problem 26596

Find the Laplacian of the Gaussian function $G(x, y) = \frac{1}{2\pi\sigma^2} e^{-\frac{x^2 + y^2}{2\sigma^2}}$ using $\nabla^2 G(x, y)$ .

See Solution

Problem 26597

Find the volume of the solid formed by rotating the area between $y=x^2+2$ and $y=4-x^2$ around $y=-3$ from $x=-1$ to $x=1$ .

See Solution

Problem 26598

Given the graph of $f(x)$ (semicircle + 2 lines), find $F(-2)$ , $F(2)$ , $F(6)$ , and the rate of change at $x=4$ .

See Solution

Problem 26599

Find the derivative of the function $f(x)=\ln \left(x \sqrt{x^{2}-1}\right)$ .

See Solution

Problem 26600

Find the derivatives of these functions: 1. $f(t)=\frac{7}{t}-\frac{5}{t^{3}}$ 2. $f(t)=\frac{14}{t}+\frac{12}{t}+\sqrt{2}$ 3. $y=\frac{6}{x^{4}}-\frac{7}{x^{3}}+\frac{3}{x}+\sqrt{5}$ 4. $y=\frac{3}{x^{4}}+\frac{1}{x^{5}}+\frac{7}{x^{2}}$ 5. $p(x)=-10 x^{-1/2}+8 x^{-1/2}$

See Solution

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Calculus

Problem 26501

Differentiate the function: y=x10−10xy=\frac{x}{10}-\frac{10}{x}y=10x​−x10​.

Problem 26502

Problem 26503

Problem 26504

Determine if the limit of f(x)=x2−6x−7x2−7xf(x)=\frac{x^{2}-6x-7}{x^{2}-7x}f(x)=x2−7xx2−6x−7​ exists at x=7x=7x=7. Round to two decimal places or enter DNE.

Problem 26505

Berechne den mittleren Benzinverbrauch über eine Strecke von 50 km mit der Funktion v(s)=−0,0000006s3+0,00006s2−0,0017s+0,06v(s) = -0,0000006 s^{3} + 0,00006 s^{2} - 0,0017 s + 0,06v(s)=−0,0000006s3+0,00006s2−0,0017s+0,06.

Problem 26506

Find the limit as xxx approaches 5 for the expression x2−25x−5\frac{x^{2}-25}{x-5}x−5x2−25​.

Problem 26507

A function hhh is differentiable. Let kkk be its inverse. Part A: Find k′k'k′ using the inverse function rule. Part B: Use the values to find k′(1)k'(1)k′(1).

Problem 26508

A 240 g ball moves at 20.0 m/s with a GPE of 70.0 J. What speed will it hit the ground considering air resistance?

Problem 26509

Problem 26510

Problem 26511

Evaluate the limit: lim⁡x→6−(∣x−6∣6−x)=\lim _{x \rightarrow 6^{-}}\left(\frac{|x-6|}{6-x}\right)=x→6−lim​(6−x∣x−6∣​)=. If DNE, enter DNE.

Problem 26512

Evaluate the limit: lim⁡h→0(13−h−13h)=\lim _{h \rightarrow 0}\left(\frac{\sqrt{13-h}-\sqrt{13}}{h}\right)=h→0lim​(h13−h​−13​​)=

Problem 26513

Find the left and right limits of f(x)f(x)f(x) as xxx approaches -5, where f(x)=∣x+5∣x+5f(x)=\frac{|x+5|}{x+5}f(x)=x+5∣x+5∣​. State the limit or DNE.

Problem 26514

Is the function f(x)=6⋅3x+4f(x)=6 \cdot 3^{x+4}f(x)=6⋅3x+4 continuous everywhere? If yes, state the domain; if no, list points of discontinuity.

Problem 26515

Find the limit: lim⁡x→5−x2x2−25\lim _{x \rightarrow 5^{-}} \frac{x^{2}}{x^{2}-25}x→5−lim​x2−25x2​. Enter DNE if it doesn't exist.

Problem 26516

Evaluate the limit: lim⁡x→4x−3−1x2+x−20=\lim _{x \rightarrow 4} \frac{\sqrt{x-3}-1}{x^{2}+x-20}=x→4lim​x2+x−20x−3​−1​= Enter DNE if it doesn't exist.

Problem 26517

Bestimme die Ableitungsfunktion von f(x)=x+2xf(x)=x+2\sqrt{x}f(x)=x+2x​.

Problem 26518

A person throws ball A up at 10 m/s10 \mathrm{~m/s}10 m/s and ball B down from 5 m5 \mathrm{~m}5 m. How long after B hits the ground does A?

Problem 26519

Use the integral test to find how many terms of the series ∑n=1∞6n4\sum_{n=1}^{\infty} \frac{6}{n^{4}}∑n=1∞​n46​ ensure an error < 0.002.

Problem 26520

Calculate the average rate of change of f(x)=2x+9f(x)=\sqrt{2x+9}f(x)=2x+9​ on the interval [−4,8][-4,8][−4,8].

Problem 26521

Find the interval of convergence for the series: ∑n=1∞(−1)n4n(x−2)nn\sum_{n=1}^{\infty} \frac{(-1)^{n} 4^{n}(x-2)^{n}}{n}n=1∑∞​n(−1)n4n(x−2)n​.

Problem 26522

Find the area of the region enclosed by y=x2+2y=x^{2}+2y=x2+2, y=2x+2y=2x+2y=2x+2, and y=4y=4y=4. Sketch and find intersections.

Problem 26523

A ball is thrown down from a 12.0 m building at 3.5 m/s. What is its speed just before hitting the ground?

Problem 26524

Find the domain of f(x)=x2+3x−4−x3+27f(x)=\frac{x^{2}+3 x-4}{-x^{3}+27}f(x)=−x3+27x2+3x−4​, check its continuity, and identify asymptotes.

Problem 26525

Find the average value of y=x2−1y = x^2 - 1y=x2−1 on [0,3][0, \sqrt{3}][0,3​] and the point where it equals this average.

Problem 26526

Find the derivative of h(t)=10log⁡4tth(t) = \frac{10 \log_{4} t}{t}h(t)=t10log4​t​.

Problem 26527

Find the derivative f′(x)f^{\prime}(x)f′(x), the partition numbers for f′f^{\prime}f′, and the critical numbers of fff for f(x)=1x+8f(x)=\frac{1}{x+8}f(x)=x+81​.

Problem 26528

Find the derivative f′(x)f^{\prime}(x)f′(x), the partition numbers for it, and the critical numbers of f(x)=2x+1f(x)=\frac{2}{x+1}f(x)=x+12​.

Problem 26529

Find (A) f′(x)f^{\prime}(x)f′(x), (B) partition numbers for f′f^{\prime}f′, (C) critical numbers of fff for f(x)=2x+1f(x)=\frac{2}{x+1}f(x)=x+12​.

Problem 26530

Determine where f(x)f(x)f(x) is increasing, decreasing, and find local extrema for f(x)=(x−3)e−9xf(x)=(x-3)e^{-9x}f(x)=(x−3)e−9x.

Problem 26531

Determine where the function f(x)f(x)f(x) is increasing, decreasing, and find local extrema for f(x)=x3+6x−27f(x)=x^{3}+6 x-27f(x)=x3+6x−27.

Problem 26532

Find where f(x)=x3+6x−27f(x)=x^{3}+6x-27f(x)=x3+6x−27 is increasing, decreasing, and its local extrema. Use interval notation for answers.

Problem 26533

Find the limit as xxx approaches 2 for 3f(x)−15x−2\frac{3 f(x)-15}{x-2}x−23f(x)−15​ given f(2)=5f(2)=5f(2)=5 and f′(2)=−5f'(2)=-5f′(2)=−5.

Problem 26534

Bestimme die Tangentengleichung an f(x)=x3f(x)=x^{3}f(x)=x3 im Punkt P(−1,f(−1))P(-1, f(-1))P(−1,f(−1)).

Problem 26535

Find the limit: lim⁡x→−∞(−x4+x2+x+4)\lim _{x \rightarrow-\infty}\left(-x^{4}+x^{2}+x+4\right)limx→−∞​(−x4+x2+x+4).

Problem 26536

Find the limit: lim⁡x→−∞(x5−4x3+4x)\lim _{x \rightarrow-\infty}\left(x^{5}-4 x^{3}+4 x\right)limx→−∞​(x5−4x3+4x).

Problem 26537

Find the limit: lim⁡x→∞3xx+1\lim _{x \rightarrow \infty} \frac{3 x}{x+1}limx→∞​x+13x​.

Problem 26538

Find the average rate of change of the function y=2x2+1y = 2x^2 + 1y=2x2+1 over the interval [−1,1][-1, 1][−1,1].

Problem 26539

Bestimme die Tangentengleichung der Funktion f(x)=2x2−x+1f(x) = 2x^2 - x + 1f(x)=2x2−x+1 im Punkt P(2,f(2))P(2, f(2))P(2,f(2)).

Problem 26540

Find the limit: lim⁡x→∞−3x22x+4\lim _{x \rightarrow \infty}-\frac{3 x^{2}}{2 x+4}limx→∞​−2x+43x2​

Problem 26541

Find the volume of the solid formed by rotating the area between y=x2+2y = x^{2} + 2y=x2+2 and y=4−x2y = 4 - x^{2}y=4−x2 around y=−3y = -3y=−3.

Problem 26542

Differentiate the function: $y=\frac{x}{10}-\frac{10}{x}$ .

Determine if the limit of $f(x)=\frac{x^{2}-6x-7}{x^{2}-7x}$ exists at $x=7$ . Round to two decimal places or enter DNE.

Berechne den mittleren Benzinverbrauch über eine Strecke von 50 km mit der Funktion $v(s) = -0,0000006 s^{3} + 0,00006 s^{2} - 0,0017 s + 0,06$ .

Find the limit as $x$ approaches 5 for the expression $\frac{x^{2}-25}{x-5}$ .

A function $h$ is differentiable. Let $k$ be its inverse.
Part A: Find $k'$ using the inverse function rule.
Part B: Use the values to find $k'(1)$ .

Evaluate the limit: $\lim _{x \rightarrow 6^{-}}\left(\frac{|x-6|}{6-x}\right)=$ . If DNE, enter DNE.

Evaluate the limit: $\lim _{h \rightarrow 0}\left(\frac{\sqrt{13-h}-\sqrt{13}}{h}\right)=$

Find the left and right limits of $f(x)$ as $x$ approaches -5, where $f(x)=\frac{|x+5|}{x+5}$ . State the limit or DNE.

Is the function $f(x)=6 \cdot 3^{x+4}$ continuous everywhere? If yes, state the domain; if no, list points of discontinuity.

Find the limit: $\lim _{x \rightarrow 5^{-}} \frac{x^{2}}{x^{2}-25}$ . Enter DNE if it doesn't exist.

Evaluate the limit: $\lim _{x \rightarrow 4} \frac{\sqrt{x-3}-1}{x^{2}+x-20}=$ Enter DNE if it doesn't exist.

Bestimme die Ableitungsfunktion von $f(x)=x+2\sqrt{x}$ .

A person throws ball A up at $10 \mathrm{~m/s}$ and ball B down from $5 \mathrm{~m}$ . How long after B hits the ground does A?

Use the integral test to find how many terms of the series $\sum_{n=1}^{\infty} \frac{6}{n^{4}}$ ensure an error < 0.002.

Calculate the average rate of change of $f(x)=\sqrt{2x+9}$ on the interval $[-4,8]$ .

Find the interval of convergence for the series: $\sum_{n=1}^{\infty} \frac{(-1)^{n} 4^{n}(x-2)^{n}}{n}$ .

Find the area of the region enclosed by $y=x^{2}+2$ , $y=2x+2$ , and $y=4$ . Sketch and find intersections.

Find the domain of $f(x)=\frac{x^{2}+3 x-4}{-x^{3}+27}$ , check its continuity, and identify asymptotes.

Find the average value of $y = x^2 - 1$ on $[0, \sqrt{3}]$ and the point where it equals this average.

Find the derivative of $h(t) = \frac{10 \log_{4} t}{t}$ .

Find the derivative $f^{\prime}(x)$ , the partition numbers for $f^{\prime}$ , and the critical numbers of $f$ for $f(x)=\frac{1}{x+8}$ .

Find the derivative $f^{\prime}(x)$ , the partition numbers for it, and the critical numbers of $f(x)=\frac{2}{x+1}$ .

Find (A) $f^{\prime}(x)$ , (B) partition numbers for $f^{\prime}$ , (C) critical numbers of $f$ for $f(x)=\frac{2}{x+1}$ .

Determine where $f(x)$ is increasing, decreasing, and find local extrema for $f(x)=(x-3)e^{-9x}$ .

Determine where the function $f(x)$ is increasing, decreasing, and find local extrema for $f(x)=x^{3}+6 x-27$ .

Find where $f(x)=x^{3}+6x-27$ is increasing, decreasing, and its local extrema. Use interval notation for answers.

Find the limit as $x$ approaches 2 for $\frac{3 f(x)-15}{x-2}$ given $f(2)=5$ and $f'(2)=-5$ .

Bestimme die Tangentengleichung an $f(x)=x^{3}$ im Punkt $P(-1, f(-1))$ .

Find the limit: $\lim _{x \rightarrow-\infty}\left(-x^{4}+x^{2}+x+4\right)$ .

Find the limit: $\lim _{x \rightarrow-\infty}\left(x^{5}-4 x^{3}+4 x\right)$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{3 x}{x+1}$ .

Find the average rate of change of the function $y = 2x^2 + 1$ over the interval $[-1, 1]$ .

Bestimme die Tangentengleichung der Funktion $f(x) = 2x^2 - x + 1$ im Punkt $P(2, f(2))$ .

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

Find the volume of the solid formed by rotating the area between $y = x^{2} + 2$ and $y = 4 - x^{2}$ around $y = -3$ .

Find the limit as $x$ approaches 2 for $\frac{3 f(x)-15}{x-2}$ given $f(0)=3$ , $f(1)=4$ , $f(2)=5$ .

Find the average rate of change of $y=\frac{1}{x-1}$ over the interval $[-1,-\frac{1}{2}]$ .

Find the derivative of the function $y = \sqrt{2x + 4}$ .

Find the tangent line equation for the curve $y = -x^3 + x^2 + 3$ at the point $(1, 3)$ .

Find the sum $\sum_{n=1}^{\infty}(-1)^{n} \frac{n !}{n^{n} b_{1} b_{2} \ldots b_{n}}$ where $b_n \to \frac{1}{2}$ .

Find the rate of change of $y$ with respect to $x$ from the equation $x+\frac{1}{2} y=5$ .

Find $c$ in the interval $(1, 3)$ such that $h'(c) = \frac{h(3) - h(1)}{2}$ . Given $h(x)=x^{3}-x$ .

Find the slope of the tangent line to $f(x)=\sqrt{x}$ when $x=25$ .

Find the slope of the tangent line to $g(x)=x^{2}$ when $x=5$ .

Find the slope of the tangent line to $g(x)=\frac{1}{x}$ at $x=6$ using limits.

Find the height $x$ (in cm) that maximizes the cross-sectional area $A = x(28 - 2x)$ . Round to the nearest cm.

Determine the slope of the tangent line to $k(x)=x^{3}$ when $x=3$ .

Find the slope of the tangent line to $g(x)=\frac{1}{x}$ at the point where $x=6$ .

Bestimme die Normalengleichung der Funktion $f(x)=x^{2}-4$ am Punkt $P(1|-3)$ .

Find the limit as $x$ approaches 8 from the left of $\frac{3 p(x)-2 x}{15-3 x}$ , given $p(8)=3$ .

Determine the half-life of a radioactive element with decay function $A(t)=A_{0} e^{-0.0375 t}$ .

Find the relative minima and maxima of the polynomial function $y = x^{4} - x^{2} + 4$ .

A bullet is shot up at $+742 \mathrm{~m/s}$ . Find its velocity at $40 \mathrm{~s}$ , $80 \mathrm{~s}$ , and $120 \mathrm{~s}$ .

Given values for a function $h$ and its derivative $h^{\prime}$ , find $g^{\prime \prime}(e^{0.7})$ , $f^{\prime}(1)$ , tangent line for $h^{-1}$ at $x=-2$ , and $\frac{d y}{d x}$ at $(2,1)$ .

Find the volume of the solid formed by rotating the area between $y=\sin \left(x^{2}\right)$ , $[0, \sqrt{\pi}]$ , and the $x$ -axis around the $y$ -axis.

Verify if $y=(x+1)-\frac{1}{3} e^{x}$ satisfies $\frac{dy}{dx}=y-x$ .

Calculate the average rate of change of $k(x)=2 x^{2}-3$ from $x=1$ to $x=6$ .

Approximate the volume of a bird's egg shell with inner radius 5 mm and outer radius 5.2 mm using $V(r)=\frac{4}{3} \pi r^{3}$ . Volume is approximately $\square \mathrm{mm}^{3}$ . (Round to nearest hundredth.)

Solve the initial value problem: $\frac{d y}{d x}=y^{2}(1-x^{2})$ , with $y(0)=1$ .

Approximate the volume of a bird's egg shell with inner radius 7 mm and outer radius 7.3 mm using $V(r)=\frac{4}{3} \pi r^{3}$ . Volume is approximately $\square \mathrm{mm}^{3}$ .

Find $\frac{d y}{d x}$ for the equation $3 x - \tan y = 4$ in terms of $y$ . Options: (A) $3 \sin^2 y$ , (B) $3 \cos^2 y$ , (C) $3 \cos y \cot y$ , (D) $\frac{3}{1 + 9 y^2}$ .

Approximate the shell volume of an egg with inner radius 8 mm and outer radius 8.4 mm using $V(r)=\frac{4}{3} \pi r^{3}$ . The volume is approximately $\square \mathrm{mm}^{3}$ .

Approximate the volume of a bird's egg shell with inner radius 8 mm and outer radius 8.2 mm using $V(r)=\frac{4}{3} \pi r^{3}$ . Volume: $\square \mathrm{mm}^{3}$ .