Calculus

Problem 24801

Find the series for $f(x)=\frac{\cos x}{x^{7}}$ , its derivative, and what function the derivative represents.

See Solution

Problem 24802

Evaluate the integral: $\int \frac{x^{4}}{\left(6-x^{5}\right)^{7}} d x$ with $\mathrm{u}>0$ for $\ln \mathrm{u}$ .

See Solution

Problem 24803

Differentiate $g(t)=\frac{t-\sqrt{t}}{t^{1/3}}$ and find $g'(t)$ .

See Solution

Problem 24804

Find the volume of a solid with a triangular base at $(0,0),(9,0),(0,9)$ and semicircular cross sections. Set up the integral.

See Solution

Problem 24805

Find the dimensions of a square-based box with volume $1000 \mathrm{~cm}^{3}$ that minimizes material use, with lengths $\geq 2 \mathrm{~cm}$ .

See Solution

Problem 24806

Sketch a function $f$ with: $f(0)=0$ , $f'(-2)=f'(1)=f'(9)=0$ , $\lim_{x \to \infty} f(x)=0$ , $\lim_{x \to 6} f(x)=-\infty$ .

See Solution

Problem 24807

Find the volume of a solid with a triangular base at vertices $(0,0),(12,0)$ , $(0,12)$ and semicircular cross sections. Set up the integral for volume.

See Solution

Problem 24808

Evaluate the integral: $\int \frac{(\sqrt{x}+1)^{4}}{2 \sqrt{x}} d x$ using a change of variables.

See Solution

Problem 24809

Find the marginal profit for the function $P(x)=1.75x-\frac{x^{2}}{25000}-2100$ at $x=15000$ .

See Solution

Problem 24810

Evaluate the integral: $\int 5 \sin x(6+5 \cos x)^{4} d x$

See Solution

Problem 24811

Find the derivative of the function $(\ln (x))^{\prime}$ .

See Solution

Problem 24812

Differentiate: $y=7 x^{2} \sin x \tan x$ , find $y'$ .

See Solution

Problem 24813

Find the volume of the solid formed by revolving the area bounded by $y=3x$ , $y=0$ , and $x=2$ around the $x$ -axis. Set up the integral for volume.

See Solution

Problem 24814

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

See Solution

Problem 24815

Approximate $\int_{-2}^{10}(e^{2} x^{2}) dx$ using 4 equal subintervals and the Trapezoidal sum method.

See Solution

Problem 24816

Find the horizontal asymptote of $y=\frac{-3 x^{3}-4 x+12}{3 x^{3}-2 x+12}$ .

See Solution

Problem 24817

Calculate the integral: $\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x$

See Solution

Problem 24818

Differentiate $f(\theta)=\frac{\sec \theta}{3+\sec \theta}$ and find $f^{\prime}(\theta)$ .

See Solution

Problem 24819

Find the integral using substitution: $\int \tan 2 x \sec ^{2} 2 x \, dx$

See Solution

Problem 24820

Find the derivative of the function $f(x)=(\ln (\csc x))^{\prime}$ .

See Solution

Problem 24821

Find the volume using the shell method for the region bounded by $y=\sqrt{x}$ , $y=0$ , and $y=\frac{x-11}{10}$ around the $x$ -axis. The volume is $\square$ .

See Solution

Problem 24822

Differentiate the function $(\ln^{4}(x))$ .

See Solution

Problem 24823

Differentiate the function $f(x)=3 \sqrt{x} \sin x$ . Find $f^{\prime}(x)$ .

See Solution

Problem 24824

Evaluate the integral from -2 to 5 of the function $x^{2}-3x-10$ .

See Solution

Problem 24825

Calculate the integral $\int_{0}^{1}(x^{2}-4 x+6) dx=\square$

See Solution

Problem 24826

A rural town started with 420 people. The growth rate is $P'(t) = 45(1+\sqrt{t})$ . Find: a. Population after 15 years. b. $P(t)$ for $t \geq 0$ .

See Solution

Problem 24827

Find the volume using the shell method for the region between $y=\sqrt{x}$ , $y=0$ , and $y=\frac{x-11}{10}$ around the $x$ -axis.

See Solution

Problem 24828

State which theorem applies: MVT, IVT, RT, or EVT for each scenario.
a) $f(x)$ is continuous on $[-1, 7]$ and differentiable on $(-1, 7)$ . $f^{\prime}(c)=\frac{f(7)-f(1)}{7-1}$ .
MVT
b) Mr. Clas starts at 3.5 mph and ends at 5.8 mph. He must have run 4.2 mph at some point.
IVT
c) Construct a box from a $14 \times 30$ inch cardboard by cutting squares from corners to maximize volume.
EVT
d) Santa speeds to $800 \mathrm{mph}$ , slows down, and stops after 3 mins, passing $800 \mathrm{mph}$ again.
RT

See Solution

Problem 24829

Let $f(x) = x^3 - 9x^2 + 24x$ on $[1, 6]$ . Find relative extrema, inflection points, increasing/decreasing & concavity intervals, and global extrema.

See Solution

Problem 24830

Find the derivative of $\left(\sin ^{3}(x)\right)^{\prime}$ .

See Solution

Problem 24831

Find the distance traveled by a particle from $t=2$ to $t=6$ given $v(t)=21 t^{2}+8 t$ .

See Solution

Problem 24832

Find the min and max values of $f(t)=t \sqrt{9-t^{2}}$ on the interval $[-1,3]$ .

See Solution

Problem 24833

Find the distance traveled by a particle from $t=2$ to $t=6$ given $v(t)=21 t^{2}+8 t$ . Answer in miles.

See Solution

Problem 24834

Determine the horizontal asymptote for the function $y=\frac{8 x^{3}-9 x+7}{-6 x^{6}+9 x+7}$ .

See Solution

Problem 24835

Eucalyptus tree growth: $0.7 + \frac{4}{(t+2)^{3}}$ ft/year. Find growth in year 2 and year 3.

See Solution

Problem 24836

Find volumes using the shell method for the region defined by $X = \frac{y^4}{4} - \frac{y^2}{2}$ and $X = \frac{y^2}{2}$ , revolving around:
(a) the $x$ -axis, (b) $y=2$ , (c) $y=6$ , (d) $y=-\frac{3}{8}$ . Volume is $\square$ cubic units.

See Solution

Problem 24837

A coffee mug cools from 210°F to $65 + 145 e^{-1.9 t}$ °F after $t$ hours. Find temp after (a) 15 min, (b) 1 hour.

See Solution

Problem 24838

Find the distance traveled by the particle from $t=2$ to $t=5$ given $v(t)=15 t^{2}+14 t$ . Answer in miles.

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Problem 24839

Find the critical numbers of the function $g(y)=\frac{y-5}{y^{2}-3y+15}$ . Enter answers as a comma-separated list or DNE.

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Problem 24840

Find the displacement of a ball with velocity $v(t)=-32t+101$ ft/s from $t=0$ to $t=3$ seconds. Displacement is $\square$ feet.

See Solution

Problem 24841

Find the derivative of the function $f(x)=\left(\tan \left(\ln \left(e^{x}+5\right)\right)\right)$ .

See Solution

Problem 24842

Find $y$ if $y=\left(\int_{0}^{x}\left(t^{3}+1\right)^{10} dt\right)^{3}$ .

See Solution

Problem 24843

Find the displacement of a ball with velocity $v(t)=-32 t+101$ from $t=0$ to $t=3$ . Then find its position at $t=3$ .

See Solution

Problem 24844

Calculate the area between the curve $f(x)=\sin x$ and the $x$ -axis from $x=-\pi / 3$ to $x=3 \pi / 4$ .

See Solution

Problem 24845

Find the critical numbers of the function $f(x)=x^{3}+12 x^{2}-27 x$ . Enter answers as a comma-separated list or DNE.

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Problem 24846

Estimate the distance traveled in cm using left and right endpoint values from the velocity data over 10 sec intervals.

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Problem 24847

Find the volume using the shell method for the region between $x = \frac{y^4}{16} - \frac{y^2}{4}$ and $x = \frac{3y^2}{4}$ , revolving around:
1. The $x$ -axis
2. The line $y=4$
3. The line $y=8$
4. The line $y=-\frac{5}{8}$
Volume = $\square$ cubic units (exact, use $\pi$ as needed).

See Solution

Problem 24848

Estimate the distance traveled in 10 seconds using left and right endpoint values for velocity data. Left: $147 \mathrm{~cm}$ , Right: $\square \mathrm{cm}$ .

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Problem 24849

Find $y$ given the integral $y=\int_{\tan x}^{0} \frac{d t}{1+t^{2}}$ .

See Solution

Problem 24850

Find the derivative of the function: $f^{\prime}(x)=\left(\frac{5}{3-(1-\sec x)^{3}}\right)^{\prime}$ .

See Solution

Problem 24851

Find the distance traveled to the right by a particle with position $s(t)=3t-2t^{2}$ at $\mathrm{t}=4$ .

See Solution

Problem 24852

Given the function $v(t)=t^{3}-8 t^{2}+15 t$ for $t \in [0,7]$ , determine:
a. When is the motion positive/negative? b. What is the displacement over the interval? c. What is the distance traveled over the interval?
Displacement: $\square \mathrm{m}$ .

See Solution

Problem 24853

Find the volume of the solid formed by revolving the region bounded by $y=\frac{1}{x^{\frac{1}{3}}}$ , $x=\frac{1}{216}$ , and $y=1$ around the x-axis using the washer and shell methods. Complete the integral for the washer method: $\int_{\frac{1}{216}}^{\square}$ .

See Solution

Problem 24854

Given $f^{\prime}(x)$ values in the table, which statements are true for $-2<x<6$ ? (A) Concave up, (B) 2 inflection points, (C) Increasing, (D) No critical points, (E) 2 relative extrema.

See Solution

Problem 24855

Find where the velocity of the particle is positive for $s(t)=3 t-2 t^{2}$ , $t \geq 0$ .

See Solution

Problem 24856

Given the function $v(t)=t^{3}-9 t^{2}+20 t$ on $[0,7]$ , find when motion is positive/negative, displacement, and distance.

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Problem 24857

Alex needs to build a fence for 80 sq ft area, costing \$2/ft for wood and \$1/ft for wire mesh. Find cost-minimizing dimensions.

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Problem 24858

Estude a série $\sum_{n=1}^{\infty} \frac{2}{7^{n}}$ e calcule a soma se convergir.

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Problem 24859

Graph $y=x^{4}-19x^{2}$ : find domain, symmetries, $y'$ , $y''$ , critical points, increasing/decreasing intervals, inflection points, concavity, asymptotes, and absolute extremes. Domain: $\square$ (in interval notation).

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Problem 24860

Differentiate $h(t)=\frac{7}{2} t^{2}-8 t+5$ with respect to $t$ .

See Solution

Problem 24861

Find the value of $C(t)$ for $t = 3$ using the formula $C(t) = 0.06\left(1 - e^{-0.2 \times 3}\right)$ .

See Solution

Problem 24862

Determine the largest interval where $f(x)=\frac{1}{x^{2}+1}$ is increasing. Options: $(-\infty, 1)$ , $(0, \infty)$ , $(-\infty, 0)$ , $(1, \infty)$ .

See Solution

Problem 24863

Find $\int_{0}^{4} f(x) d x$ given $f(-x)=f(x)$ , $\int_{0}^{2} f(x) d x=C$ , and $\int_{-4}^{-2} f(x) d x=D$ .

See Solution

Problem 24864

Determine for which functions does $\lim _{x \rightarrow \infty} f(x)=0$ : I. $f(x)=\frac{\ln x}{x^{99}}$ , II. $f(x)=\frac{e^{x}}{\ln x}$ , III. $f(x)=\frac{x^{99}}{e^{x}}$ .

See Solution

Problem 24865

If $f x^{2} y^{3}=4 / 27$ and $\frac{dy}{dt}=1/2$ , find $\frac{dx}{dt}$ when $x=2$ .

See Solution

Problem 24866

Find the inflection points of $f(x)=6 e^{-x^{2}}$ . Choose from: $(-0.71,3.64)$ , $(0.71,3.64)$ , $(-0.71,3.16)$ , $(0.71,3.16)$ , $(-0.57,3.16)$ , $(0.57,3.16)$ , $(-0.57,3.64)$ , $(0.57,3.64)$ .

See Solution

Problem 24867

Given $x^{2} y^{3}=4 / 27$ and $\frac{dy}{dt}=1/2$ , find $\frac{dx}{dt}$ when $x=2$ .

See Solution

Problem 24868

Evaluate the third derivative of $x^{8}$ at $x=-1$ .

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Problem 24869

Find the rate of change of air volume if an adult breathes $37 \mathrm{~L}$ every 5 minutes.

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Problem 24870

A teenager's heart pumps $7200 \mathrm{~L}$ of blood in 24 hours. Find the rate of change of blood volume.
A hummingbird flaps its wings 1800 times in 30 seconds. Find the rate of change of wing flaps.

See Solution

Problem 24871

Find the derivatives of the function $f(x)=5 \sqrt{x^{5}}$ : (a) $f^{\prime}(x)$ , (b) $f^{\prime \prime}(x)$ , (c) $f^{\prime \prime \prime}(x)$ , (d) $f^{(4)}(x)$ .

See Solution

Problem 24872

Use implicit differentiation on the equation $x p^{5}=49$ to find $\frac{d p}{d x}$ .

See Solution

Problem 24873

Differentiate $f(x)=5(2-x^{2})^{3}$ .

See Solution

Problem 24874

Differentiate $f(x)=a x^{21}$ with respect to $x$ , where $a$ is a constant.

See Solution

Problem 24875

Differentiate $g(N)=r N(1-\frac{N}{K})$ with respect to $N$ , where $K$ and $r$ are positive constants.

See Solution

Problem 24876

Find $\frac{d p}{d x}$ for the demand equation $x=\sqrt{2000-p^{2}}$ at $p=40$ .

See Solution

Problem 24877

Find $\frac{d p}{d x}$ when $p=40$ for the demand equation $x=\sqrt{2000-p^{2}}$ . Interpret your answer.

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Problem 24878

Find the area between the curves $y=x^{2}-x$ and $y=2x$ over the interval $[-2,3]$ .

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Problem 24879

Find the derivative of $f(x)=x^{3}$ at $x=-6$ . What is $\left.\frac{d f}{d x}\right|_{x=-6}=\square$ ?

See Solution

Problem 24880

Find the time $t$ (0 ≤ $t$ ≤ 50) when the forest value $V(t)=444 \sqrt{t}-37 t$ is maximized. $t=$

See Solution

Problem 24881

A company produces flash drives at a cost of \$7 each and \$80 daily fixed costs.
(a) Find the average cost function $A C(x)=\frac{C(x)}{x}$ .
(b) Find the marginal average cost function $M A C(x)$ .
(c) Evaluate $M A C(x)$ at $x=22$ and interpret the result.

See Solution

Problem 24882

Find intervals where the function $f(x)=x^{3}+39 x^{2}-81 x$ is increasing and concave down.

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Problem 24883

Find intervals where the function $f(x)=2 x^{3}-6 x^{2}$ is decreasing and concave up.

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Problem 24884

Find intervals where the function $f(x)=4 x^{3}-36 x^{2}+60 x$ is increasing and concave down.

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Problem 24885

Find the limit of $y=\frac{-2 x^{2}+5}{3 x^{3}-1}$ as $x$ approaches infinity.

See Solution

Problem 24886

Find the limit of $y=\frac{-2 x^{2}+5}{3 x^{3}-1}$ as $x \rightarrow \infty$ . What does it approach?

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Problem 24887

Find the derivative of the Phillips curve $y=9.638 x^{-1.394}-0.900$ at $x=2\%$ and $x=7\%$ . Interpret results.

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Problem 24888

Find the derivatives of the function $f(x)=x^{4}-2 x^{3}-9 x^{2}+4 x-9$ : (a) $f^{\prime}(x)$ , (b) $f^{\prime \prime}(x)$ , (c) $f^{\prime \prime \prime}(x)$ , (d) $f^{(4)}(x)$ .

See Solution

Problem 24889

Find the tangent line approximation for $f(4.15)$ given $f(4)=5$ and $f'(4)=-10$ .

See Solution

Problem 24890

Find the derivative of the Phillips curve $y=9.638 x^{-1.394}-0.900$ at $x=2\%$ and $x=7\%$ . Round to two decimals. Interpret results.

See Solution

Problem 24891

Find points on $f(x) = x^3 - 13x^2 + 47x - 25$ at $x=0$ and $x=8$ , then find $c$ satisfying MVT on $[0, 8]$ .

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Problem 24892

A study shows that a smoker's chance of quitting increases with education. Use $f(x)=0.831 x^{2}-18.1 x+137.3$ for $10 \leq x \leq 16$ .
(a) Find $f(12)$ and interpret it. A high school graduate has a $\%$ chance of quitting.
Also, find $f^{\prime}(12)$ and interpret it. The quitting chance increases $\%$ per year of education for high school graduates.

See Solution

Problem 24893

Plot the line segment between $f(0)$ and $f(8)$ for $f(x)=x^{3}-13 x^{2}+47 x-25$ . Find $c$ satisfying $f'(c)=\frac{f(8)-f(0)}{8-0}$ .

See Solution

Problem 24894

Plot the line segment between $f(-1)$ and $f(7)$ , then find all $c$ satisfying the Mean Value Theorem on $[-1, 7]$ .

See Solution

Problem 24895

Find the average value of $f(x)$ on $[4,10]$ if $\int_{4}^{10} f(x) dx=5$ .

See Solution

Problem 24896

Find the absolute max and min of $f(x)=x^{4/3}$ on $[-1, 8]$ and their $x$ locations.

See Solution

Problem 24897

Differentiate $y=7^{4x+2}$ with respect to $x$ .

See Solution

Problem 24898

Differentiate $y=\ln(10-6x^{3})$ with respect to $x$ .

See Solution

Problem 24899

Evaluate the integral $I = \int_{0}^{\frac{\pi}{4}} \frac{\sin \theta}{\cos ^{4} \theta} d \theta$ .

See Solution

Problem 24900

Plot the line segment on $f(x)=x^{3}-7x^{2}+4x+12$ between $x=3$ and $x=6$ . Find values of $c$ for the Mean Value Theorem on $[3,6]$ .

See Solution

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Calculus

Problem 24801

Find the series for f(x)=cos⁡xx7f(x)=\frac{\cos x}{x^{7}}f(x)=x7cosx​, its derivative, and what function the derivative represents.

Problem 24802

Evaluate the integral: ∫x4(6−x5)7dx\int \frac{x^{4}}{\left(6-x^{5}\right)^{7}} d x∫(6−x5)7x4​dx with u>0\mathrm{u}>0u>0 for ln⁡u\ln \mathrm{u}lnu.

Problem 24803

Differentiate g(t)=t−tt1/3g(t)=\frac{t-\sqrt{t}}{t^{1/3}}g(t)=t1/3t−t​​ and find g′(t)g'(t)g′(t).

Problem 24804

Find the volume of a solid with a triangular base at (0,0),(9,0),(0,9)(0,0),(9,0),(0,9)(0,0),(9,0),(0,9) and semicircular cross sections. Set up the integral.

Problem 24805

Find the dimensions of a square-based box with volume 1000 cm31000 \mathrm{~cm}^{3}1000 cm3 that minimizes material use, with lengths ≥2 cm\geq 2 \mathrm{~cm}≥2 cm.

Problem 24806

Sketch a function fff with: f(0)=0f(0)=0f(0)=0, f′(−2)=f′(1)=f′(9)=0f'(-2)=f'(1)=f'(9)=0f′(−2)=f′(1)=f′(9)=0, lim⁡x→∞f(x)=0\lim_{x \to \infty} f(x)=0limx→∞​f(x)=0, lim⁡x→6f(x)=−∞\lim_{x \to 6} f(x)=-\inftylimx→6​f(x)=−∞.

Problem 24807

Find the volume of a solid with a triangular base at vertices (0,0),(12,0)(0,0),(12,0)(0,0),(12,0), (0,12)(0,12)(0,12) and semicircular cross sections. Set up the integral for volume.

Problem 24808

Evaluate the integral: ∫(x+1)42xdx\int \frac{(\sqrt{x}+1)^{4}}{2 \sqrt{x}} d x∫2x​(x​+1)4​dx using a change of variables.

Problem 24809

Find the marginal profit for the function P(x)=1.75x−x225000−2100P(x)=1.75x-\frac{x^{2}}{25000}-2100P(x)=1.75x−25000x2​−2100 at x=15000x=15000x=15000.

Problem 24810

Evaluate the integral: ∫5sin⁡x(6+5cos⁡x)4dx\int 5 \sin x(6+5 \cos x)^{4} d x∫5sinx(6+5cosx)4dx

Problem 24811

Find the derivative of the function (ln⁡(x))′(\ln (x))^{\prime}(ln(x))′.

Problem 24812

Differentiate: y=7x2sin⁡xtan⁡xy=7 x^{2} \sin x \tan xy=7x2sinxtanx, find y′y'y′.

Problem 24813

Find the volume of the solid formed by revolving the area bounded by y=3xy=3xy=3x, y=0y=0y=0, and x=2x=2x=2 around the xxx-axis. Set up the integral for volume.

Problem 24814

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

Problem 24815

Approximate ∫−210(e2x2)dx\int_{-2}^{10}(e^{2} x^{2}) dx∫−210​(e2x2)dx using 4 equal subintervals and the Trapezoidal sum method.

Problem 24816

Find the horizontal asymptote of y=−3x3−4x+123x3−2x+12y=\frac{-3 x^{3}-4 x+12}{3 x^{3}-2 x+12}y=3x3−2x+12−3x3−4x+12​.

Problem 24817

Calculate the integral: ∫ex−e−xex+e−xdx\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x∫ex+e−xex−e−x​dx

Problem 24818

Differentiate f(θ)=sec⁡θ3+sec⁡θf(\theta)=\frac{\sec \theta}{3+\sec \theta}f(θ)=3+secθsecθ​ and find f′(θ)f^{\prime}(\theta)f′(θ).

Problem 24819

Find the integral using substitution: ∫tan⁡2xsec⁡22x dx\int \tan 2 x \sec ^{2} 2 x \, dx∫tan2xsec22xdx

Problem 24820

Find the derivative of the function f(x)=(ln⁡(csc⁡x))′f(x)=(\ln (\csc x))^{\prime}f(x)=(ln(cscx))′.

Problem 24821

Find the volume using the shell method for the region bounded by y=xy=\sqrt{x}y=x​, y=0y=0y=0, and y=x−1110y=\frac{x-11}{10}y=10x−11​ around the xxx-axis. The volume is □\square□.

Problem 24822

Differentiate the function (ln⁡4(x))(\ln^{4}(x))(ln4(x)).

Problem 24823

Differentiate the function f(x)=3xsin⁡xf(x)=3 \sqrt{x} \sin xf(x)=3x​sinx. Find f′(x)f^{\prime}(x)f′(x).

Problem 24824

Evaluate the integral from -2 to 5 of the function x2−3x−10x^{2}-3x-10x2−3x−10.

Problem 24825

Calculate the integral ∫01(x2−4x+6)dx=□\int_{0}^{1}(x^{2}-4 x+6) dx=\square∫01​(x2−4x+6)dx=□

Problem 24826

A rural town started with 420 people. The growth rate is P′(t)=45(1+t)P'(t) = 45(1+\sqrt{t})P′(t)=45(1+t​). Find: a. Population after 15 years. b. P(t)P(t)P(t) for t≥0t \geq 0t≥0.

Problem 24827

Find the volume using the shell method for the region between y=xy=\sqrt{x}y=x​, y=0y=0y=0, and y=x−1110y=\frac{x-11}{10}y=10x−11​ around the xxx-axis.

Problem 24828

Problem 24829

Let f(x)=x3−9x2+24xf(x) = x^3 - 9x^2 + 24xf(x)=x3−9x2+24x on [1,6][1, 6][1,6]. Find relative extrema, inflection points, increasing/decreasing & concavity intervals, and global extrema.

Problem 24830

Find the derivative of (sin⁡3(x))′\left(\sin ^{3}(x)\right)^{\prime}(sin3(x))′.

Problem 24831

Find the distance traveled by a particle from t=2t=2t=2 to t=6t=6t=6 given v(t)=21t2+8tv(t)=21 t^{2}+8 tv(t)=21t2+8t.

Problem 24832

Find the min and max values of f(t)=t9−t2f(t)=t \sqrt{9-t^{2}}f(t)=t9−t2​ on the interval [−1,3][-1,3][−1,3].

Problem 24833

Find the distance traveled by a particle from t=2t=2t=2 to t=6t=6t=6 given v(t)=21t2+8tv(t)=21 t^{2}+8 tv(t)=21t2+8t. Answer in miles.

Problem 24834

Determine the horizontal asymptote for the function y=8x3−9x+7−6x6+9x+7y=\frac{8 x^{3}-9 x+7}{-6 x^{6}+9 x+7}y=−6x6+9x+78x3−9x+7​.

Problem 24835

Eucalyptus tree growth: 0.7+4(t+2)30.7 + \frac{4}{(t+2)^{3}}0.7+(t+2)34​ ft/year. Find growth in year 2 and year 3.

Problem 24836

Find volumes using the shell method for the region defined by X=y44−y22X = \frac{y^4}{4} - \frac{y^2}{2}X=4y4​−2y2​ and X=y22X = \frac{y^2}{2}X=2y2​, revolving around:(a) the xxx-axis, (b) y=2y=2y=2, (c) y=6y=6y=6, (d) y=−38y=-\frac{3}{8}y=−83​. Volume is □\square□ cubic units.

Problem 24837

A coffee mug cools from 210°F to 65+145e−1.9t65 + 145 e^{-1.9 t}65+145e−1.9t°F after ttt hours. Find temp after (a) 15 min, (b) 1 hour.

Problem 24838

Find the distance traveled by the particle from t=2t=2t=2 to t=5t=5t=5 given v(t)=15t2+14tv(t)=15 t^{2}+14 tv(t)=15t2+14t. Answer in miles.

Problem 24839

Find the critical numbers of the function g(y)=y−5y2−3y+15g(y)=\frac{y-5}{y^{2}-3y+15}g(y)=y2−3y+15y−5​. Enter answers as a comma-separated list or DNE.

Problem 24840

Find the displacement of a ball with velocity v(t)=−32t+101v(t)=-32t+101v(t)=−32t+101 ft/s from t=0t=0t=0 to t=3t=3t=3 seconds. Displacement is □\square□ feet.

Find the series for $f(x)=\frac{\cos x}{x^{7}}$ , its derivative, and what function the derivative represents.

Evaluate the integral: $\int \frac{x^{4}}{\left(6-x^{5}\right)^{7}} d x$ with $\mathrm{u}>0$ for $\ln \mathrm{u}$ .

Differentiate $g(t)=\frac{t-\sqrt{t}}{t^{1/3}}$ and find $g'(t)$ .

Find the volume of a solid with a triangular base at $(0,0),(9,0),(0,9)$ and semicircular cross sections. Set up the integral.

Find the dimensions of a square-based box with volume $1000 \mathrm{~cm}^{3}$ that minimizes material use, with lengths $\geq 2 \mathrm{~cm}$ .

Sketch a function $f$ with: $f(0)=0$ , $f'(-2)=f'(1)=f'(9)=0$ , $\lim_{x \to \infty} f(x)=0$ , $\lim_{x \to 6} f(x)=-\infty$ .

Find the volume of a solid with a triangular base at vertices $(0,0),(12,0)$ , $(0,12)$ and semicircular cross sections. Set up the integral for volume.

Evaluate the integral: $\int \frac{(\sqrt{x}+1)^{4}}{2 \sqrt{x}} d x$ using a change of variables.

Find the marginal profit for the function $P(x)=1.75x-\frac{x^{2}}{25000}-2100$ at $x=15000$ .

Evaluate the integral: $\int 5 \sin x(6+5 \cos x)^{4} d x$

Find the derivative of the function $(\ln (x))^{\prime}$ .

Differentiate: $y=7 x^{2} \sin x \tan x$ , find $y'$ .

Find the volume of the solid formed by revolving the area bounded by $y=3x$ , $y=0$ , and $x=2$ around the $x$ -axis. Set up the integral for volume.

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

Approximate $\int_{-2}^{10}(e^{2} x^{2}) dx$ using 4 equal subintervals and the Trapezoidal sum method.

Find the horizontal asymptote of $y=\frac{-3 x^{3}-4 x+12}{3 x^{3}-2 x+12}$ .

Calculate the integral: $\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x$

Differentiate $f(\theta)=\frac{\sec \theta}{3+\sec \theta}$ and find $f^{\prime}(\theta)$ .

Find the integral using substitution: $\int \tan 2 x \sec ^{2} 2 x \, dx$

Find the derivative of the function $f(x)=(\ln (\csc x))^{\prime}$ .

Find the volume using the shell method for the region bounded by $y=\sqrt{x}$ , $y=0$ , and $y=\frac{x-11}{10}$ around the $x$ -axis. The volume is $\square$ .

Differentiate the function $(\ln^{4}(x))$ .

Differentiate the function $f(x)=3 \sqrt{x} \sin x$ . Find $f^{\prime}(x)$ .

Evaluate the integral from -2 to 5 of the function $x^{2}-3x-10$ .

Calculate the integral $\int_{0}^{1}(x^{2}-4 x+6) dx=\square$

A rural town started with 420 people. The growth rate is $P'(t) = 45(1+\sqrt{t})$ . Find: a. Population after 15 years. b. $P(t)$ for $t \geq 0$ .

Find the volume using the shell method for the region between $y=\sqrt{x}$ , $y=0$ , and $y=\frac{x-11}{10}$ around the $x$ -axis.

Let $f(x) = x^3 - 9x^2 + 24x$ on $[1, 6]$ . Find relative extrema, inflection points, increasing/decreasing & concavity intervals, and global extrema.

Find the derivative of $\left(\sin ^{3}(x)\right)^{\prime}$ .

Find the distance traveled by a particle from $t=2$ to $t=6$ given $v(t)=21 t^{2}+8 t$ .

Find the min and max values of $f(t)=t \sqrt{9-t^{2}}$ on the interval $[-1,3]$ .

Find the distance traveled by a particle from $t=2$ to $t=6$ given $v(t)=21 t^{2}+8 t$ . Answer in miles.

Determine the horizontal asymptote for the function $y=\frac{8 x^{3}-9 x+7}{-6 x^{6}+9 x+7}$ .

Eucalyptus tree growth: $0.7 + \frac{4}{(t+2)^{3}}$ ft/year. Find growth in year 2 and year 3.

Find volumes using the shell method for the region defined by $X = \frac{y^4}{4} - \frac{y^2}{2}$ and $X = \frac{y^2}{2}$ , revolving around:
(a) the $x$ -axis, (b) $y=2$ , (c) $y=6$ , (d) $y=-\frac{3}{8}$ . Volume is $\square$ cubic units.

A coffee mug cools from 210°F to $65 + 145 e^{-1.9 t}$ °F after $t$ hours. Find temp after (a) 15 min, (b) 1 hour.

Find the distance traveled by the particle from $t=2$ to $t=5$ given $v(t)=15 t^{2}+14 t$ . Answer in miles.

Find the critical numbers of the function $g(y)=\frac{y-5}{y^{2}-3y+15}$ . Enter answers as a comma-separated list or DNE.

Find the displacement of a ball with velocity $v(t)=-32t+101$ ft/s from $t=0$ to $t=3$ seconds. Displacement is $\square$ feet.

Find the derivative of the function $f(x)=\left(\tan \left(\ln \left(e^{x}+5\right)\right)\right)$ .

Find $y$ if $y=\left(\int_{0}^{x}\left(t^{3}+1\right)^{10} dt\right)^{3}$ .

Find the displacement of a ball with velocity $v(t)=-32 t+101$ from $t=0$ to $t=3$ . Then find its position at $t=3$ .

Calculate the area between the curve $f(x)=\sin x$ and the $x$ -axis from $x=-\pi / 3$ to $x=3 \pi / 4$ .

Find the critical numbers of the function $f(x)=x^{3}+12 x^{2}-27 x$ . Enter answers as a comma-separated list or DNE.

Estimate the distance traveled in 10 seconds using left and right endpoint values for velocity data. Left: $147 \mathrm{~cm}$ , Right: $\square \mathrm{cm}$ .

Find $y$ given the integral $y=\int_{\tan x}^{0} \frac{d t}{1+t^{2}}$ .

Find the derivative of the function: $f^{\prime}(x)=\left(\frac{5}{3-(1-\sec x)^{3}}\right)^{\prime}$ .

Find the distance traveled to the right by a particle with position $s(t)=3t-2t^{2}$ at $\mathrm{t}=4$ .

Given $f^{\prime}(x)$ values in the table, which statements are true for $-2<x<6$ ? (A) Concave up, (B) 2 inflection points, (C) Increasing, (D) No critical points, (E) 2 relative extrema.

Find where the velocity of the particle is positive for $s(t)=3 t-2 t^{2}$ , $t \geq 0$ .

Given the function $v(t)=t^{3}-9 t^{2}+20 t$ on $[0,7]$ , find when motion is positive/negative, displacement, and distance.

Estude a série $\sum_{n=1}^{\infty} \frac{2}{7^{n}}$ e calcule a soma se convergir.

Graph $y=x^{4}-19x^{2}$ : find domain, symmetries, $y'$ , $y''$ , critical points, increasing/decreasing intervals, inflection points, concavity, asymptotes, and absolute extremes. Domain: $\square$ (in interval notation).

Differentiate $h(t)=\frac{7}{2} t^{2}-8 t+5$ with respect to $t$ .

Find the value of $C(t)$ for $t = 3$ using the formula $C(t) = 0.06\left(1 - e^{-0.2 \times 3}\right)$ .

Determine the largest interval where $f(x)=\frac{1}{x^{2}+1}$ is increasing. Options: $(-\infty, 1)$ , $(0, \infty)$ , $(-\infty, 0)$ , $(1, \infty)$ .

Find $\int_{0}^{4} f(x) d x$ given $f(-x)=f(x)$ , $\int_{0}^{2} f(x) d x=C$ , and $\int_{-4}^{-2} f(x) d x=D$ .

Determine for which functions does $\lim _{x \rightarrow \infty} f(x)=0$ : I. $f(x)=\frac{\ln x}{x^{99}}$ , II. $f(x)=\frac{e^{x}}{\ln x}$ , III. $f(x)=\frac{x^{99}}{e^{x}}$ .

If $f x^{2} y^{3}=4 / 27$ and $\frac{dy}{dt}=1/2$ , find $\frac{dx}{dt}$ when $x=2$ .

Given $x^{2} y^{3}=4 / 27$ and $\frac{dy}{dt}=1/2$ , find $\frac{dx}{dt}$ when $x=2$ .

Evaluate the third derivative of $x^{8}$ at $x=-1$ .

Find the rate of change of air volume if an adult breathes $37 \mathrm{~L}$ every 5 minutes.

A teenager's heart pumps $7200 \mathrm{~L}$ of blood in 24 hours. Find the rate of change of blood volume.
A hummingbird flaps its wings 1800 times in 30 seconds. Find the rate of change of wing flaps.

Find the derivatives of the function $f(x)=5 \sqrt{x^{5}}$ : (a) $f^{\prime}(x)$ , (b) $f^{\prime \prime}(x)$ , (c) $f^{\prime \prime \prime}(x)$ , (d) $f^{(4)}(x)$ .

Use implicit differentiation on the equation $x p^{5}=49$ to find $\frac{d p}{d x}$ .

Differentiate $f(x)=5(2-x^{2})^{3}$ .

Differentiate $f(x)=a x^{21}$ with respect to $x$ , where $a$ is a constant.