Calculus

Problem 15801

Find the derivative $g^{\prime}(1)$ for the function $g(t)=(t^{3}-2)^{7}$ .

See Solution

Problem 15802

Find the linearization $L(x)$ of $f(x)=\sqrt{4x+9}$ at $x=0$ . Show your work.

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Problem 15803

Find $p^{\prime}(5)$ for $p(x)=f(x) \cdot g(f(x))$ using values from $f$ and $g$ .

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Problem 15804

Find the tangent line slope to the ellipse $x^{2}+4y^{2}=16$ at the point $(4,0)$ .

See Solution

Problem 15805

Find the derivative $\frac{d y}{d x}$ for the equation $y^{2}=\frac{1}{2 x+5}$ .

See Solution

Problem 15806

Differentiate the function $\frac{4 t}{0.5+t^{2}}$ with respect to $t$ .

See Solution

Problem 15807

An isotope with a half-life of 65 days starts at 1 kg. How much remains after 120 and 240 days? Round to 3 decimals.

See Solution

Problem 15808

Differentiate the function $\frac{2-4 t^{2}}{(0.5+t^{2})^{2}}$ with respect to $t$ .

See Solution

Problem 15809

Find the slope of the tangent line to the ellipse $x^{2}+4y^{2}=16$ at the point $(4,0)$ .

See Solution

Problem 15810

Find $\frac{d^{2} y}{d x^{2}}$ for the circle $x^{2}+y^{2}=25$ at the point $(3,-4)$ .

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Problem 15811

Find the limit: $\lim _{h \rightarrow 0} \frac{\frac{1}{(2+h)}-\frac{1}{2}}{h}$ .

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Problem 15812

Find the limit as $x$ approaches 2: $\lim _{x \rightarrow 2} \frac{(x-2)^4}{x^{4}-16}$ .

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Problem 15813

Find when the drug concentration $C(t)=\frac{4 t}{0.5+t^{2}}$ is at its maximum. Round to two decimal places.

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Problem 15814

Differentiate using the product rule: a) i) $y=(x^{2}-3x)^{2}$ , ii) $y=(2x^{3}+x)^{2}$ , iii) $y=(-x^{4}+5x^{2})^{2}$ . b) Conjecture $\frac{d}{dx}([f(x)]^{2})$ . c) Verify by replacing $g(x)$ with $f(x)$ in the product rule. d) Compare derivatives from a) and c). What do you notice?

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Problem 15815

Find $f^{\prime}(1)$ if $f(x)=(x^{2}-3)^{4}$ .

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Problem 15816

Differentiate using the product rule: a) i) $y=(x^{2}-3x)^{2}$ , ii) $y=(2x^{3}+x)^{2}$ , iii) $y=(-x^{4}+5x^{2})^{2}$ . b) Conjecture $\frac{d}{dx}([f(x)]^{2})$ . c) Verify by replacing $g(x)$ with $f(x)$ in the product rule. d) Compare derivatives from part a) with those from part c).

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Problem 15817

An observer on a cliff sees a boat approaching the shore. If the telescope is $250^{\prime}$ high and the boat moves at $20 \mathrm{ft/sec}$ , find the angle change rate when the boat is $250^{\prime}$ from shore.

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Problem 15818

Find $\frac{d y}{d x}$ if $y=\left(\frac{x}{x+1}\right)^{5}$ . Choose from options (A) to (E).

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Problem 15819

Find the derivative of the function $f(x)=e^{1/x}$ , which is $f^{\prime}(x)=$ .

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Problem 15820

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

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Problem 15821

Find the limit of $f(x) = \frac{x-16}{\sqrt{x}-4}$ as $x$ approaches 16.

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Problem 15822

Tim threw a ball to point $B$ in the water, $10 \mathrm{~m}$ from point $C$ , $15 \mathrm{~m}$ from $A$ . Find $x$ where Elvis entered to minimize time.

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Problem 15823

Given $h(g(x))=x$ , find $g^{\prime}(7)$ using $h(3)=7$ , $h(7)=22$ , $h^{\prime}(3)=5$ , $h^{\prime}(7)=10$ . Choices: (A) $-1/10$ , (B) $1/10$ , (C) $1/5$ , (D) $7/5$ .

See Solution

Problem 15824

Find the limit: $\lim _{x \rightarrow 2} \frac{2 x^{2}-5 x+2}{5 x^{2}-7 x-6}$ .

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Problem 15825

Calculate the left and right Riemann sums for $f(x)=x+4$ on $[0,5]$ with $n=5$ .

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Problem 15826

Trouvez la dérivée de $y$ par rapport à $x$ pour l'équation $y^{7} \ln (y)-x^{4} \ln (x)=5$ . Réponse: $y^{\prime}=$

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Problem 15827

Find the function $f$ where $f^{\prime}(x)=\frac{5}{\sqrt{x}}$ and $f(9)=45$ . What is $f(x)$ ?

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Problem 15828

Evaluate the integral: $\int_{-5}^{1}(x-5) \, dx$ .

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Problem 15829

Calculate the left and right Riemann sums for $f(x)=\frac{2}{x}$ on $[1,5]$ with $n=4$ .

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Problem 15830

Differentiate the function $y=(4x^4-x+1)(-x^5+4)$ . Find $y'=\square$ .

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Problem 15831

Find the differential $d y$ for the function $y=\sin(2x^2)$ . Show your work.

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Problem 15832

Find the tangent line equation for $f(x)=x(1-2x)^{3}$ at $(1,-1)$ . Choose from options (A)-(E).

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Problem 15833

Find the derivative $g^{\prime}(x)$ for the function $g(x)=5 x e^{2 x}$ .

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Problem 15834

Find the antiderivative of $\frac{d R}{d t}=\frac{81}{t^{4}}$ with $R(3)=75$ . What is $R$ ?

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Problem 15835

Find $f^{\prime}(-1)$ for the function $f(x)=\sqrt{x^{4}+8}$ .

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Problem 15836

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{e^{x}}{7 x^{2}+1}$ .

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Problem 15837

Find intervals where $f(x)=x^{3}-18 x^{2}+11 x+2$ is concave up, concave down, and its inflection points.

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Problem 15838

Find the antiderivatives of $f(x)=22 x^{21}$ and verify by differentiating. $F(x)=\square$ .

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Problem 15839

Find the area between the curve $y=1-x^{2}$ , the $x$ -axis, and lines $x=-1$ and $x=2$ , treating below $x$ -axis as zero.

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Problem 15840

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{2 x+3}{3 x+2}$ .

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Problem 15841

Evaluate the integral: $\int_{0}^{1} 9 e^{x} dx$

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Problem 15842

Find $f^{\prime}(x)$ and the tangent line equation at $x=2$ for $f(x)=\frac{x-9}{6x-1}$ .

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Problem 15843

Find the second derivative of $y=-3 \cos (2 x)$ . Choices: (A) $-12 \cos (2 x)$ , (B) $12 \cos (2 x)$ , (C) $-3 \cos (2 x)$ , (D) $3 \cos (2 x)$ .

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Problem 15844

Find the tangent lines for $f(x)=\sqrt{x-1}+2$ at $x=2$ and $f(x)=x^{3}+2$ at $x=-1$ .

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Problem 15845

Deondra wants to invest for an account to reach \ $30,000 in 10 years at a continuous interest rate of$ 4.7\%$. How much?

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Problem 15846

Determine where the function $f(x)=2x^{4}+8x^{3}+25$ is increasing, decreasing, and find local extrema.

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Problem 15847

Find the derivative of $f(x)=(\ln (x))^{15}$ and evaluate it at $x=2$ : $f^{\prime}(2)=$

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Problem 15848

Given the function $f(x)=x^{3}-2 x^{2}$ on $[0,1]$ , does the Mean Value Theorem apply? Find the point(s) if it does.

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Problem 15849

Find the derivative of $y=\tan x-\cot x$ , i.e., compute $\frac{dy}{dx}$ .

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Problem 15850

Jeriel wants to invest to reach \$137,000 in 5 years at a 6.5% continuous interest rate. How much should he invest?

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Problem 15851

Find the instantaneous rate of change of $f(x)=2x^{3}-5x^{2}+8x+1$ at $x=1$ on $(0,3)$ . Choices: (A) 2, (B) 4, (C) 6, (D) 8.

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Problem 15852

Find $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\tan x - \tan\left(\frac{\pi}{4}\right)}{x - \frac{\pi}{4}}$ .

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Problem 15853

Determine where $f(x)$ is increasing, decreasing, and find local extrema for $f(x)=2 x^{4}+8 x^{3}+25$ .

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Problem 15854

Sketch $f(x)=2x^{2}+3$ on $[3,8]$ , find $\Delta x$ , grid points, and calculate left/right Riemann sums for $n=5$ .

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Problem 15855

Find the cost increase for producing 300 to 720 bikes using $C^{\prime}(x)=300-\frac{x}{3}$ . Evaluate the integral. The increase is \$ \square.

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Problem 15856

Transform the graphs of $f$ , find tangent line equations, and graph them for: 29. $(x-2)^{2}$ at 3; 30. $(x+2)^{2}$ at -1; 31. $\sqrt{x+1}-2$ at 0; 32. $\sqrt{x-1}+2$ at 2; 33. $x^{3}+2$ at -1; 34. $x^{3}-2$ at 1; 35. $-\frac{1}{x+3}$ at -2; 36. $-\frac{1}{x-2}$ at 3.

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Problem 15857

Given $f(x)=\sin ^{3}(x)$ , calculate $f^{\prime}(2)$ where $f^{\prime}(x)=3 \sin (x)^{2} \cos x$ .

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Problem 15858

Find points $(x, y)$ on the graph of $f(x)=\frac{x}{x+2}$ where the tangent line has slope $\frac{1}{2}$ .

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Problem 15859

Arianys invested \ $61,000 at$ 7 \frac{1}{8} \% $continuous interest. Yusuf invested \$61,000 at$ 7 \frac{5}{8} \%$ monthly. After 14 years, how much more does Yusuf have?

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Problem 15860

Find the critical points of the function $f(x)=45 x^{3}-3 x^{5}$ using calculus methods. Show your work.

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Problem 15861

Find the increase in total cost when production changes from $x = 30$ to $x = 90$ using $MC(x) = 4 + \frac{x^2}{1000}$ .

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Problem 15862

Explain how to use l'Hôpital's Rule for limits of the form $\frac{0}{0}$ . Choose the correct answer: A, B, C, or D.

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Problem 15863

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

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Problem 15864

Find the rate of area increase of an isosceles triangle with sides $20 \mathrm{~cm}$ and vertex angle $\frac{\pi}{6}$ rads, given the angle increases at 1 rads/min.

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Problem 15865

Find the antiderivatives of $f(x)=1$ . Choose the correct option for $F(x)$ .

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Problem 15866

Find $g^{\prime}(9)$ for $g(x)=\frac{f(x)}{\sqrt{x}}$ given $f(9)=18$ and $f^{\prime}(9)=7$ .

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Problem 15867

Given $f(1)=3$ , $f'(1)=-2$ , $g(1)=-3$ , $g'(1)=4$ , find $h'(1)$ for $h(x)=(2f(x)+3)(1+g(x))$ . Choices: A) -28 B) -16 C) 40 D) 44 E) 47

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Problem 15868

Find all antiderivatives of $f(x)=-8 \sec ^{2} x$ and verify by differentiating. Antiderivatives are $F(x)=\square$ .

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Problem 15869

Find $f'(x)$ , partition numbers for $f'$ , and critical numbers of $f$ where $f(x)=x^{3}-27x+3$ .

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Problem 15870

Find the increase in cost from producing 0 to 900 bikes using the marginal cost function $C^{\prime}(x)=600-\frac{x}{3}$ . Set up and evaluate the integral. The increase in cost is \$ \square.

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Problem 15871

Given the function $f(x)=\frac{4 e^{x}}{4 e^{x}+5}$ , find critical values, concavity intervals, and inflection points.

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Problem 15872

Find $f'(x)$ , critical numbers of $f$ , and partition numbers for $f'$ where $f(x)=x^{3}-27x+3$ .

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Problem 15873

Find the derivative of the function $f(x)=\cos^{3}(4x)$ . What is $f^{\prime}(x)$ ?

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Problem 15874

Find $\lim _{x \rightarrow 2 \pi} \frac{\sin(2 \pi)-\sin x}{x-2 \pi}$ .

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Problem 15875

Solve the initial value problem: $f'(x) = 6x - 5$ with $f(0) = 9$ .

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Problem 15876

A 6' tall person walks toward an 18' streetlight at 5 ft/sec. Find the rate of angle change when 9' away from the light.

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Problem 15877

Find the derivative of the function $y=\frac{1}{2} x^{4 / 5}-\frac{3}{x^{5}}$ . What is $\frac{d y}{d x}$ ?

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Problem 15878

Find $y^{\prime}$ for $y=\ln((x^{6}+6)^{3/2})$ . What is $y^{\prime}=\square$ ?

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Problem 15879

Find the x and y coordinates of the inflection points for $f(x)=x^{3}+24x^{2}$ . What are the inflection point(s)? A. $\square$ . B. No inflection points.

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Problem 15880

Find $f^{\prime}(x)$ for $f(x)=3 e^{x^{2}-7 x+5}$ at $x=0$ and where the tangent line is horizontal.

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Problem 15881

Find the indefinite integral: $\int\left(\sqrt[5]{x^{2}}-\sqrt{x^{7}}\right) d x$

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Problem 15882

Calculate total income over 3 years from the flow rate $f(t)=300 e^{0.05 t}$ . What is the total income earned? \$ \square

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Problem 15883

Find the derivative of each function using two methods: chain rule then simplify, or simplify then differentiate. a) $f(x)=(2 x)^{3}$ b) $g(x)=\left(-4 x^{2}\right)^{2}$ c) $p(x)=\sqrt{9 x^{2}}$ d) $f(x)=\left(16 x^{2}\right)^{\frac{3}{4}}$ e) $q(x)=(8 x)^{\frac{2}{3}}$

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Problem 15884

Find $f^{\prime}(x)$ for $f(x)=3 e^{x^{2}-7 x+5}$ , the tangent line at $x=0$ , and where it's horizontal.

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Problem 15885

Find all critical points of the function $f(x)=\frac{-2 x}{x+3}$ using calculus methods. Show your work.

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Problem 15886

Find the radius and interval of convergence for the series $\sum_{n=1}^{\infty} \frac{(n !)^{2} x^{n}}{2^{n}(2 n) !}$ . Check endpoints for absolute and conditional convergence.

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Problem 15887

Find the derivative of $f(x)=x^{\frac{3}{2}}$ and evaluate it at $x=4$ . What is $f^{\prime}(4)$ ?

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Problem 15888

Find the limit: $\lim _{h \rightarrow 0} \frac{(x+h)^{2}+4(x+h)-x^{2}-4 x}{h}$ .

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Problem 15889

Find $\frac{d \theta}{d t}$ from the equation $3 \sqrt{100} \frac{d \theta}{d t}=\frac{2}{200}$ .

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Problem 15890

Find the antiderivative of $6 x^{-4}+2 x^{-1}-1$ with $y(1)=1$ . What is $y(x)=\square$ ?

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Problem 15891

Find the value(s) of $c$ for $f^{\prime}(c)=\frac{f(2)-f(-3)}{2-(-3)}$ with $f(x)=x^{2}+2x+2$ .

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Problem 15892

Find the useful life of a photocopying machine and its total profit, given $C'(t)=\frac{1}{28} t^{5}$ and $R'(t)=5 t^{5} e^{-t^{6}}$ .

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Problem 15893

Evaluate the limit: $\lim _{x \rightarrow 0} \frac{3 \sin 9 x}{7 x}$ using l'Hôpital's Rule. What is the exact answer?

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Problem 15894

Find the limit: $\lim _{u \rightarrow \frac{\pi}{4}} \frac{7 \tan u-7 \cot u}{2 u-\frac{\pi}{2}}$ using IHôpital's Rule.

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Problem 15895

Evaluate the double integral: $\int_{-9}^{9} \int_{0}^{\pi / 2}(y+y^{2} \cos x) \, dx \, dy$ .

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Problem 15896

Find the machine's useful life in years and total profit, given $C'(t)=\frac{1}{28} t^{5}$ and $R'(t)=5 t^{5} e^{-t^{8}}$ .

See Solution

Problem 15897

Find $f^{\prime}(-1)$ for the piecewise function $f(x)=\{2x+5 \text{ if } x<-1, -x^{2}+6 \text{ if } x \geq -1\}$ .

See Solution

Problem 15898

Find the limit: $\lim _{x \rightarrow 0} \frac{3 \sin 9 x}{7 x}$ using l'Hôpital's Rule as needed.

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Problem 15899

Solve the equation $100 \sqrt{3} \frac{d \theta}{d t}=\frac{1}{100}$ .

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Problem 15900

Find the slope of the graph of $y=e^{\tan x}-2$ at the $x$ -axis crossing in $[0,1]$ . Options: (A) 0.606 (B) 2 (C) 2.242 (D) 2.961 (E) 3.747

See Solution

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Calculus

Problem 15801

Find the derivative g′(1)g^{\prime}(1)g′(1) for the function g(t)=(t3−2)7g(t)=(t^{3}-2)^{7}g(t)=(t3−2)7.

Problem 15802

Find the linearization L(x)L(x)L(x) of f(x)=4x+9f(x)=\sqrt{4x+9}f(x)=4x+9​ at x=0x=0x=0. Show your work.

Problem 15803

Find p′(5)p^{\prime}(5)p′(5) for p(x)=f(x)⋅g(f(x))p(x)=f(x) \cdot g(f(x))p(x)=f(x)⋅g(f(x)) using values from fff and ggg.

Problem 15804

Find the tangent line slope to the ellipse x2+4y2=16x^{2}+4y^{2}=16x2+4y2=16 at the point (4,0)(4,0)(4,0).

Problem 15805

Find the derivative dydx\frac{d y}{d x}dxdy​ for the equation y2=12x+5y^{2}=\frac{1}{2 x+5}y2=2x+51​.

Problem 15806

Differentiate the function 4t0.5+t2\frac{4 t}{0.5+t^{2}}0.5+t24t​ with respect to ttt.

Problem 15807

An isotope with a half-life of 65 days starts at 1 kg. How much remains after 120 and 240 days? Round to 3 decimals.

Problem 15808

Differentiate the function 2−4t2(0.5+t2)2\frac{2-4 t^{2}}{(0.5+t^{2})^{2}}(0.5+t2)22−4t2​ with respect to ttt.

Problem 15809

Find the slope of the tangent line to the ellipse x2+4y2=16x^{2}+4y^{2}=16x2+4y2=16 at the point (4,0)(4,0)(4,0).

Problem 15810

Find d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​ for the circle x2+y2=25x^{2}+y^{2}=25x2+y2=25 at the point (3,−4)(3,-4)(3,−4).

Problem 15811

Find the limit: lim⁡h→01(2+h)−12h\lim _{h \rightarrow 0} \frac{\frac{1}{(2+h)}-\frac{1}{2}}{h}limh→0​h(2+h)1​−21​​.

Problem 15812

Find the limit as xxx approaches 2: lim⁡x→2(x−2)4x4−16\lim _{x \rightarrow 2} \frac{(x-2)^4}{x^{4}-16}limx→2​x4−16(x−2)4​.

Problem 15813

Find when the drug concentration C(t)=4t0.5+t2C(t)=\frac{4 t}{0.5+t^{2}}C(t)=0.5+t24t​ is at its maximum. Round to two decimal places.

Problem 15814

Problem 15815

Find f′(1)f^{\prime}(1)f′(1) if f(x)=(x2−3)4f(x)=(x^{2}-3)^{4}f(x)=(x2−3)4.

Problem 15816

Problem 15817

An observer on a cliff sees a boat approaching the shore. If the telescope is 250′250^{\prime}250′ high and the boat moves at 20ft/sec20 \mathrm{ft/sec}20ft/sec, find the angle change rate when the boat is 250′250^{\prime}250′ from shore.

Problem 15818

Find dydx\frac{d y}{d x}dxdy​ if y=(xx+1)5y=\left(\frac{x}{x+1}\right)^{5}y=(x+1x​)5. Choose from options (A) to (E).

Problem 15819

Find the derivative of the function f(x)=e1/xf(x)=e^{1/x}f(x)=e1/x, which is f′(x)=f^{\prime}(x)=f′(x)=.

Problem 15820

Find the limit: lim⁡h→−2h3+8h+2\lim _{h \rightarrow-2} \frac{h^{3}+8}{h+2}limh→−2​h+2h3+8​.

Problem 15821

Find the limit of f(x)=x−16x−4f(x) = \frac{x-16}{\sqrt{x}-4}f(x)=x​−4x−16​ as xxx approaches 16.

Problem 15822

Tim threw a ball to point BBB in the water, 10 m10 \mathrm{~m}10 m from point CCC, 15 m15 \mathrm{~m}15 m from AAA. Find xxx where Elvis entered to minimize time.

Problem 15823

Given h(g(x))=xh(g(x))=xh(g(x))=x, find g′(7)g^{\prime}(7)g′(7) using h(3)=7h(3)=7h(3)=7, h(7)=22h(7)=22h(7)=22, h′(3)=5h^{\prime}(3)=5h′(3)=5, h′(7)=10h^{\prime}(7)=10h′(7)=10. Choices: (A) −1/10-1/10−1/10, (B) 1/101/101/10, (C) 1/51/51/5, (D) 7/57/57/5.

Problem 15824

Find the limit: lim⁡x→22x2−5x+25x2−7x−6\lim _{x \rightarrow 2} \frac{2 x^{2}-5 x+2}{5 x^{2}-7 x-6}limx→2​5x2−7x−62x2−5x+2​.

Problem 15825

Calculate the left and right Riemann sums for f(x)=x+4f(x)=x+4f(x)=x+4 on [0,5][0,5][0,5] with n=5n=5n=5.

Problem 15826

Trouvez la dérivée de yyy par rapport à xxx pour l'équation y7ln⁡(y)−x4ln⁡(x)=5y^{7} \ln (y)-x^{4} \ln (x)=5y7ln(y)−x4ln(x)=5. Réponse: y′=y^{\prime}=y′=

Problem 15827

Find the function fff where f′(x)=5xf^{\prime}(x)=\frac{5}{\sqrt{x}}f′(x)=x​5​ and f(9)=45f(9)=45f(9)=45. What is f(x)f(x)f(x)?

Problem 15828

Evaluate the integral: ∫−51(x−5) dx\int_{-5}^{1}(x-5) \, dx∫−51​(x−5)dx.

Problem 15829

Calculate the left and right Riemann sums for f(x)=2xf(x)=\frac{2}{x}f(x)=x2​ on [1,5][1,5][1,5] with n=4n=4n=4.

Problem 15830

Differentiate the function y=(4x4−x+1)(−x5+4)y=(4x^4-x+1)(-x^5+4)y=(4x4−x+1)(−x5+4). Find y′=□y'=\squarey′=□.

Problem 15831

Find the differential dyd ydy for the function y=sin⁡(2x2)y=\sin(2x^2)y=sin(2x2). Show your work.

Problem 15832

Find the tangent line equation for f(x)=x(1−2x)3f(x)=x(1-2x)^{3}f(x)=x(1−2x)3 at (1,−1)(1,-1)(1,−1). Choose from options (A)-(E).

Problem 15833

Find the derivative g′(x)g^{\prime}(x)g′(x) for the function g(x)=5xe2xg(x)=5 x e^{2 x}g(x)=5xe2x.

Problem 15834

Find the antiderivative of dRdt=81t4\frac{d R}{d t}=\frac{81}{t^{4}}dtdR​=t481​ with R(3)=75R(3)=75R(3)=75. What is RRR?

Problem 15835

Find f′(−1)f^{\prime}(-1)f′(−1) for the function f(x)=x4+8f(x)=\sqrt{x^{4}+8}f(x)=x4+8​.

Problem 15836

Find the derivative f′(x)f^{\prime}(x)f′(x) for the function f(x)=ex7x2+1f(x)=\frac{e^{x}}{7 x^{2}+1}f(x)=7x2+1ex​.

Problem 15837

Find intervals where f(x)=x3−18x2+11x+2f(x)=x^{3}-18 x^{2}+11 x+2f(x)=x3−18x2+11x+2 is concave up, concave down, and its inflection points.

Problem 15838

Find the antiderivatives of f(x)=22x21f(x)=22 x^{21}f(x)=22x21 and verify by differentiating. F(x)=□F(x)=\squareF(x)=□.

Problem 15839

Find the area between the curve y=1−x2y=1-x^{2}y=1−x2, the xxx-axis, and lines x=−1x=-1x=−1 and x=2x=2x=2, treating below xxx-axis as zero.

Problem 15840

Find the derivative dydx\frac{d y}{d x}dxdy​ for the function y=2x+33x+2y=\frac{2 x+3}{3 x+2}y=3x+22x+3​.

Problem 15841

Find the derivative $g^{\prime}(1)$ for the function $g(t)=(t^{3}-2)^{7}$ .

Find the linearization $L(x)$ of $f(x)=\sqrt{4x+9}$ at $x=0$ . Show your work.

Find $p^{\prime}(5)$ for $p(x)=f(x) \cdot g(f(x))$ using values from $f$ and $g$ .

Find the tangent line slope to the ellipse $x^{2}+4y^{2}=16$ at the point $(4,0)$ .

Find the derivative $\frac{d y}{d x}$ for the equation $y^{2}=\frac{1}{2 x+5}$ .

Differentiate the function $\frac{4 t}{0.5+t^{2}}$ with respect to $t$ .

Differentiate the function $\frac{2-4 t^{2}}{(0.5+t^{2})^{2}}$ with respect to $t$ .

Find the slope of the tangent line to the ellipse $x^{2}+4y^{2}=16$ at the point $(4,0)$ .

Find $\frac{d^{2} y}{d x^{2}}$ for the circle $x^{2}+y^{2}=25$ at the point $(3,-4)$ .

Find the limit: $\lim _{h \rightarrow 0} \frac{\frac{1}{(2+h)}-\frac{1}{2}}{h}$ .

Find the limit as $x$ approaches 2: $\lim _{x \rightarrow 2} \frac{(x-2)^4}{x^{4}-16}$ .

Find when the drug concentration $C(t)=\frac{4 t}{0.5+t^{2}}$ is at its maximum. Round to two decimal places.

Find $f^{\prime}(1)$ if $f(x)=(x^{2}-3)^{4}$ .

An observer on a cliff sees a boat approaching the shore. If the telescope is $250^{\prime}$ high and the boat moves at $20 \mathrm{ft/sec}$ , find the angle change rate when the boat is $250^{\prime}$ from shore.

Find $\frac{d y}{d x}$ if $y=\left(\frac{x}{x+1}\right)^{5}$ . Choose from options (A) to (E).

Find the derivative of the function $f(x)=e^{1/x}$ , which is $f^{\prime}(x)=$ .

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

Find the limit of $f(x) = \frac{x-16}{\sqrt{x}-4}$ as $x$ approaches 16.

Tim threw a ball to point $B$ in the water, $10 \mathrm{~m}$ from point $C$ , $15 \mathrm{~m}$ from $A$ . Find $x$ where Elvis entered to minimize time.

Given $h(g(x))=x$ , find $g^{\prime}(7)$ using $h(3)=7$ , $h(7)=22$ , $h^{\prime}(3)=5$ , $h^{\prime}(7)=10$ . Choices: (A) $-1/10$ , (B) $1/10$ , (C) $1/5$ , (D) $7/5$ .

Find the limit: $\lim _{x \rightarrow 2} \frac{2 x^{2}-5 x+2}{5 x^{2}-7 x-6}$ .

Calculate the left and right Riemann sums for $f(x)=x+4$ on $[0,5]$ with $n=5$ .

Trouvez la dérivée de $y$ par rapport à $x$ pour l'équation $y^{7} \ln (y)-x^{4} \ln (x)=5$ . Réponse: $y^{\prime}=$

Find the function $f$ where $f^{\prime}(x)=\frac{5}{\sqrt{x}}$ and $f(9)=45$ . What is $f(x)$ ?

Evaluate the integral: $\int_{-5}^{1}(x-5) \, dx$ .

Calculate the left and right Riemann sums for $f(x)=\frac{2}{x}$ on $[1,5]$ with $n=4$ .

Differentiate the function $y=(4x^4-x+1)(-x^5+4)$ . Find $y'=\square$ .

Find the differential $d y$ for the function $y=\sin(2x^2)$ . Show your work.

Find the tangent line equation for $f(x)=x(1-2x)^{3}$ at $(1,-1)$ . Choose from options (A)-(E).

Find the derivative $g^{\prime}(x)$ for the function $g(x)=5 x e^{2 x}$ .

Find the antiderivative of $\frac{d R}{d t}=\frac{81}{t^{4}}$ with $R(3)=75$ . What is $R$ ?

Find $f^{\prime}(-1)$ for the function $f(x)=\sqrt{x^{4}+8}$ .

Find the derivative $f^{\prime}(x)$ for the function $f(x)=\frac{e^{x}}{7 x^{2}+1}$ .

Find intervals where $f(x)=x^{3}-18 x^{2}+11 x+2$ is concave up, concave down, and its inflection points.

Find the antiderivatives of $f(x)=22 x^{21}$ and verify by differentiating. $F(x)=\square$ .

Find the area between the curve $y=1-x^{2}$ , the $x$ -axis, and lines $x=-1$ and $x=2$ , treating below $x$ -axis as zero.

Find the derivative $\frac{d y}{d x}$ for the function $y=\frac{2 x+3}{3 x+2}$ .

Evaluate the integral: $\int_{0}^{1} 9 e^{x} dx$

Find $f^{\prime}(x)$ and the tangent line equation at $x=2$ for $f(x)=\frac{x-9}{6x-1}$ .

Find the second derivative of $y=-3 \cos (2 x)$ . Choices: (A) $-12 \cos (2 x)$ , (B) $12 \cos (2 x)$ , (C) $-3 \cos (2 x)$ , (D) $3 \cos (2 x)$ .

Find the tangent lines for $f(x)=\sqrt{x-1}+2$ at $x=2$ and $f(x)=x^{3}+2$ at $x=-1$ .

Deondra wants to invest for an account to reach \ $30,000 in 10 years at a continuous interest rate of$ 4.7\%$. How much?

Determine where the function $f(x)=2x^{4}+8x^{3}+25$ is increasing, decreasing, and find local extrema.

Find the derivative of $f(x)=(\ln (x))^{15}$ and evaluate it at $x=2$ : $f^{\prime}(2)=$

Given the function $f(x)=x^{3}-2 x^{2}$ on $[0,1]$ , does the Mean Value Theorem apply? Find the point(s) if it does.

Find the derivative of $y=\tan x-\cot x$ , i.e., compute $\frac{dy}{dx}$ .

Find the instantaneous rate of change of $f(x)=2x^{3}-5x^{2}+8x+1$ at $x=1$ on $(0,3)$ . Choices: (A) 2, (B) 4, (C) 6, (D) 8.

Find $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\tan x - \tan\left(\frac{\pi}{4}\right)}{x - \frac{\pi}{4}}$ .

Determine where $f(x)$ is increasing, decreasing, and find local extrema for $f(x)=2 x^{4}+8 x^{3}+25$ .

Sketch $f(x)=2x^{2}+3$ on $[3,8]$ , find $\Delta x$ , grid points, and calculate left/right Riemann sums for $n=5$ .

Find the cost increase for producing 300 to 720 bikes using $C^{\prime}(x)=300-\frac{x}{3}$ . Evaluate the integral. The increase is \$ \square.

Given $f(x)=\sin ^{3}(x)$ , calculate $f^{\prime}(2)$ where $f^{\prime}(x)=3 \sin (x)^{2} \cos x$ .

Find points $(x, y)$ on the graph of $f(x)=\frac{x}{x+2}$ where the tangent line has slope $\frac{1}{2}$ .

Arianys invested \ $61,000 at$ 7 \frac{1}{8} \% $continuous interest. Yusuf invested \$61,000 at$ 7 \frac{5}{8} \%$ monthly. After 14 years, how much more does Yusuf have?

Find the critical points of the function $f(x)=45 x^{3}-3 x^{5}$ using calculus methods. Show your work.

Find the increase in total cost when production changes from $x = 30$ to $x = 90$ using $MC(x) = 4 + \frac{x^2}{1000}$ .

Explain how to use l'Hôpital's Rule for limits of the form $\frac{0}{0}$ . Choose the correct answer: A, B, C, or D.

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

Find the rate of area increase of an isosceles triangle with sides $20 \mathrm{~cm}$ and vertex angle $\frac{\pi}{6}$ rads, given the angle increases at 1 rads/min.

Find the antiderivatives of $f(x)=1$ . Choose the correct option for $F(x)$ .

Find $g^{\prime}(9)$ for $g(x)=\frac{f(x)}{\sqrt{x}}$ given $f(9)=18$ and $f^{\prime}(9)=7$ .

Given $f(1)=3$ , $f'(1)=-2$ , $g(1)=-3$ , $g'(1)=4$ , find $h'(1)$ for $h(x)=(2f(x)+3)(1+g(x))$ . Choices: A) -28 B) -16 C) 40 D) 44 E) 47

Find all antiderivatives of $f(x)=-8 \sec ^{2} x$ and verify by differentiating. Antiderivatives are $F(x)=\square$ .

Find $f'(x)$ , partition numbers for $f'$ , and critical numbers of $f$ where $f(x)=x^{3}-27x+3$ .

Find the increase in cost from producing 0 to 900 bikes using the marginal cost function $C^{\prime}(x)=600-\frac{x}{3}$ . Set up and evaluate the integral. The increase in cost is \$ \square.

Given the function $f(x)=\frac{4 e^{x}}{4 e^{x}+5}$ , find critical values, concavity intervals, and inflection points.

Find $f'(x)$ , critical numbers of $f$ , and partition numbers for $f'$ where $f(x)=x^{3}-27x+3$ .

Find the derivative of the function $f(x)=\cos^{3}(4x)$ . What is $f^{\prime}(x)$ ?

Find $\lim _{x \rightarrow 2 \pi} \frac{\sin(2 \pi)-\sin x}{x-2 \pi}$ .