Calculus

Problem 31701

Find the derivative $s^{\prime}(x)$ of $s(x)=6x-7$ and calculate $s^{\prime}(1)$ , $s^{\prime}(2)$ , $s^{\prime}(3)$ .

See Solution

Problem 31702

Find $f^{\prime}(x)$ for $f(x)=x^{2}+8x-2$ and calculate $f^{\prime}(1)$ , $f^{\prime}(2)$ , $f^{\prime}(3)$ .

See Solution

Problem 31703

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{3x}{x+6}$ , where $h \neq 0$ .

See Solution

Problem 31704

Find the first derivative $f'(x)$ and the second derivative $f''(4)$ for the function $f(x)=2 \sqrt{x}(x^{3}-2 \sqrt{x}+6)$ .

See Solution

Problem 31705

Calculate the average rate of change of $f(x)=12 x^{3}$ from $x=1$ to $x=3$ .

See Solution

Problem 31706

Find the sales function $S(t)=9 \sqrt{t+1}$ , then: (A) Calculate $S^{\prime}(t)$ . (B) Determine $S(15)$ and $S^{\prime}(15)$ . (C) Estimate $S(16)$ and $S(17)$ .

See Solution

Problem 31707

Evaluate $\pi\left(\frac{-2 y^{3}}{3}+y\right)$ from $y=\sqrt{-2}$ to $y=\sqrt{2}$ .

See Solution

Problem 31708

Find the limit as $x$ approaches $\infty$ for $\frac{9x+4}{4x-2}$ . What is the result?

See Solution

Problem 31709

Find the end behavior of the function $f(x) = \frac{4x+16}{2x-18}$ using limits: $\lim_{{x \to +\infty}} f(x)$ and $\lim_{{x \to -\infty}} f(x)$ .

See Solution

Problem 31710

Evaluate the expression $\pi\left(\frac{-2 y^{3}}{3}+y\right)$ from $y = -2$ to $y = 2$ .

See Solution

Problem 31711

Find the end behavior using limits for the function $p(x)=-\frac{x^{3}-5 x^{2}+2 x}{x^{2}-3 x+5}$ .

See Solution

Problem 31712

Find the second derivative $f^{\prime \prime}(x)$ for the function $f(x)=\sqrt[6]{(x^{2}+1)^{5}}$ .

See Solution

Problem 31713

Find the vertical asymptote of $p(x)=\frac{4 x+4}{x+3}$ using limits.

See Solution

Problem 31714

Find the derivative $f'(x)$ of the function $f(x)=2 \sqrt{x}(x^{3}-2 \sqrt{x}+6)$ and calculate $f'(4)$ .

See Solution

Problem 31715

Find the value of $x$ where the first derivative of $g(x)=6x-9\ln(x)$ equals zero (the critical number).

See Solution

Problem 31716

Find the limits: $\lim_{{x \to \infty}} \left(\frac{16x^2 - 32}{8x^2 - 32}\right)$ and $\lim_{{x \to -\infty}} \left(\frac{16x^2 - 32}{8x^2 - 32}\right)$ .

See Solution

Problem 31717

Find the derivative of $y=6 \pi^{7}$ , i.e., calculate $y^{\prime}$ .

See Solution

Problem 31718

Find the nonzero $x$ where the second derivative of $f(x) = 2x^6 - 8x^5$ is zero.

See Solution

Problem 31719

Find the derivative of the function $f(u)=\sqrt{5} u+\sqrt{8 u}$ . What is $f^{\prime}(u)$ ?

See Solution

Problem 31720

Find the general antiderivative of $f(x)=x^{3}-7$ and $F(3)=8$ . What is $F(0)$ as a decimal?

See Solution

Problem 31721

Find $f^{\prime}(x)$ for $f(x)=2x^{2}+x-1$ and calculate $f^{\prime}(-1)$ , $f^{\prime}(6)$ , $f^{\prime}(7)$ .

See Solution

Problem 31722

Find the vertical and horizontal asymptotes of $f(x)=\frac{x^{3}+x}{3 x^{2}-6 x-9}$ and describe their behavior with limits.

See Solution

Problem 31723

Find the derivative of $y=x^{-2}$ with respect to $x$ : $\frac{d y}{d x}=$

See Solution

Problem 31724

Find the limit $\lim _{x \rightarrow \infty} G(x)$ for the antiderivative $G(x)$ of $g(x)=e^{-8x}$ with $G(0)=16$ .

See Solution

Problem 31725

Evaluate the integral: $\int_{-1}^{1} \frac{\pi}{2}\left(\frac{\frac{6}{1+x^{2}}-3}{2}\right)^{2}$ .

See Solution

Problem 31726

Solve the equation $s^{\prime \prime}(t)=\cos(t)-\sin(t)$ with $s(0)=10$ , $s^{\prime}(0)=5$ , then find $s(\pi)$ .

See Solution

Problem 31727

Find the derivative of $y=\frac{1}{x^{3}}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 31728

Find the marginal cost function for $C(x)=190+4.5 x-0.02 x^{2}$ . What is $C^{\prime}(x)=\square$ ?

See Solution

Problem 31729

Find the marginal revenue function for $R(x)=x(20-0.02 x)$ . What is $R^{\prime}(x)=\square$ ?

See Solution

Problem 31730

Find the vertical asymptote of $f(x)=\frac{x^{2}-2x-8}{x^{2}-1}$ and calculate $\lim$ .

See Solution

Problem 31731

Differentiate $\frac{3 x^{3}}{7}-\frac{2}{5 x^{3}}$ with respect to $x$ . What is the result?

See Solution

Problem 31732

Find the limit as $x$ approaches infinity for $p(x)=\frac{9 x-6}{x^{2}-5 x-18}$ .

See Solution

Problem 31733

Find the limits of $g(x)=-\frac{9}{x^{2}-5 x}$ as $x \rightarrow 0$ and $x \rightarrow 5$ to identify discontinuities.

See Solution

Problem 31734

Given the sales function $S(t)=0.02 t^{3}+0.6 t^{2}+2 t+5$ , find $S^{\prime}(t)$ , $S(5)$ , and interpret $S(11)=126.22$ and $S^{\prime}(11)=22.46$ .

See Solution

Problem 31735

Find the limit as $x$ approaches infinity for the function $\frac{9x^{4}-12x^{3}-20x^{2}+x-35}{4x^{4}+18x^{3}+25x^{2}-18x-45}$ .

See Solution

Problem 31736

Find the derivative of the function $f(x)=\frac{4 x^{2}+8 x+29}{\sqrt{x}}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 31737

Find $f(x)=2 x^{4}-16 x^{3}+5$ . (A) Find $f^{\prime}(x)$ . (B) Slope at $x=-2$ . (C) Tangent line equation at $x=-2$ . (D) Where is the tangent line horizontal?

See Solution

Problem 31738

A 3D printer costs \ $1200, and chair materials cost \$125 each. Find$ \lim_{x \rightarrow 0} g(x) $and$ \lim_{x \rightarrow \infty} g(x) $for$ g(x)=\frac{125x+1200}{x}$.

See Solution

Problem 31739

Find the limit as $x \rightarrow 3$ to determine the type of discontinuity for $g(x)=\frac{4x^{3}+8x^{2}-32x}{2x^{3}-4x^{2}-6x}$ .

See Solution

Problem 31740

Find the derivative $y^{\prime}$ for the function $y=2 x^{-2}-7 x^{-1}$ .

See Solution

Problem 31741

Find the derivative of the function $f(t)=\frac{\sin t}{4}+\frac{3}{t}$ . What is $f^{\prime}(t)$ ?

See Solution

Problem 31742

Find the derivative of $\frac{16 x+18}{x}$ . What is $\frac{d}{d x} \frac{16 x+18}{x}=\square$ ?

See Solution

Problem 31743

Find and explain $\lim_{x \rightarrow \infty} \frac{100 x}{0.5 x^{2}+150}$ for the electric field strength function.

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Problem 31744

Find the derivative of the function $f(t)=3 \sqrt{t}+\frac{12}{\sqrt{t}}$ . What is $f^{\prime}(t)$ ?

See Solution

Problem 31745

Find the derivative of the function $f(x)=5 x^{5} \sqrt{x}+\frac{6}{x^{3} \sqrt{x}}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 31746

Find the derivative of the function $f(x)=\frac{-3 x^{5}+8 x^{4}-4 x^{3}}{x^{4}}$ . What is $f^{\prime}(x)$ ?

See Solution

Problem 31747

Find the average cost per unit for 400 frames with $C(x)=80,000+700x$ . Also, find the marginal average cost and estimate for 401 frames.

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Problem 31748

Find the area of the region $R$ in the first quadrant between $y=\sqrt{x}$ and $y=\frac{x}{3}$ . Then, write an integral for the volume when $R$ is rotated about $y=-1$ , and calculate the volume when rotated about the $y$ -axis.

See Solution

Problem 31749

Find the derivative of the function $f(t)=\sqrt[3]{t^{2}}+2 \sqrt{t^{3}}$ . What is $f^{\prime}(t)$ ?

See Solution

Problem 31750

Find the derivative of $y=x^{4} \cdot x^{6}$ using the Product Rule and by multiplying first.

See Solution

Problem 31751

Find the tangent line equation for $y=2 \sin x$ at $(\pi / 6,1)$ in the form $y=m x+b$ where $m=$ and $b=$ .

See Solution

Problem 31752

Find the derivative of $y=\frac{x^{8}}{x^{3}}$ using the Quotient Rule. Select the correct answer below.

See Solution

Problem 31753

Find the exact value of $F(2)$ if $F$ is the antiderivative of $f(x)=2x+8$ and $F(1)=3$ .

See Solution

Problem 31754

Find the derivative of $y=\frac{x^{7}}{x^{3}}$ using the Quotient Rule and choose the correct form.

See Solution

Problem 31755

Find the derivative of $y=\frac{x^{8}}{x^{3}}$ using the Quotient Rule. Choose the correct answer: A, B, C, or D.

See Solution

Problem 31756

Find the average velocity of a particle from $t=0$ to $t=6$ given its positions: $2$ at $t=0$ and $8$ at $t=6$ .

See Solution

Problem 31757

Find the derivative of $y=\frac{x^{5}}{x^{3}}$ using the Quotient Rule or by simplifying first.

See Solution

Problem 31758

A moped slows down from $44 \mathrm{ft} / \mathrm{s}$ at $6 \mathrm{ft} / \mathrm{s}^{2}$ . Find time $t$ when $v(t)=0$ . Round to two decimal places.

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Problem 31759

Find the derivative of $y=\frac{x^{9}}{x^{7}}$ using the Quotient Rule or by dividing first. Select the correct answer.

See Solution

Problem 31760

Find the exact value of $F(2)$ if $F$ is the antiderivative of $f(x)=2x+4$ with $F(1)=6$ .

See Solution

Problem 31761

Differentiate $\ln y^{3}-\tan^{-1} y$ and set it equal to $\frac{d}{dx}(x+1)$ .

See Solution

Problem 31762

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule or by dividing first.

See Solution

Problem 31763

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule or by dividing first.

See Solution

Problem 31764

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule and select the correct answer.

See Solution

Problem 31765

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule and by dividing first.

See Solution

Problem 31766

Find the derivative $\frac{dy}{dx}$ for the equation $x^{3}+2xy^{2}+y^{3}=3$ .

See Solution

Problem 31767

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule and by dividing first.

See Solution

Problem 31768

Evaluate the integral I = ∫ₑ⁸ (ln(x) / x²) dx.

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Problem 31769

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule and choose the correct answer.

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Problem 31770

Differentiate $ye^x + 2(x + y)$ and set it equal to the derivative of $\ln 3$ . What do you get?

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Problem 31771

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule and by simplifying first. Check if both methods match.

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Problem 31772

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule. Choose the correct answer below.

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Problem 31773

Find the derivative of $y=\frac{x^{6}}{x^{4}}$ using the Quotient Rule and select the correct answer.

See Solution

Problem 31774

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{7}{x^{2}}$ , where $h \neq 0$ .

See Solution

Problem 31775

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule or by simplifying first.

See Solution

Problem 31776

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{6}{x^{2}}$ , where $h \neq 0$ .

See Solution

Problem 31777

Find the antiderivative $F(x)$ of $f(x)=x^{3}+8 \sqrt{x}$ given $F(1)=-8$ .

See Solution

Problem 31778

Find $\frac{dy}{dx}$ for the equation $x + 2y + 4\ln(xy) = 60$ .

See Solution

Problem 31779

Find the derivative of $y=\frac{x^{8}}{x^{3}}$ using the Quotient Rule or by dividing the expressions first.

See Solution

Problem 31780

Find the derivative of $y=\frac{x^{8}}{x^{3}}$ using the Quotient Rule or by dividing the expressions.

See Solution

Problem 31781

Find the derivative of $y=\frac{x^{8}}{x^{3}}$ using the Quotient Rule or by dividing first. Choose the correct answer.

See Solution

Problem 31782

Find the derivative of $y=\frac{x^{8}}{x^{3}}$ using the Quotient Rule or by dividing first.

See Solution

Problem 31783

Find the derivative of $y=\frac{x^{8}}{x^{3}}$ using the Quotient Rule or by dividing first.

See Solution

Problem 31784

Find the derivative of $y=\frac{x^{9}}{x^{7}}$ using the Quotient Rule and choose the correct answer.

See Solution

Problem 31785

Differentiate the function $f(t) = \frac{1}{4} \sin \left(400 t+\frac{\pi}{4}\right)+50 \sqrt{2} t$ .

See Solution

Problem 31786

Find the derivative of $y=\frac{x^{8}}{x^{3}}$ using the Quotient Rule. Choose the correct option for the derivative.

See Solution

Problem 31787

Find the limit as $x$ approaches 2 for the expression $(2x + 3) - (x^2 + 1)$ .

See Solution

Problem 31788

Find the derivative of $y=\frac{x^{6}}{x^{3}}$ using the Quotient Rule and select the correct answer.

See Solution

Problem 31789

Find the derivative of $y=\frac{x^{6}}{x^{3}}$ using the Quotient Rule. Choose the correct answer from the options.

See Solution

Problem 31790

Find the power $P$ from the work equation $W=\frac{1}{4} \sin \left(400 t+\frac{\pi}{4}\right)+50 \sqrt{2} t$ .

See Solution

Problem 31791

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

See Solution

Problem 31792

Find the limit: $\lim _{x \rightarrow 3} \frac{4 x}{2 x-6}$ .

See Solution

Problem 31793

Find the antiderivative $F(t)$ of $f(t)=9 \sec^2(t)-5 t^2$ with $F(0)=0$ . Calculate $F(1.4)$ .

See Solution

Problem 31794

Find slopes of secant lines for $f(x)=-4 x^{2}$ and conjecture the tangent slope at $x=3$ . Complete the table.

See Solution

Problem 31795

Find $f(2)$ for the function with $f^{\prime \prime}(x)=10 x+4 \sin (x)$ , given $f(0)=3$ and $f^{\prime}(0)=3$ .

See Solution

Problem 31796

For the function $f(x)=-4 x^{2}$ , find slopes of secant lines and conjecture the tangent slope at $x=3$ .

See Solution

Problem 31797

For $f(x)=-4x^{2}$ , find slopes of secant lines for intervals and conjecture the tangent slope at $x=3$ .

See Solution

Problem 31798

For $f(x)=-4 x^{2}$ , find secant line slopes for intervals near $x=3$ and predict the tangent slope at that point.

See Solution

Problem 31799

Find the slope of the secant line for $f(x)=17 \cos x$ on the interval $\left[\frac{\pi}{2}, \pi\right]$ .

See Solution

Problem 31800

Find the area between $y = \ln x$ and $y = \ln(2x)$ from $x = 1$ to $x = 5$ . Area = $\int_{1}^{5} (\ln(2x) - \ln x) \, dx$ .

See Solution

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Calculus

Problem 31701

Find the derivative s′(x)s^{\prime}(x)s′(x) of s(x)=6x−7s(x)=6x-7s(x)=6x−7 and calculate s′(1)s^{\prime}(1)s′(1), s′(2)s^{\prime}(2)s′(2), s′(3)s^{\prime}(3)s′(3).

Problem 31702

Find f′(x)f^{\prime}(x)f′(x) for f(x)=x2+8x−2f(x)=x^{2}+8x-2f(x)=x2+8x−2 and calculate f′(1)f^{\prime}(1)f′(1), f′(2)f^{\prime}(2)f′(2), f′(3)f^{\prime}(3)f′(3).

Problem 31703

Find the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ for f(x)=3xx+6f(x)=\frac{3x}{x+6}f(x)=x+63x​, where h≠0h \neq 0h=0.

Problem 31704

Find the first derivative f′(x)f'(x)f′(x) and the second derivative f′′(4)f''(4)f′′(4) for the function f(x)=2x(x3−2x+6)f(x)=2 \sqrt{x}(x^{3}-2 \sqrt{x}+6)f(x)=2x​(x3−2x​+6).

Problem 31705

Calculate the average rate of change of f(x)=12x3f(x)=12 x^{3}f(x)=12x3 from x=1x=1x=1 to x=3x=3x=3.

Problem 31706

Find the sales function S(t)=9t+1S(t)=9 \sqrt{t+1}S(t)=9t+1​, then: (A) Calculate S′(t)S^{\prime}(t)S′(t). (B) Determine S(15)S(15)S(15) and S′(15)S^{\prime}(15)S′(15). (C) Estimate S(16)S(16)S(16) and S(17)S(17)S(17).

Problem 31707

Evaluate π(−2y33+y)\pi\left(\frac{-2 y^{3}}{3}+y\right)π(3−2y3​+y) from y=−2y=\sqrt{-2}y=−2​ to y=2y=\sqrt{2}y=2​.

Problem 31708

Find the limit as xxx approaches ∞\infty∞ for 9x+44x−2\frac{9x+4}{4x-2}4x−29x+4​. What is the result?

Problem 31709

Find the end behavior of the function f(x)=4x+162x−18f(x) = \frac{4x+16}{2x-18}f(x)=2x−184x+16​ using limits: lim⁡x→+∞f(x)\lim_{{x \to +\infty}} f(x)limx→+∞​f(x) and lim⁡x→−∞f(x)\lim_{{x \to -\infty}} f(x)limx→−∞​f(x).

Problem 31710

Evaluate the expression π(−2y33+y)\pi\left(\frac{-2 y^{3}}{3}+y\right)π(3−2y3​+y) from y=−2y = -2y=−2 to y=2y = 2y=2.

Problem 31711

Find the end behavior using limits for the function p(x)=−x3−5x2+2xx2−3x+5p(x)=-\frac{x^{3}-5 x^{2}+2 x}{x^{2}-3 x+5}p(x)=−x2−3x+5x3−5x2+2x​.

Problem 31712

Find the second derivative f′′(x)f^{\prime \prime}(x)f′′(x) for the function f(x)=(x2+1)56f(x)=\sqrt[6]{(x^{2}+1)^{5}}f(x)=6(x2+1)5​.

Problem 31713

Find the vertical asymptote of p(x)=4x+4x+3p(x)=\frac{4 x+4}{x+3}p(x)=x+34x+4​ using limits.

Problem 31714

Find the derivative f′(x)f'(x)f′(x) of the function f(x)=2x(x3−2x+6)f(x)=2 \sqrt{x}(x^{3}-2 \sqrt{x}+6)f(x)=2x​(x3−2x​+6) and calculate f′(4)f'(4)f′(4).

Problem 31715

Find the value of xxx where the first derivative of g(x)=6x−9ln⁡(x)g(x)=6x-9\ln(x)g(x)=6x−9ln(x) equals zero (the critical number).

Problem 31716

Find the limits: lim⁡x→∞(16x2−328x2−32) \lim_{{x \to \infty}} \left(\frac{16x^2 - 32}{8x^2 - 32}\right) limx→∞​(8x2−3216x2−32​) and lim⁡x→−∞(16x2−328x2−32) \lim_{{x \to -\infty}} \left(\frac{16x^2 - 32}{8x^2 - 32}\right) limx→−∞​(8x2−3216x2−32​).

Problem 31717

Find the derivative of y=6π7y=6 \pi^{7}y=6π7, i.e., calculate y′y^{\prime}y′.

Problem 31718

Find the nonzero xxx where the second derivative of f(x)=2x6−8x5f(x) = 2x^6 - 8x^5f(x)=2x6−8x5 is zero.

Problem 31719

Find the derivative of the function f(u)=5u+8uf(u)=\sqrt{5} u+\sqrt{8 u}f(u)=5​u+8u​. What is f′(u)f^{\prime}(u)f′(u)?

Problem 31720

Find the general antiderivative of f(x)=x3−7f(x)=x^{3}-7f(x)=x3−7 and F(3)=8F(3)=8F(3)=8. What is F(0)F(0)F(0) as a decimal?

Problem 31721

Find f′(x)f^{\prime}(x)f′(x) for f(x)=2x2+x−1f(x)=2x^{2}+x-1f(x)=2x2+x−1 and calculate f′(−1)f^{\prime}(-1)f′(−1), f′(6)f^{\prime}(6)f′(6), f′(7)f^{\prime}(7)f′(7).

Problem 31722

Find the vertical and horizontal asymptotes of f(x)=x3+x3x2−6x−9f(x)=\frac{x^{3}+x}{3 x^{2}-6 x-9}f(x)=3x2−6x−9x3+x​ and describe their behavior with limits.

Problem 31723

Find the derivative of y=x−2y=x^{-2}y=x−2 with respect to xxx: dydx=\frac{d y}{d x}=dxdy​=

Problem 31724

Find the limit lim⁡x→∞G(x)\lim _{x \rightarrow \infty} G(x)limx→∞​G(x) for the antiderivative G(x)G(x)G(x) of g(x)=e−8xg(x)=e^{-8x}g(x)=e−8x with G(0)=16G(0)=16G(0)=16.

Problem 31725

Evaluate the integral: ∫−11π2(61+x2−32)2\int_{-1}^{1} \frac{\pi}{2}\left(\frac{\frac{6}{1+x^{2}}-3}{2}\right)^{2}∫−11​2π​(21+x26​−3​)2.

Problem 31726

Solve the equation s′′(t)=cos⁡(t)−sin⁡(t)s^{\prime \prime}(t)=\cos(t)-\sin(t)s′′(t)=cos(t)−sin(t) with s(0)=10s(0)=10s(0)=10, s′(0)=5s^{\prime}(0)=5s′(0)=5, then find s(π)s(\pi)s(π).

Problem 31727

Find the derivative of y=1x3y=\frac{1}{x^{3}}y=x31​. What is dydx\frac{d y}{d x}dxdy​?

Problem 31728

Find the marginal cost function for C(x)=190+4.5x−0.02x2C(x)=190+4.5 x-0.02 x^{2}C(x)=190+4.5x−0.02x2. What is C′(x)=□C^{\prime}(x)=\squareC′(x)=□?

Problem 31729

Find the marginal revenue function for R(x)=x(20−0.02x)R(x)=x(20-0.02 x)R(x)=x(20−0.02x). What is R′(x)=□R^{\prime}(x)=\squareR′(x)=□?

Problem 31730

Find the vertical asymptote of f(x)=x2−2x−8x2−1f(x)=\frac{x^{2}-2x-8}{x^{2}-1}f(x)=x2−1x2−2x−8​ and calculate lim⁡\limlim.

Problem 31731

Differentiate 3x37−25x3\frac{3 x^{3}}{7}-\frac{2}{5 x^{3}}73x3​−5x32​ with respect to xxx. What is the result?

Problem 31732

Find the limit as xxx approaches infinity for p(x)=9x−6x2−5x−18p(x)=\frac{9 x-6}{x^{2}-5 x-18}p(x)=x2−5x−189x−6​.

Problem 31733

Find the limits of g(x)=−9x2−5xg(x)=-\frac{9}{x^{2}-5 x}g(x)=−x2−5x9​ as x→0x \rightarrow 0x→0 and x→5x \rightarrow 5x→5 to identify discontinuities.

Problem 31734

Given the sales function S(t)=0.02t3+0.6t2+2t+5S(t)=0.02 t^{3}+0.6 t^{2}+2 t+5S(t)=0.02t3+0.6t2+2t+5, find S′(t)S^{\prime}(t)S′(t), S(5)S(5)S(5), and interpret S(11)=126.22S(11)=126.22S(11)=126.22 and S′(11)=22.46S^{\prime}(11)=22.46S′(11)=22.46.

Problem 31735

Find the limit as xxx approaches infinity for the function 9x4−12x3−20x2+x−354x4+18x3+25x2−18x−45\frac{9x^{4}-12x^{3}-20x^{2}+x-35}{4x^{4}+18x^{3}+25x^{2}-18x-45}4x4+18x3+25x2−18x−459x4−12x3−20x2+x−35​.

Problem 31736

Find the derivative of the function f(x)=4x2+8x+29xf(x)=\frac{4 x^{2}+8 x+29}{\sqrt{x}}f(x)=x​4x2+8x+29​. What is f′(x)f^{\prime}(x)f′(x)?

Problem 31737

Find f(x)=2x4−16x3+5f(x)=2 x^{4}-16 x^{3}+5f(x)=2x4−16x3+5. (A) Find f′(x)f^{\prime}(x)f′(x). (B) Slope at x=−2x=-2x=−2. (C) Tangent line equation at x=−2x=-2x=−2. (D) Where is the tangent line horizontal?

Problem 31738

A 3D printer costs \1200,andchairmaterialscost$125each.Find1200, and chair materials cost \$125 each. Find 1200,andchairmaterialscost$125each.Find\lim_{x \rightarrow 0} g(x)and and and\lim_{x \rightarrow \infty} g(x)for for forg(x)=\frac{125x+1200}{x}$.

Problem 31739

Find the limit as x→3x \rightarrow 3x→3 to determine the type of discontinuity for g(x)=4x3+8x2−32x2x3−4x2−6xg(x)=\frac{4x^{3}+8x^{2}-32x}{2x^{3}-4x^{2}-6x}g(x)=2x3−4x2−6x4x3+8x2−32x​.

Problem 31740

Find the derivative $s^{\prime}(x)$ of $s(x)=6x-7$ and calculate $s^{\prime}(1)$ , $s^{\prime}(2)$ , $s^{\prime}(3)$ .

Find $f^{\prime}(x)$ for $f(x)=x^{2}+8x-2$ and calculate $f^{\prime}(1)$ , $f^{\prime}(2)$ , $f^{\prime}(3)$ .

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{3x}{x+6}$ , where $h \neq 0$ .

Find the first derivative $f'(x)$ and the second derivative $f''(4)$ for the function $f(x)=2 \sqrt{x}(x^{3}-2 \sqrt{x}+6)$ .

Calculate the average rate of change of $f(x)=12 x^{3}$ from $x=1$ to $x=3$ .

Find the sales function $S(t)=9 \sqrt{t+1}$ , then: (A) Calculate $S^{\prime}(t)$ . (B) Determine $S(15)$ and $S^{\prime}(15)$ . (C) Estimate $S(16)$ and $S(17)$ .

Evaluate $\pi\left(\frac{-2 y^{3}}{3}+y\right)$ from $y=\sqrt{-2}$ to $y=\sqrt{2}$ .

Find the limit as $x$ approaches $\infty$ for $\frac{9x+4}{4x-2}$ . What is the result?

Find the end behavior of the function $f(x) = \frac{4x+16}{2x-18}$ using limits: $\lim_{{x \to +\infty}} f(x)$ and $\lim_{{x \to -\infty}} f(x)$ .

Evaluate the expression $\pi\left(\frac{-2 y^{3}}{3}+y\right)$ from $y = -2$ to $y = 2$ .

Find the end behavior using limits for the function $p(x)=-\frac{x^{3}-5 x^{2}+2 x}{x^{2}-3 x+5}$ .

Find the second derivative $f^{\prime \prime}(x)$ for the function $f(x)=\sqrt[6]{(x^{2}+1)^{5}}$ .

Find the vertical asymptote of $p(x)=\frac{4 x+4}{x+3}$ using limits.

Find the derivative $f'(x)$ of the function $f(x)=2 \sqrt{x}(x^{3}-2 \sqrt{x}+6)$ and calculate $f'(4)$ .

Find the value of $x$ where the first derivative of $g(x)=6x-9\ln(x)$ equals zero (the critical number).

Find the limits: $\lim_{{x \to \infty}} \left(\frac{16x^2 - 32}{8x^2 - 32}\right)$ and $\lim_{{x \to -\infty}} \left(\frac{16x^2 - 32}{8x^2 - 32}\right)$ .

Find the derivative of $y=6 \pi^{7}$ , i.e., calculate $y^{\prime}$ .

Find the nonzero $x$ where the second derivative of $f(x) = 2x^6 - 8x^5$ is zero.

Find the derivative of the function $f(u)=\sqrt{5} u+\sqrt{8 u}$ . What is $f^{\prime}(u)$ ?

Find the general antiderivative of $f(x)=x^{3}-7$ and $F(3)=8$ . What is $F(0)$ as a decimal?

Find $f^{\prime}(x)$ for $f(x)=2x^{2}+x-1$ and calculate $f^{\prime}(-1)$ , $f^{\prime}(6)$ , $f^{\prime}(7)$ .

Find the vertical and horizontal asymptotes of $f(x)=\frac{x^{3}+x}{3 x^{2}-6 x-9}$ and describe their behavior with limits.

Find the derivative of $y=x^{-2}$ with respect to $x$ : $\frac{d y}{d x}=$

Find the limit $\lim _{x \rightarrow \infty} G(x)$ for the antiderivative $G(x)$ of $g(x)=e^{-8x}$ with $G(0)=16$ .

Evaluate the integral: $\int_{-1}^{1} \frac{\pi}{2}\left(\frac{\frac{6}{1+x^{2}}-3}{2}\right)^{2}$ .

Solve the equation $s^{\prime \prime}(t)=\cos(t)-\sin(t)$ with $s(0)=10$ , $s^{\prime}(0)=5$ , then find $s(\pi)$ .

Find the derivative of $y=\frac{1}{x^{3}}$ . What is $\frac{d y}{d x}$ ?

Find the marginal cost function for $C(x)=190+4.5 x-0.02 x^{2}$ . What is $C^{\prime}(x)=\square$ ?

Find the marginal revenue function for $R(x)=x(20-0.02 x)$ . What is $R^{\prime}(x)=\square$ ?

Find the vertical asymptote of $f(x)=\frac{x^{2}-2x-8}{x^{2}-1}$ and calculate $\lim$ .

Differentiate $\frac{3 x^{3}}{7}-\frac{2}{5 x^{3}}$ with respect to $x$ . What is the result?

Find the limit as $x$ approaches infinity for $p(x)=\frac{9 x-6}{x^{2}-5 x-18}$ .

Find the limits of $g(x)=-\frac{9}{x^{2}-5 x}$ as $x \rightarrow 0$ and $x \rightarrow 5$ to identify discontinuities.

Given the sales function $S(t)=0.02 t^{3}+0.6 t^{2}+2 t+5$ , find $S^{\prime}(t)$ , $S(5)$ , and interpret $S(11)=126.22$ and $S^{\prime}(11)=22.46$ .

Find the limit as $x$ approaches infinity for the function $\frac{9x^{4}-12x^{3}-20x^{2}+x-35}{4x^{4}+18x^{3}+25x^{2}-18x-45}$ .

Find the derivative of the function $f(x)=\frac{4 x^{2}+8 x+29}{\sqrt{x}}$ . What is $f^{\prime}(x)$ ?

Find $f(x)=2 x^{4}-16 x^{3}+5$ . (A) Find $f^{\prime}(x)$ . (B) Slope at $x=-2$ . (C) Tangent line equation at $x=-2$ . (D) Where is the tangent line horizontal?

A 3D printer costs \ $1200, and chair materials cost \$125 each. Find$ \lim_{x \rightarrow 0} g(x) $and$ \lim_{x \rightarrow \infty} g(x) $for$ g(x)=\frac{125x+1200}{x}$.

Find the limit as $x \rightarrow 3$ to determine the type of discontinuity for $g(x)=\frac{4x^{3}+8x^{2}-32x}{2x^{3}-4x^{2}-6x}$ .

Find the derivative $y^{\prime}$ for the function $y=2 x^{-2}-7 x^{-1}$ .

Find the derivative of the function $f(t)=\frac{\sin t}{4}+\frac{3}{t}$ . What is $f^{\prime}(t)$ ?

Find the derivative of $\frac{16 x+18}{x}$ . What is $\frac{d}{d x} \frac{16 x+18}{x}=\square$ ?

Find and explain $\lim_{x \rightarrow \infty} \frac{100 x}{0.5 x^{2}+150}$ for the electric field strength function.

Find the derivative of the function $f(t)=3 \sqrt{t}+\frac{12}{\sqrt{t}}$ . What is $f^{\prime}(t)$ ?

Find the derivative of the function $f(x)=5 x^{5} \sqrt{x}+\frac{6}{x^{3} \sqrt{x}}$ . What is $f^{\prime}(x)$ ?

Find the derivative of the function $f(x)=\frac{-3 x^{5}+8 x^{4}-4 x^{3}}{x^{4}}$ . What is $f^{\prime}(x)$ ?

Find the average cost per unit for 400 frames with $C(x)=80,000+700x$ . Also, find the marginal average cost and estimate for 401 frames.

Find the area of the region $R$ in the first quadrant between $y=\sqrt{x}$ and $y=\frac{x}{3}$ . Then, write an integral for the volume when $R$ is rotated about $y=-1$ , and calculate the volume when rotated about the $y$ -axis.

Find the derivative of the function $f(t)=\sqrt[3]{t^{2}}+2 \sqrt{t^{3}}$ . What is $f^{\prime}(t)$ ?

Find the derivative of $y=x^{4} \cdot x^{6}$ using the Product Rule and by multiplying first.

Find the tangent line equation for $y=2 \sin x$ at $(\pi / 6,1)$ in the form $y=m x+b$ where $m=$ and $b=$ .

Find the derivative of $y=\frac{x^{8}}{x^{3}}$ using the Quotient Rule. Select the correct answer below.

Find the exact value of $F(2)$ if $F$ is the antiderivative of $f(x)=2x+8$ and $F(1)=3$ .

Find the derivative of $y=\frac{x^{7}}{x^{3}}$ using the Quotient Rule and choose the correct form.

Find the derivative of $y=\frac{x^{8}}{x^{3}}$ using the Quotient Rule. Choose the correct answer: A, B, C, or D.

Find the average velocity of a particle from $t=0$ to $t=6$ given its positions: $2$ at $t=0$ and $8$ at $t=6$ .

Find the derivative of $y=\frac{x^{5}}{x^{3}}$ using the Quotient Rule or by simplifying first.

A moped slows down from $44 \mathrm{ft} / \mathrm{s}$ at $6 \mathrm{ft} / \mathrm{s}^{2}$ . Find time $t$ when $v(t)=0$ . Round to two decimal places.

Find the derivative of $y=\frac{x^{9}}{x^{7}}$ using the Quotient Rule or by dividing first. Select the correct answer.

Find the exact value of $F(2)$ if $F$ is the antiderivative of $f(x)=2x+4$ with $F(1)=6$ .

Differentiate $\ln y^{3}-\tan^{-1} y$ and set it equal to $\frac{d}{dx}(x+1)$ .

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule or by dividing first.

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule or by dividing first.

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule and select the correct answer.

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule and by dividing first.

Find the derivative $\frac{dy}{dx}$ for the equation $x^{3}+2xy^{2}+y^{3}=3$ .

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule and by dividing first.

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule and choose the correct answer.

Differentiate $ye^x + 2(x + y)$ and set it equal to the derivative of $\ln 3$ . What do you get?

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule and by simplifying first. Check if both methods match.

Find the derivative of $y=\frac{x^{8}}{x^{5}}$ using the Quotient Rule. Choose the correct answer below.

Find the derivative of $y=\frac{x^{6}}{x^{4}}$ using the Quotient Rule and select the correct answer.

Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=\frac{7}{x^{2}}$ , where $h \neq 0$ .