Calculus

Problem 1601

Minimize the cost function $C(x, y) = 3000 + 600x^2 + 700y^2$ for pounds of sulfur ( $x$ ) and lead ( $y$ ) removed daily.

See Solution

Problem 1602

Find the implicit derivative of $y - \cos y = x + 1$ .

See Solution

Problem 1603

Find $f^{\prime}(-4)$ for $f(x)=4 x^{2}-3 x$ using the difference quotient and limit as $h \rightarrow 0$ .

See Solution

Problem 1604

Find the limit: $\lim _{x \rightarrow 1} \frac{x}{\ln x}$ .

See Solution

Problem 1605

Find the limit: $\lim _{x \rightarrow e}(x \ln x - x)$ .

See Solution

Problem 1606

Calculate the limit: $\lim _{x \rightarrow e}(x \ln x - x)$

See Solution

Problem 1607

Find the limit: $\lim _{x \rightarrow 0} \ln \left(e^{x+1}\right)$ .

See Solution

Problem 1608

Find the limit: $\lim _{x \rightarrow \infty} \frac{x+1}{\sqrt{4 x^{2}+2}}$ .

See Solution

Problem 1609

Find the velocity function $v(t)$ for the hummingbird's position $s(t)=-10 t^{3}-t-6$ in feet.

See Solution

Problem 1610

Determine if the car is moving left or right at $t=8$ for the function $s(t)=-4 t^{3}-10 t^{2}+t-4$ .

See Solution

Problem 1611

Find the acceleration function $a(t)$ for the particle with position $s(t)=-3t^{2}-t+1$ (in meters) at time $t$ .

See Solution

Problem 1612

Find the acceleration of the particle at $t=3$ seconds for the function $s(t)=t^{3}-3 t^{2}-8 t+1$ . Answer in $\mathrm{ft} / \mathrm{s}^{2}$ .

See Solution

Problem 1613

Determine if the train, described by $s(t)=t^{4}-2 t^{3}-5 t^{2}+3$ , is speeding up or slowing down at $t=2$ seconds.

See Solution

Problem 1614

Find the interval(s) where the hummingbird slows down for $s(t)=-t^{3}+9 t^{2}-24 t-4$ , $t \geq 0$ .

See Solution

Problem 1615

Find the limit $\lim _{x \rightarrow a}\left\{[h(x)]^{2}-f(x) g(x)\right\}$ given $\lim _{x \rightarrow a} f(x)=3$ , $\lim _{x \rightarrow a} g(x)=5$ , $\lim _{x \rightarrow a} h(x)=-2$ .

See Solution

Problem 1616

Find the interval(s) where the car moves left for $s(t)=-t^{3}+6 t^{2}+36 t+4$ with $t \geq 0$ .

See Solution

Problem 1617

Find the ball's velocity when it first reaches $880 \mathrm{ft}$ , given $s(t)=-16 t^{2}+256 t$ .

See Solution

Problem 1618

Find the ball's velocity when it first reaches $880 \mathrm{ft}$ , given $s(t)=-16 t^{2}+256 t$ .

See Solution

Problem 1619

Find the limit as $x$ approaches $\pi$ for $\sqrt{3+\cos 4x}$ .

See Solution

Problem 1620

Find the car's acceleration at $t=2$ seconds given $s(t)=2 t^{4}-10 t^{3}-5 t^{2}+9$ . Answer in $\mathrm{m} / \mathrm{s}^{2}$ .

See Solution

Problem 1621

Find the car's acceleration at $t=2$ seconds, given $s(t)=2 t^{4}-9 t^{3}-6 t^{2}-8$ .

See Solution

Problem 1622

Find the interval(s) where the car slows down given $s(t)=-\frac{2 t^{3}}{3}+3 t^{2}+20 t-3$ , $t \geq 0$ .

See Solution

Problem 1623

Find $\lim _{x \rightarrow -1} f(x)$ given that $1 \leq f(x) \leq x^{2} + 2x + 2$ for all $x$ .

See Solution

Problem 1624

Find the limit as $x$ approaches 2 for the expression $\frac{x^{2}+x-6}{x^{2}-4}$ .

See Solution

Problem 1625

Find the limit as $x$ approaches 0 for the expression $x^{4}-\frac{4^{x}}{8000}$ .

See Solution

Problem 1626

Determine if the particle described by $s(t)=t^{4}-9 t^{3}-9 t^{2}-7$ is speeding up or slowing down at $t=4$ seconds.

See Solution

Problem 1627

Determine if the car is speeding up or slowing down at $t=4$ seconds for the function $s(t)=2 t^{4}-4 t^{3}-4 t^{2}+6$ .

See Solution

Problem 1628

Find $g(13)$ given $f(13)=6$ and $\lim _{x \rightarrow 13}[2 f(x)-g(x)]=13$ .

See Solution

Problem 1629

Determine if the particle defined by $s(t)=t^{3}-9 t^{2}-9 t-4$ is speeding up or slowing down at $t=4$ seconds.

See Solution

Problem 1630

Determine if the particle described by $s(t)=t^{4}-7 t^{3}-7 t^{2}-9$ is speeding up or slowing down at $t=2$ .

See Solution

Problem 1631

Determine if the hummingbird is speeding up or slowing down at $t=2$ for the position function $s(t)=2 t^{4}-8 t^{3}-6 t^{2}+7$ .

See Solution

Problem 1632

Determine the interval(s) where the particle defined by $s(t)=\frac{2 t^{3}}{3}-10 t^{2}+48 t$ is slowing down.

See Solution

Problem 1633

Find where the function $f(x)=|6-x|$ is not differentiable and explain the reason.

See Solution

Problem 1634

1. Find the derivative $\frac{d y}{d t}$ for $y=5 t^{3}+t^{2}+4 t-1$ .
2. Find the second derivative $\frac{d^{2} y}{d u^{2}}$ for $y=2 u^{4}+4 u^{3}+4$ .
3. Find the fifth derivative $\frac{d^{5} y}{d \theta^{5}}$ for $y=2 \sin (\theta)+5 \cos (\theta)$ .

See Solution

Problem 1635

Find $\delta$ for $\varepsilon=0.01$ in the limit $\lim _{x \rightarrow 4} 8 x=32$ .

See Solution

Problem 1636

Find $\delta$ so that if $|x-2|<\delta$ , then $|4x-8|<0.9$ .

See Solution

Problem 1637

Determine where the function $f(x)$ is discontinuous and explain using continuity's definition. $f(x)=\left\{\begin{array}{l} \frac{1}{x-5}, \quad x \neq 5 \\ 5, \quad x=5 \end{array}\right.$

See Solution

Problem 1638

Find the derivative $f^{\prime}(5)$ for the function $f(x)=x^{3}-2x$ using the definition of the derivative.

See Solution

Problem 1639

Find the slope of the tangent line to $y=3x^{2}$ at the point $(-4,48)$ .

See Solution

Problem 1640

Find the instantaneous rate of change of $f(x)=x^{3}+2x$ at $x=-3$ .

See Solution

Problem 1641

Find the tangent line equation for $f(x)=\frac{3}{x}$ at the point (3,1).

See Solution

Problem 1642

Explain the Intermediate Value Theorem and provide an example of how it can be applied.

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Problem 1643

Compute the integral: $\int e^{x}(\cos x-\sin x) d x$ .

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Problem 1644

Air is pumped into a balloon at 10 cm³/sec. Find the diameter's increase rate when the radius is $5 \mathrm{~cm}$ .

See Solution

Problem 1645

Find the derivative of $g(x)=(2 x-3)(1-5 x)$ using the Product Rule.

See Solution

Problem 1646

Find higher-order derivatives for: 105. $f^{\prime \prime}(x)$ , 106. $f^{\prime \prime \prime}(x)$ , 107. $f^{(4)}(x)$ , 108. $f^{(6)}(x)$ . Then, find $f^{\prime}(2)$ for 109-112 using $g(2)=3$ , $g^{\prime}(2)=-2$ , $h(2)=-1$ , $h^{\prime}(2)=4$ .

See Solution

Problem 1647

Calculate the derivative of the function $y = 8x + 1$ .

See Solution

Problem 1648

Find the derivative of the function $g(x) = 5x$ .

See Solution

Problem 1649

Find the derivative of the function $g(x)=5x+1$ .

See Solution

Problem 1650

Find the derivative of the function $f(x)=8 x^{2}$ .

See Solution

Problem 1651

Find the derivative of the function $g(x)=x^{\frac{4}{7}}$ .

See Solution

Problem 1652

Find the derivative of the function $f(x)=-8 x^{5}$ .

See Solution

Problem 1653

Find the derivative of the function $y=-2 x^{9.9}$ .

See Solution

Problem 1654

Find the derivative of the function $f(x)=8 x^{\frac{1}{8}}$ .

See Solution

Problem 1655

Find the derivative of the function $g(x)=\frac{5}{x^{3}}$ .

See Solution

Problem 1656

Find the derivative of the function $g(x)=\frac{4}{x^{3}}$ .

See Solution

Problem 1657

Evaluate the integral $\int_{1}^{\infty} \frac{\tan^{-1}(x)}{x^{2}} \, dx$ .

See Solution

Problem 1658

Find the derivative of the function $f(x)=5 x^{5}-5 x^{4}$ .

See Solution

Problem 1659

Find the derivative of the function $g(x)=-8 x^{5}+5 x^{4}+6$ .

See Solution

Problem 1660

Find the velocity of a particle at $t=7$ given its position function $s(t)=2.1 t^{2}+12 t$ for $0 \leq t \leq 17$ .

See Solution

Problem 1661

Find the slopes of the tangent lines for the function $f(x)=7+8x-5x^{2}$ at $x=0$ , $x=1$ , and $x=2$ .

See Solution

Problem 1662

Rewrite $g(x)=\frac{3 x^{6}-6 x^{7}}{x^{2}}$ as a sum or difference and find $g^{\prime}(x)$ .

See Solution

Problem 1663

Find the derivative of $g(x)=-7 x^{4}-8.8+5 x^{5}$ .

See Solution

Problem 1664

Find the slope of the tangent line for the function $f(x)=-6 x^{2}-9 x+4 x^{3}$ at $x=2$ .

See Solution

Problem 1665

Determine the asymptotes of $f(x)=\left(\frac{x+a}{x-a}\right)^{x}$ , including horizontal and vertical ones.

See Solution

Problem 1666

Calculate the limit: $\lim _{x \rightarrow \infty}\left(\frac{x+a}{x-a}\right)^{x}$ .

See Solution

Problem 1667

Find the fuel consumption rate at 1 PM using $f(t)=0.4 t^{2}-0.2 t^{0.5}+19$ . Round to 2 decimal places.

See Solution

Problem 1668

What is the average fuel consumption rate from 6 AM to 1 PM using $f(t)=0.8 t^{3}-0.3 t^{0.4}+27$ ? Round to 2 decimal places. Answer in barrels per hour.

See Solution

Problem 1669

Find the fuel consumption rate at 4 PM using $f(t)=0.8 t^{3}-0.3 t^{0.4}+27$ . Round to 2 decimal places.

See Solution

Problem 1670

Find the average fuel consumption rate from 6 AM to 3 PM using $f(t)=0.9 t^{2}-0.2 t^{0.4}+10$ . Round to 2 decimal places.

See Solution

Problem 1671

Find the average fuel consumption rate from 6 AM to noon using $f(t)=0.9 t^{3}-0.1 t^{0.3}+14$ . Round to 2 decimal places.

See Solution

Problem 1672

Find the secant line equation for $y=9 \sqrt{x}$ at $x=9$ and $x=16$ , and the tangent line at $x=9$ .

See Solution

Problem 1673

Find the secant line for $f(x)=\frac{5}{x}$ at $x=4$ and $x=5$ , then find the tangent line at $x=4$ .

See Solution

Problem 1674

Find the secant line for $f(x)=\frac{8}{x}$ at $x=2$ and $x=8$ , and the tangent line at $x=2$ .

See Solution

Problem 1675

Find the secant line between $x=9$ and $x=16$ for $y=9 \sqrt{x}$ , and the tangent line at $x=9$ .

See Solution

Problem 1676

Find the secant line between $x=9$ and $x=16$ for $y=9 \sqrt{x}$ , then find the tangent line at $x=9$ .

See Solution

Problem 1677

Find $f^{\prime}(5)$ , $f^{\prime}(15)$ , and $f^{\prime}(-4)$ for $f(x)=2 e^{x}$ when the derivative exists.

See Solution

Problem 1678

Find $f^{\prime}(2)$ , $f^{\prime}(15)$ , and $f^{\prime}(-4)$ for $f(x)=3 e^{x}$ . Round to three decimal places.

See Solution

Problem 1679

Find $f^{\prime}(5)$ , $f^{\prime}(15)$ , and $f^{\prime}(-3)$ for $f(x)=3 e^{x}$ when the derivative exists.

See Solution

Problem 1680

Find $f^{\prime}(5)$ , $f^{\prime}(16)$ , and $f^{\prime}(-4)$ for $f(x)=e^{x}$ . Round to three decimal places.

See Solution

Problem 1681

Find the derivative of the function $g(x)=-8+4x^{5}-8x^{2}$ .

See Solution

Problem 1682

Find $f^{\prime}(2)$ , $f^{\prime}(16)$ , and $f^{\prime}(-6)$ for $f(x)=\frac{-2}{x}$ when the derivative exists.

See Solution

Problem 1683

Find the derivative of the function $f(x)=7 x^{4}+5 x^{2}$ .

See Solution

Problem 1684

Find $f^{\prime}(7)$ , $f^{\prime}(36)$ , and $f^{\prime}(-5)$ for $f(x)=\sqrt{x}$ when the derivative exists. Round $f^{\prime}(7)$ to four decimal places.

See Solution

Problem 1685

Find $f^{\prime}(7)$ for $f(x)=\sqrt{x}$ . Choose A or B: A. $f^{\prime}(7)=\square$ (4 decimal places) B. Does not exist.

See Solution

Problem 1686

Find $f^{\prime}(19)$ , $f^{\prime}(4)$ , and $f^{\prime}(-5)$ for $f(x)=\sqrt{x}$ . Round $f^{\prime}(19)$ to four decimal places.

See Solution

Problem 1687

Find the derivative of the function $f(x) = 9x^2 - 3.2 + 8x^3$ .

See Solution

Problem 1688

Rewrite $g(x)=(3 x+4)(5 x+5)$ as a sum; then calculate $g^{\prime}(x)$ .

See Solution

Problem 1689

Find the derivative of $f(x)=4$ . Choose A if it's a value or B if it doesn't exist.

See Solution

Problem 1690

Find the derivative of the function $f(x)=x$ . Is it A. The derivative is or B. The derivative does not exist?

See Solution

Problem 1691

Find the slopes of the tangent lines for $f(x)=1+8x-3x^{2}$ at $x=1$ , $x=2$ , and $x=3$ .

See Solution

Problem 1692

Approximate the derivative of $f(x)=5 x^{x}$ at $a=2$ using small $h$ . Find the slope of the line on a graphing calculator.

See Solution

Problem 1693

Find the velocity of a particle at $t=3$ seconds, given its position $s(t)=1.6 t^{2}+20 t$ feet.

See Solution

Problem 1694

Find k and m so that the piecewise function $g(x)=\left\{\begin{array}{ll} k \sqrt{x+5} & \text { for } 0 \leq x \leq 4 \\ m x+1 & \text { for } 4<x \leqslant 8 \end{array}\right.$ is differentiable at x=4.

See Solution

Problem 1695

Explain why $\frac{f(x+h)-f(x-h)}{2 h}$ approximates $f^{\prime}(x)$ for small $h$ . Choose the correct answer.

See Solution

Problem 1696

Find the rate of change of demand $D(p)=-4 p^{2}-7 p+300$ with respect to price and at $p=\$10$ .

See Solution

Problem 1697

Expand and differentiate $g(x)=\frac{2 x^{\frac{12}{3}}-7 x^{\frac{4}{3}}+6}{\sqrt[3]{x}}$ . Simplify using exponents first.

See Solution

Problem 1698

Find the derivative $D^{\prime}(p)$ of the demand function $D(p)=\frac{100,000}{p^{2}+10 p+50}$ for $4 \leq p \leq 20$ .

See Solution

Problem 1699

Revenue from selling $x$ picnic tables is $R(x)=55x-\frac{x^{2}}{250}$ . Find marginal revenue, $R'(5000)$ , and actual revenue for the 5001st table. Compare results.

See Solution

Problem 1700

Expand $g(x)=\frac{2 x^{\frac{13}{3}}-7 x^{\frac{4}{3}}+6}{\sqrt[3]{x}}$ and find its derivative by separating $2 x^{4}-7 x+6 x^{-\frac{1}{3}}$ .

See Solution

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Calculus

Problem 1601

Minimize the cost function C(x,y)=3000+600x2+700y2C(x, y) = 3000 + 600x^2 + 700y^2C(x,y)=3000+600x2+700y2 for pounds of sulfur (xxx) and lead (yyy) removed daily.

Problem 1602

Find the implicit derivative of y−cos⁡y=x+1y - \cos y = x + 1y−cosy=x+1.

Problem 1603

Find f′(−4)f^{\prime}(-4)f′(−4) for f(x)=4x2−3xf(x)=4 x^{2}-3 xf(x)=4x2−3x using the difference quotient and limit as h→0h \rightarrow 0h→0.

Problem 1604

Find the limit: lim⁡x→1xln⁡x\lim _{x \rightarrow 1} \frac{x}{\ln x}limx→1​lnxx​.

Problem 1605

Find the limit: lim⁡x→e(xln⁡x−x)\lim _{x \rightarrow e}(x \ln x - x)limx→e​(xlnx−x).

Problem 1606

Calculate the limit: lim⁡x→e(xln⁡x−x)\lim _{x \rightarrow e}(x \ln x - x)limx→e​(xlnx−x)

Problem 1607

Find the limit: lim⁡x→0ln⁡(ex+1)\lim _{x \rightarrow 0} \ln \left(e^{x+1}\right)limx→0​ln(ex+1).

Problem 1608

Find the limit: lim⁡x→∞x+14x2+2\lim _{x \rightarrow \infty} \frac{x+1}{\sqrt{4 x^{2}+2}}limx→∞​4x2+2​x+1​.

Problem 1609

Find the velocity function v(t)v(t)v(t) for the hummingbird's position s(t)=−10t3−t−6s(t)=-10 t^{3}-t-6s(t)=−10t3−t−6 in feet.

Problem 1610

Determine if the car is moving left or right at t=8t=8t=8 for the function s(t)=−4t3−10t2+t−4s(t)=-4 t^{3}-10 t^{2}+t-4s(t)=−4t3−10t2+t−4.

Problem 1611

Find the acceleration function a(t)a(t)a(t) for the particle with position s(t)=−3t2−t+1s(t)=-3t^{2}-t+1s(t)=−3t2−t+1 (in meters) at time ttt.

Problem 1612

Find the acceleration of the particle at t=3t=3t=3 seconds for the function s(t)=t3−3t2−8t+1s(t)=t^{3}-3 t^{2}-8 t+1s(t)=t3−3t2−8t+1. Answer in ft/s2\mathrm{ft} / \mathrm{s}^{2}ft/s2.

Problem 1613

Determine if the train, described by s(t)=t4−2t3−5t2+3s(t)=t^{4}-2 t^{3}-5 t^{2}+3s(t)=t4−2t3−5t2+3, is speeding up or slowing down at t=2t=2t=2 seconds.

Problem 1614

Find the interval(s) where the hummingbird slows down for s(t)=−t3+9t2−24t−4s(t)=-t^{3}+9 t^{2}-24 t-4s(t)=−t3+9t2−24t−4, t≥0t \geq 0t≥0.

Problem 1615

Problem 1616

Find the interval(s) where the car moves left for s(t)=−t3+6t2+36t+4s(t)=-t^{3}+6 t^{2}+36 t+4s(t)=−t3+6t2+36t+4 with t≥0t \geq 0t≥0.

Problem 1617

Find the ball's velocity when it first reaches 880ft880 \mathrm{ft}880ft, given s(t)=−16t2+256ts(t)=-16 t^{2}+256 ts(t)=−16t2+256t.

Problem 1618

Find the ball's velocity when it first reaches 880ft880 \mathrm{ft}880ft, given s(t)=−16t2+256ts(t)=-16 t^{2}+256 ts(t)=−16t2+256t.

Problem 1619

Find the limit as xxx approaches π\piπ for 3+cos⁡4x\sqrt{3+\cos 4x}3+cos4x​.

Problem 1620

Find the car's acceleration at t=2t=2t=2 seconds given s(t)=2t4−10t3−5t2+9s(t)=2 t^{4}-10 t^{3}-5 t^{2}+9s(t)=2t4−10t3−5t2+9. Answer in m/s2\mathrm{m} / \mathrm{s}^{2}m/s2.

Problem 1621

Find the car's acceleration at t=2t=2t=2 seconds, given s(t)=2t4−9t3−6t2−8s(t)=2 t^{4}-9 t^{3}-6 t^{2}-8s(t)=2t4−9t3−6t2−8.

Problem 1622

Find the interval(s) where the car slows down given s(t)=−2t33+3t2+20t−3s(t)=-\frac{2 t^{3}}{3}+3 t^{2}+20 t-3s(t)=−32t3​+3t2+20t−3, t≥0t \geq 0t≥0.

Problem 1623

Find lim⁡x→−1f(x)\lim _{x \rightarrow -1} f(x)limx→−1​f(x) given that 1≤f(x)≤x2+2x+21 \leq f(x) \leq x^{2} + 2x + 21≤f(x)≤x2+2x+2 for all xxx.

Problem 1624

Find the limit as xxx approaches 2 for the expression x2+x−6x2−4\frac{x^{2}+x-6}{x^{2}-4}x2−4x2+x−6​.

Problem 1625

Find the limit as xxx approaches 0 for the expression x4−4x8000x^{4}-\frac{4^{x}}{8000}x4−80004x​.

Problem 1626

Determine if the particle described by s(t)=t4−9t3−9t2−7s(t)=t^{4}-9 t^{3}-9 t^{2}-7s(t)=t4−9t3−9t2−7 is speeding up or slowing down at t=4t=4t=4 seconds.

Problem 1627

Determine if the car is speeding up or slowing down at t=4t=4t=4 seconds for the function s(t)=2t4−4t3−4t2+6s(t)=2 t^{4}-4 t^{3}-4 t^{2}+6s(t)=2t4−4t3−4t2+6.

Problem 1628

Find g(13)g(13)g(13) given f(13)=6f(13)=6f(13)=6 and lim⁡x→13[2f(x)−g(x)]=13\lim _{x \rightarrow 13}[2 f(x)-g(x)]=13limx→13​[2f(x)−g(x)]=13.

Problem 1629

Determine if the particle defined by s(t)=t3−9t2−9t−4s(t)=t^{3}-9 t^{2}-9 t-4s(t)=t3−9t2−9t−4 is speeding up or slowing down at t=4t=4t=4 seconds.

Problem 1630

Determine if the particle described by s(t)=t4−7t3−7t2−9s(t)=t^{4}-7 t^{3}-7 t^{2}-9s(t)=t4−7t3−7t2−9 is speeding up or slowing down at t=2t=2t=2.

Problem 1631

Determine if the hummingbird is speeding up or slowing down at t=2t=2t=2 for the position function s(t)=2t4−8t3−6t2+7s(t)=2 t^{4}-8 t^{3}-6 t^{2}+7s(t)=2t4−8t3−6t2+7.

Problem 1632

Determine the interval(s) where the particle defined by s(t)=2t33−10t2+48ts(t)=\frac{2 t^{3}}{3}-10 t^{2}+48 ts(t)=32t3​−10t2+48t is slowing down.

Problem 1633

Find where the function f(x)=∣6−x∣f(x)=|6-x|f(x)=∣6−x∣ is not differentiable and explain the reason.

Problem 1634

Problem 1635

Find δ\deltaδ for ε=0.01\varepsilon=0.01ε=0.01 in the limit lim⁡x→48x=32\lim _{x \rightarrow 4} 8 x=32limx→4​8x=32.

Problem 1636

Find δ\deltaδ so that if ∣x−2∣<δ|x-2|<\delta∣x−2∣<δ, then ∣4x−8∣<0.9|4x-8|<0.9∣4x−8∣<0.9.

Problem 1637

Determine where the function f(x)f(x)f(x) is discontinuous and explain using continuity's definition. f(x)={1x−5,x≠55,x=5 f(x)=\left\{\begin{array}{l} \frac{1}{x-5}, \quad x \neq 5 \\ 5, \quad x=5 \end{array}\right. f(x)={x−51​,x=55,x=5​

Problem 1638

Find the derivative f′(5)f^{\prime}(5)f′(5) for the function f(x)=x3−2xf(x)=x^{3}-2xf(x)=x3−2x using the definition of the derivative.

Problem 1639

Find the slope of the tangent line to y=3x2y=3x^{2}y=3x2 at the point (−4,48)(-4,48)(−4,48).

Problem 1640

Find the instantaneous rate of change of f(x)=x3+2xf(x)=x^{3}+2xf(x)=x3+2x at x=−3x=-3x=−3.

Problem 1641

Minimize the cost function $C(x, y) = 3000 + 600x^2 + 700y^2$ for pounds of sulfur ( $x$ ) and lead ( $y$ ) removed daily.

Find the implicit derivative of $y - \cos y = x + 1$ .

Find $f^{\prime}(-4)$ for $f(x)=4 x^{2}-3 x$ using the difference quotient and limit as $h \rightarrow 0$ .

Find the limit: $\lim _{x \rightarrow 1} \frac{x}{\ln x}$ .

Find the limit: $\lim _{x \rightarrow e}(x \ln x - x)$ .

Calculate the limit: $\lim _{x \rightarrow e}(x \ln x - x)$

Find the limit: $\lim _{x \rightarrow 0} \ln \left(e^{x+1}\right)$ .

Find the limit: $\lim _{x \rightarrow \infty} \frac{x+1}{\sqrt{4 x^{2}+2}}$ .

Find the velocity function $v(t)$ for the hummingbird's position $s(t)=-10 t^{3}-t-6$ in feet.

Determine if the car is moving left or right at $t=8$ for the function $s(t)=-4 t^{3}-10 t^{2}+t-4$ .

Find the acceleration function $a(t)$ for the particle with position $s(t)=-3t^{2}-t+1$ (in meters) at time $t$ .

Find the acceleration of the particle at $t=3$ seconds for the function $s(t)=t^{3}-3 t^{2}-8 t+1$ . Answer in $\mathrm{ft} / \mathrm{s}^{2}$ .

Determine if the train, described by $s(t)=t^{4}-2 t^{3}-5 t^{2}+3$ , is speeding up or slowing down at $t=2$ seconds.

Find the interval(s) where the hummingbird slows down for $s(t)=-t^{3}+9 t^{2}-24 t-4$ , $t \geq 0$ .

Find the interval(s) where the car moves left for $s(t)=-t^{3}+6 t^{2}+36 t+4$ with $t \geq 0$ .

Find the ball's velocity when it first reaches $880 \mathrm{ft}$ , given $s(t)=-16 t^{2}+256 t$ .

Find the ball's velocity when it first reaches $880 \mathrm{ft}$ , given $s(t)=-16 t^{2}+256 t$ .

Find the limit as $x$ approaches $\pi$ for $\sqrt{3+\cos 4x}$ .

Find the car's acceleration at $t=2$ seconds given $s(t)=2 t^{4}-10 t^{3}-5 t^{2}+9$ . Answer in $\mathrm{m} / \mathrm{s}^{2}$ .

Find the car's acceleration at $t=2$ seconds, given $s(t)=2 t^{4}-9 t^{3}-6 t^{2}-8$ .

Find the interval(s) where the car slows down given $s(t)=-\frac{2 t^{3}}{3}+3 t^{2}+20 t-3$ , $t \geq 0$ .

Find $\lim _{x \rightarrow -1} f(x)$ given that $1 \leq f(x) \leq x^{2} + 2x + 2$ for all $x$ .

Find the limit as $x$ approaches 2 for the expression $\frac{x^{2}+x-6}{x^{2}-4}$ .

Find the limit as $x$ approaches 0 for the expression $x^{4}-\frac{4^{x}}{8000}$ .

Determine if the particle described by $s(t)=t^{4}-9 t^{3}-9 t^{2}-7$ is speeding up or slowing down at $t=4$ seconds.

Determine if the car is speeding up or slowing down at $t=4$ seconds for the function $s(t)=2 t^{4}-4 t^{3}-4 t^{2}+6$ .

Find $g(13)$ given $f(13)=6$ and $\lim _{x \rightarrow 13}[2 f(x)-g(x)]=13$ .

Determine if the particle defined by $s(t)=t^{3}-9 t^{2}-9 t-4$ is speeding up or slowing down at $t=4$ seconds.

Determine if the particle described by $s(t)=t^{4}-7 t^{3}-7 t^{2}-9$ is speeding up or slowing down at $t=2$ .

Determine if the hummingbird is speeding up or slowing down at $t=2$ for the position function $s(t)=2 t^{4}-8 t^{3}-6 t^{2}+7$ .

Determine the interval(s) where the particle defined by $s(t)=\frac{2 t^{3}}{3}-10 t^{2}+48 t$ is slowing down.

Find where the function $f(x)=|6-x|$ is not differentiable and explain the reason.

Find $\delta$ for $\varepsilon=0.01$ in the limit $\lim _{x \rightarrow 4} 8 x=32$ .

Find $\delta$ so that if $|x-2|<\delta$ , then $|4x-8|<0.9$ .

Determine where the function $f(x)$ is discontinuous and explain using continuity's definition. $f(x)=\left\{\begin{array}{l} \frac{1}{x-5}, \quad x \neq 5 \\ 5, \quad x=5 \end{array}\right.$

Find the derivative $f^{\prime}(5)$ for the function $f(x)=x^{3}-2x$ using the definition of the derivative.

Find the slope of the tangent line to $y=3x^{2}$ at the point $(-4,48)$ .

Find the instantaneous rate of change of $f(x)=x^{3}+2x$ at $x=-3$ .

Find the tangent line equation for $f(x)=\frac{3}{x}$ at the point (3,1).

Compute the integral: $\int e^{x}(\cos x-\sin x) d x$ .

Air is pumped into a balloon at 10 cm³/sec. Find the diameter's increase rate when the radius is $5 \mathrm{~cm}$ .

Find the derivative of $g(x)=(2 x-3)(1-5 x)$ using the Product Rule.

Calculate the derivative of the function $y = 8x + 1$ .

Find the derivative of the function $g(x) = 5x$ .

Find the derivative of the function $g(x)=5x+1$ .

Find the derivative of the function $f(x)=8 x^{2}$ .

Find the derivative of the function $g(x)=x^{\frac{4}{7}}$ .

Find the derivative of the function $f(x)=-8 x^{5}$ .

Find the derivative of the function $y=-2 x^{9.9}$ .

Find the derivative of the function $f(x)=8 x^{\frac{1}{8}}$ .

Find the derivative of the function $g(x)=\frac{5}{x^{3}}$ .

Find the derivative of the function $g(x)=\frac{4}{x^{3}}$ .

Evaluate the integral $\int_{1}^{\infty} \frac{\tan^{-1}(x)}{x^{2}} \, dx$ .

Find the derivative of the function $f(x)=5 x^{5}-5 x^{4}$ .

Find the derivative of the function $g(x)=-8 x^{5}+5 x^{4}+6$ .

Find the velocity of a particle at $t=7$ given its position function $s(t)=2.1 t^{2}+12 t$ for $0 \leq t \leq 17$ .

Find the slopes of the tangent lines for the function $f(x)=7+8x-5x^{2}$ at $x=0$ , $x=1$ , and $x=2$ .

Rewrite $g(x)=\frac{3 x^{6}-6 x^{7}}{x^{2}}$ as a sum or difference and find $g^{\prime}(x)$ .

Find the derivative of $g(x)=-7 x^{4}-8.8+5 x^{5}$ .

Find the slope of the tangent line for the function $f(x)=-6 x^{2}-9 x+4 x^{3}$ at $x=2$ .

Determine the asymptotes of $f(x)=\left(\frac{x+a}{x-a}\right)^{x}$ , including horizontal and vertical ones.

Calculate the limit: $\lim _{x \rightarrow \infty}\left(\frac{x+a}{x-a}\right)^{x}$ .

Find the fuel consumption rate at 1 PM using $f(t)=0.4 t^{2}-0.2 t^{0.5}+19$ . Round to 2 decimal places.

What is the average fuel consumption rate from 6 AM to 1 PM using $f(t)=0.8 t^{3}-0.3 t^{0.4}+27$ ? Round to 2 decimal places. Answer in barrels per hour.

Find the fuel consumption rate at 4 PM using $f(t)=0.8 t^{3}-0.3 t^{0.4}+27$ . Round to 2 decimal places.

Find the average fuel consumption rate from 6 AM to 3 PM using $f(t)=0.9 t^{2}-0.2 t^{0.4}+10$ . Round to 2 decimal places.

Find the average fuel consumption rate from 6 AM to noon using $f(t)=0.9 t^{3}-0.1 t^{0.3}+14$ . Round to 2 decimal places.

Find the secant line equation for $y=9 \sqrt{x}$ at $x=9$ and $x=16$ , and the tangent line at $x=9$ .

Find the secant line for $f(x)=\frac{5}{x}$ at $x=4$ and $x=5$ , then find the tangent line at $x=4$ .

Find the secant line for $f(x)=\frac{8}{x}$ at $x=2$ and $x=8$ , and the tangent line at $x=2$ .