Derivatives

Problem 1701

4. SpongeBob wants to go to point $D$ from point $A$ on an island. He can swim to any point $C$ on the beach. He can swim at $4 \mathrm{~km} / \mathrm{hr}$ and run at $5 \mathrm{~km} / \mathrm{hr}$ . (a) Find analytically the location of $C$ between $B$ and $D$ that will take the least amount of time. (b) Find the time it would take to swim from A to C and then run from C to D using the result of ) (c) Find the time it would take if Spongebob swam from A to B, and then run from B to D (d) Find the time if Spongebob swam directly from A to D, and compare the results with those of (b) and (c).

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Problem 1702

The value of an investment (in dollars) after $t$ years is gives by $A(t)=100(1.03)^{t}$
Find the average rate of change of the value (in dollars per year) over the first 5 years, that is, on the interval $[0,5]$ .
Round to the nearest cent, and do not include the units or a dollar sign; just type in a qumber.

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Problem 1703

Find the derivative $f^{\prime}\left(\frac{\pi}{4}\right)$ for the function $f(x)=\sin x(2+\cos x)$ .

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Problem 1704

Find the derivative $\frac{d y}{d x}$ for $y=\sin x-\cos 3 x$ .

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Problem 1705

Find $\frac{d y}{d x}$ for $y=\sin x-\cos 3 x$ at $x=45^{\circ}$ .

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Problem 1706

Find $\left.\frac{d y}{d x}\right|_{45^{\circ}}$ for $y=\sin x-\cos 3 x$ .

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Problem 1707

Cari turunan $f^{\prime}(x)$ dari fungsi $f(x)=\sin ^{2} x+\cos 3 x$ .

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Problem 1708

Find the derivative of $y=\sin 3x - \cos 3x$ at $x=45^{\circ}$ .

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Problem 1709

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

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Problem 1710

Find the derivative of the function $y=\frac{\sin x+\cos x}{x}$ .

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Problem 1711

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

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Problem 1712

Determine the slope, $m$ , of the tangent line to the curve $y=7+5 x^{2}-2 x^{3}$ at $x=a$ .

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Problem 1713

Compute $\Delta y / \Delta x$ for $y=4x-7$ over $[2,6]$ and find the instantaneous rate of change at $x=2$ .

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Problem 1714

Find the velocity of a particle with displacement $s=\frac{2}{t^{2}}$ at $t=a, t=1, t=2, t=3$ (in $\mathrm{m/s}$ ).

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Problem 1715

Find the critical points of $f(x, y)=x^{2}+2xy+2y^{2}-8x+2y$ and classify each as max, min, or saddle.

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Problem 1716

Find critical points of $t(x, y)=x^{3}-12xy+y^{3}$ and classify as max, min, or saddle.

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Problem 1717

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

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Problem 1718

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

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Problem 1719

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

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Problem 1720

Let $g$ be a.function defined for all $x \neq 0$ , such that $g(5)=-3$ , and the derivative of $g$ is given by $g^{\prime}(x)=\frac{x^{2}-x-2}{x}$ for all $x \neq 0$ . A. Find all values of $x$ for which the graph of $g$ has a horizontal tangent, and determine whether $g$ has a local maximum a local minimum, or neither at each of these values. Justify your answers. B. On what intervals, if any, is the graph of $g$ concave up? Justify your answer. C. Write an equation for the line tangent to the graph of $g$ at $x=5$ . D. Does the line tangent to the graph of $g$ at $x=5$ lie above or below the graph of $g$ for $x>5$ ? Why?

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Problem 1721

1 - Derivatives of Polynomials and Exponential Functions: point)
At a time $t$ seconds after it is thrown up in the air, a tomato is at a height (in meters) of $f(t)=-4.9 t^{2}+60 t+4 \mathrm{~m}$ . A. What is the average velocity of the tomato during the first 5 seconds? (Include help (units) .) $\square$ B. Find (exactly) the instantaneous velocity of the tomato at $t=5$ . (Include help (units) .) $\square$ C. What is the acceleration at $t=5$ ? (Include help (units).) $\square$ D. How high does the tomato go? (Include help (units).) $\square$ E. How long is the tomato in the air? (Include help (units).) ote: You can earn partial credit on this problem.
Preview My Answers Submit Answers ou have attempted this problem 0 times. ou have unlimited attempts remaining.

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Problem 1722

5.3 - ine Fundamental ineorem ot (1 point)
Suppose that $F(x)=\int_{1}^{x} f(t) d t$ , where $f(t)=\int_{1}^{t^{4}} \frac{\sqrt{6+u^{4}}}{u} d u$
Find $F^{\prime \prime}(2)$ . $F^{\prime \prime}(2)=$ $\square$ Preview My Answers Submit Answers

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Problem 1723

Problems (1 point)
Let $f(x)=\int_{1}^{x} t^{8} d t$ . Evaluate the following. $\begin{array}{l} f^{\prime}(x)=\square \\ f^{\prime}(-4)=\square \end{array}$
Note: You can earn partial credit on this problem.

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Problem 1724

For the following equation, what is the instantaneous rate of change at $x=-1 ?$ $f(x)=-2 x^{3}-x^{2}$

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Problem 1725

Find $d y / d x$ if (1) $y=\frac{e^{4 x^{2}+2 x}}{2 x+1}$

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Problem 1726

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

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Problem 1727

Find $\frac{d y}{d x}$ of the following finctions: if $y=2 \cos 2 x-1$

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Problem 1728

Find $\frac{d y}{d x}$ of the following function (1) if $y=\frac{x^{2}}{x+1}$

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Problem 1729

Defferentiate with respect to $x: 4-x^{2}$

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Problem 1730

Find $d y / d x$ by implicit differentiation $x^{3} y^{5}+3 x=8 y^{3}+1$

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Problem 1731

If $f(x, y)=2 x^{2}-y e^{y}$ and the maximum rate of change of $f$ at $(1,0)$ is $a$ then $a^{2}$ equal
Answer: $\square$

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Problem 1732

If $f(x, y)=2 x^{2}-y e^{y}$ and the maximum rate of change of $f$ at $(1,0)$ is $a$ then $a^{2}$ equal
Answer: $\square$

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Problem 1733

Exercice $\mathrm{n}^{\circ} 9$ : Soit $a \in] 0,1$ . Montrer à l'aide des accroissements finis que pour tout $n \in \mathbb{N}^{*}$ , $\frac{a}{(n+1)^{1-a}} \leqslant(n+1)^{a}-n^{a} \leqslant \frac{a}{n^{1-a}} .$

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Problem 1734

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

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Problem 1735

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

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Problem 1736

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 $\mathrm{ft} / \mathrm{s}^{2}$ .

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Problem 1737

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.

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Problem 1738

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

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Problem 1739

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

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Problem 1740

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

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Problem 1741

$f(x)=x e^{x}-x$ is concave up ou?

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Problem 1742

8. If $f(x)=\left\{x^{2}+3\right.$ (A) $a=1, b=3$ (B) $\frac{1}{7}$ (C) $\frac{1}{3}$ (D) $\frac{1}{3}$ (B) $\left.a=\begin{array}{c}x<0 \\ 3, b=0\end{array}\right\}$ and $f(x)$ is differentiable at $x=0$ , then: (C) $a=3, b=1$ (D) $a=0, b=3$ $\begin{array}{lll}\text { (A) } x=\ln 2 & \text { (B) } x=\ln 5 & \text { (C) } x=\ln 3\end{array}$ (D) $x=2 \ln 5$

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Problem 1743

$\qquad$ (D) $y=0$

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Problem 1744

- A është vlerësues i pazhvendosur? Pse? 5.24. Shqyrtojmë shpërndarjen eksponenciale të zhvendosur me densitet $f(x)=\lambda e^{-\lambda(x-s)}$ $k u X \geq 5$ . Gjeni me anë të metodës së përgjasisë maksimale një vlerësues për $\lambda$ ?

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Problem 1745

B) If the function $f(x)=a x^{3}-3 x^{2}+b x+c$ has a local minimum at -5 and a local maximum at $x=\frac{-1}{2}$ and inflection point at $x=\frac{1}{4}$ find $a, b, c \in R$ .

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Problem 1746

Q3: A) Let M be a point moving on the curve $y^{2}=x^{3}$ such that rate of change the point $M$ getting away from of the origin point is 4 unit $/ \mathrm{s}$ . Find the rate of chang of the x -coordinate for M when $x=2$

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Derivatives

Problem 1701

Problem 1702

Problem 1703

Find the derivative f′(π4)f^{\prime}\left(\frac{\pi}{4}\right)f′(4π​) for the function f(x)=sin⁡x(2+cos⁡x)f(x)=\sin x(2+\cos x)f(x)=sinx(2+cosx).

Problem 1704

Find the derivative dydx\frac{d y}{d x}dxdy​ for y=sin⁡x−cos⁡3xy=\sin x-\cos 3 xy=sinx−cos3x.

Problem 1705

Find dydx\frac{d y}{d x}dxdy​ for y=sin⁡x−cos⁡3xy=\sin x-\cos 3 xy=sinx−cos3x at x=45∘x=45^{\circ}x=45∘.

Problem 1706

Find dydx∣45∘\left.\frac{d y}{d x}\right|_{45^{\circ}}dxdy​∣∣​45∘​ for y=sin⁡x−cos⁡3xy=\sin x-\cos 3 xy=sinx−cos3x.

Problem 1707

Cari turunan f′(x)f^{\prime}(x)f′(x) dari fungsi f(x)=sin⁡2x+cos⁡3xf(x)=\sin ^{2} x+\cos 3 xf(x)=sin2x+cos3x.

Problem 1708

Find the derivative of y=sin⁡3x−cos⁡3xy=\sin 3x - \cos 3xy=sin3x−cos3x at x=45∘x=45^{\circ}x=45∘.

Problem 1709

Find the derivative of the function f(x)=xcos⁡3xf(x)=\sqrt{x} \cos^{3} xf(x)=x​cos3x. What is f′(x)f^{\prime}(x)f′(x)?

Problem 1710

Find the derivative of the function y=sin⁡x+cos⁡xxy=\frac{\sin x+\cos x}{x}y=xsinx+cosx​.

Problem 1711

Find the derivative f′(π)f^{\prime}(\pi)f′(π) for the function f(x)=11+cos⁡xf(x)=\frac{1}{1+\cos x}f(x)=1+cosx1​.

Problem 1712

Determine the slope, mmm, of the tangent line to the curve y=7+5x2−2x3y=7+5 x^{2}-2 x^{3}y=7+5x2−2x3 at x=ax=ax=a.

Problem 1713

Compute Δy/Δx\Delta y / \Delta xΔy/Δx for y=4x−7y=4x-7y=4x−7 over [2,6][2,6][2,6] and find the instantaneous rate of change at x=2x=2x=2.

Problem 1714

Find the velocity of a particle with displacement s=2t2s=\frac{2}{t^{2}}s=t22​ at t=a,t=1,t=2,t=3t=a, t=1, t=2, t=3t=a,t=1,t=2,t=3 (in m/s\mathrm{m/s}m/s).

Problem 1715

Find the critical points of f(x,y)=x2+2xy+2y2−8x+2yf(x, y)=x^{2}+2xy+2y^{2}-8x+2yf(x,y)=x2+2xy+2y2−8x+2y and classify each as max, min, or saddle.

Problem 1716

Find critical points of t(x,y)=x3−12xy+y3t(x, y)=x^{3}-12xy+y^{3}t(x,y)=x3−12xy+y3 and classify as max, min, or saddle.

Problem 1717

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 1718

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

Problem 1719

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 1720

Problem 1721

Problem 1722

Problem 1723

Problems (1 point)Let f(x)=∫1xt8dtf(x)=\int_{1}^{x} t^{8} d tf(x)=∫1x​t8dt. Evaluate the following. f′(x)=□f′(−4)=□\begin{array}{l} f^{\prime}(x)=\square \\ f^{\prime}(-4)=\square \end{array}f′(x)=□f′(−4)=□​Note: You can earn partial credit on this problem.

Problem 1724

For the following equation, what is the instantaneous rate of change at x=−1?x=-1 ?x=−1? f(x)=−2x3−x2f(x)=-2 x^{3}-x^{2}f(x)=−2x3−x2

Problem 1725

Find dy/dxd y / d xdy/dx if (1) y=e4x2+2x2x+1y=\frac{e^{4 x^{2}+2 x}}{2 x+1}y=2x+1e4x2+2x​

Problem 1726

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 1727

Find dydx\frac{d y}{d x}dxdy​ of the following finctions: if y=2cos⁡2x−1y=2 \cos 2 x-1y=2cos2x−1

Problem 1728

Find dydx\frac{d y}{d x}dxdy​ of the following function (1) if y=x2x+1y=\frac{x^{2}}{x+1}y=x+1x2​

Problem 1729

Defferentiate with respect to x:4−x2x: 4-x^{2}x:4−x2

Problem 1730

Find dy/dxd y / d xdy/dx by implicit differentiation x3y5+3x=8y3+1x^{3} y^{5}+3 x=8 y^{3}+1x3y5+3x=8y3+1

Problem 1731

If f(x,y)=2x2−yeyf(x, y)=2 x^{2}-y e^{y}f(x,y)=2x2−yey and the maximum rate of change of fff at (1,0)(1,0)(1,0) is aaa then a2a^{2}a2 equalAnswer: □\square□

Problem 1732

If f(x,y)=2x2−yeyf(x, y)=2 x^{2}-y e^{y}f(x,y)=2x2−yey and the maximum rate of change of fff at (1,0)(1,0)(1,0) is aaa then a2a^{2}a2 equalAnswer: □\square□

Problem 1733

Problem 1734

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 1735

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 1736

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 1737

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 1738

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 1739

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 1740

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 1741

f(x)=xex−xf(x)=x e^{x}-xf(x)=xex−x is concave up ou?

Problem 1742

Problem 1743

\qquad (D) y=0y=0y=0

Find the derivative $f^{\prime}\left(\frac{\pi}{4}\right)$ for the function $f(x)=\sin x(2+\cos x)$ .

Find the derivative $\frac{d y}{d x}$ for $y=\sin x-\cos 3 x$ .

Find $\frac{d y}{d x}$ for $y=\sin x-\cos 3 x$ at $x=45^{\circ}$ .

Find $\left.\frac{d y}{d x}\right|_{45^{\circ}}$ for $y=\sin x-\cos 3 x$ .

Cari turunan $f^{\prime}(x)$ dari fungsi $f(x)=\sin ^{2} x+\cos 3 x$ .

Find the derivative of $y=\sin 3x - \cos 3x$ at $x=45^{\circ}$ .

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

Find the derivative of the function $y=\frac{\sin x+\cos x}{x}$ .

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

Determine the slope, $m$ , of the tangent line to the curve $y=7+5 x^{2}-2 x^{3}$ at $x=a$ .

Compute $\Delta y / \Delta x$ for $y=4x-7$ over $[2,6]$ and find the instantaneous rate of change at $x=2$ .

Find the velocity of a particle with displacement $s=\frac{2}{t^{2}}$ at $t=a, t=1, t=2, t=3$ (in $\mathrm{m/s}$ ).

Find the critical points of $f(x, y)=x^{2}+2xy+2y^{2}-8x+2y$ and classify each as max, min, or saddle.

Find critical points of $t(x, y)=x^{3}-12xy+y^{3}$ and classify as max, min, or saddle.

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 $h \rightarrow 0$ .

Problems (1 point)
Let $f(x)=\int_{1}^{x} t^{8} d t$ . Evaluate the following. $\begin{array}{l} f^{\prime}(x)=\square \\ f^{\prime}(-4)=\square \end{array}$
Note: You can earn partial credit on this problem.

For the following equation, what is the instantaneous rate of change at $x=-1 ?$ $f(x)=-2 x^{3}-x^{2}$

Find $d y / d x$ if (1) $y=\frac{e^{4 x^{2}+2 x}}{2 x+1}$

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

Find $\frac{d y}{d x}$ of the following finctions: if $y=2 \cos 2 x-1$

Find $\frac{d y}{d x}$ of the following function (1) if $y=\frac{x^{2}}{x+1}$

Defferentiate with respect to $x: 4-x^{2}$

Find $d y / d x$ by implicit differentiation $x^{3} y^{5}+3 x=8 y^{3}+1$

If $f(x, y)=2 x^{2}-y e^{y}$ and the maximum rate of change of $f$ at $(1,0)$ is $a$ then $a^{2}$ equal
Answer: $\square$

If $f(x, y)=2 x^{2}-y e^{y}$ and the maximum rate of change of $f$ at $(1,0)$ is $a$ then $a^{2}$ equal
Answer: $\square$

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 $s(t)=-16 t^{2}+256 t$ .

$f(x)=x e^{x}-x$ is concave up ou?

$\qquad$ (D) $y=0$

B) If the function $f(x)=a x^{3}-3 x^{2}+b x+c$ has a local minimum at -5 and a local maximum at $x=\frac{-1}{2}$ and inflection point at $x=\frac{1}{4}$ find $a, b, c \in R$ .

Q3: A) Let M be a point moving on the curve $y^{2}=x^{3}$ such that rate of change the point $M$ getting away from of the origin point is 4 unit $/ \mathrm{s}$ . Find the rate of chang of the x -coordinate for M when $x=2$