Derivatives

Problem 1401

11. [5 marks] Find the absolute extrema of f(x)=x312xf(x) = x^3 - 12x on [1,3][-1, 3]. f(x)=3x212f'(x) = 3x^2 - 12 3(x24)3(x^2 - 4) (x2)(x+2)(x-2)(x+2) 2 f(2)f(2) f(1)f(-1) f(3)f(3)

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Problem 1402

13. III To throw a discus, the thrower holds it with a fully outstretched arm. Starting from rest, he begins to turn with a constant angular acceleration, releasing the discus after making one complete revolution. The diameter of the circle in which the discus moves is about 1.8 m . If the thrower takes 1.0 s to complete one revolution, starting from rest, what will be the speed of the discus at release?

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Problem 1403

If u=f(x)u=f(x), then du=f(x)dxd u=f^{\prime}(x) d x, and so it is helpful to look for some expression in ex36+exdx\int e^{x} \sqrt{36+e^{x}} d x for which the derivative is also present.
We see that 36+ex36+e^{x} is part of this integral, and the derivative of 36+ex36+e^{x} is \square , which is also present. Submit Skip (you cannot come back)

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Problem 1404

Find the derivative of excosxe^{x} \cos x.

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Problem 1405

Find the derivative of f(x)=4cos2x+logx+xf(x)=4 \cos 2 x+\log x+x.

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Problem 1406

A stone thrown from a 120 ft cliff at 148 ft/s, with gravity -32 ft/s². Find time to highest point, max height, time to beach, and impact velocity.

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Problem 1407

Find fx\frac{\partial f}{\partial x}, fy\frac{\partial f}{\partial y}, and evaluate at (1,-1) for f(x,y)=6xe5xyf(x, y)=6 x e^{5 x y}.

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Problem 1408

Find the second partial derivatives fxx(x,y),fyy(x,y),fxy(x,y),fyx(x,y)f_{xx}(x, y), f_{yy}(x, y), f_{xy}(x, y), f_{yx}(x, y) for f(x,y)=7x2yf(x, y)=7x^2y and evaluate at (1,1)(1,-1).

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Problem 1409

Find the partial derivatives fx\frac{\partial f}{\partial x}, fy\frac{\partial f}{\partial y}, and evaluate at (1,1)(1,-1) for f(x,y)=13,00030x+20y+8xyf(x, y)=13,000-30 x+20 y+8 x y.

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Problem 1410

Find the function f(x)f(x) given f(x)=exf^{\prime \prime \prime}(x)=e^{x}, f(0)=6f^{\prime \prime}(0)=6, f(0)=10f^{\prime}(0)=10.

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Problem 1411

Find the function f(x)f(x) where f(x)=9xf'(x)=9^{x} and f(4)=5f(4)=-5. What is f(x)f(x)?

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Problem 1412

Find f(x)f(x) given that f(x)=41x2f'(x)=\frac{4}{\sqrt{1-x^{2}}} and f(12)=8f\left(\frac{1}{2}\right)=8.

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Problem 1413

Use differentials to estimate the value of 1.24\sqrt[4]{1.2}. Compare the answer to the value of 1.24\sqrt[4]{1.2} found using a calculator.
Round your answers to six decimal places, if required. You can use a calculator, spreadsheet, browser, etc. to calculate the value. estimate using differentials == \square value found using a calculator == \square

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Problem 1414

Find the indicated derivative for the function. h(x) for h(x)=2x32x6h(x)=\begin{array}{l} h^{\prime \prime}(x) \text { for } h(x)=2 x^{-3}-2 x^{-6} \\ h^{\prime \prime}(x)=\square \end{array}

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Problem 1415

Find the intervals on which the graph of ff is concave upward, the intervals on which the graph of ff is concave downward, and the inflection points. f(x)=ln(x26x+34)f(x)=\ln \left(x^{2}-6 x+34\right)
For what interval(s) of xx is the graph of ff concave upward? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \square (Type your answer in interval notation. Type an exact answer. Use a comma to separate answers as needed.). B. The graph is never concave upward.

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Problem 1416

The graph of a function gg is shown below. Use the graph of the function to find its average rate of change from x=0x = 0 to x=2x = 2. Simplify your answer as much as possible.

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Problem 1417

1313 y=(tan3x)esin3xy = (\tan{3x}) \cdot e^{\sin{3x}}

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Problem 1418

Describe the graph of ff at the given point relative to the existence of a local maximum or minimum. Assume that f(x)f(x) is continuous on (,)(-\infty, \infty). (6,f(6))(6,f(6)) if f(6)=0f'(6) = 0 and f(6)=0f''(6) = 0
What is the best description of the graph of ff at the point (6,f(6))(6,f(6))?
A. Local minimum B. Neither C. Unable to determine from the given information D. Local maximum

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Problem 1419

Let A(x)=xx+7A(x) = x\sqrt{x} + 7. Answer the following questions.
1. Find the interval(s) on which AA is increasing. Answer (in interval notation):
2. Find the interval(s) on which AA is decreasing. Answer (in interval notation):
3. Find the local maxima of AA. List your answers as points in the form (a,b)(a, b). Answer (separate by commas):
4. Find the local minima of AA. List your answers as points in the form (a,b)(a, b). Answer (separate by commas):
5. Find the interval(s) on which AA is concave upward. Answer (in interval notation):
6. Find the interval(s) on which AA is concave downward. Answer (in interval notation):

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Problem 1420

Use curl to determine whether the vector field F=3x6y,x7+3y8z,y9 \mathbf{F} = \langle 3x^6 y, x^7 + 3y^8 z, y^9 \rangle is conservative.
(a) Find curl F\mathbf{F}.
curl F\mathbf{F} = a,a,a \langle \hphantom{a}, \hphantom{a}, \hphantom{a} \rangle .
Question Help: Video Message instructor Submit Part

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Problem 1421

2. Вычислить производную функции: a) y=10x3+3xx45x14y=10 x^{3}+\frac{3}{x \sqrt{x}}-\frac{4}{5} x^{-\frac{1}{4}} b) y=sinxlnxy=\sin x \cdot \ln x

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Problem 1422

fa(x)=ax3f_a(x) = ax^3 ga(x)=a2xg_a(x) = a^2x aR,a0a \in \mathbb{R}, a \neq 0
1.1 x=2,a=3x = 2, a = 3 1.2 x=2,a=1x = -2, a = 1 1.3 1.4 fa(x)=3ax2f'_a(x) = 3ax^2 0=3ax02    x0=00 = 3ax_0^2 \iff x_0 = 0 fa(x)=6axf''_a(x) = 6ax fa(0)=0f''_a(0) = 0 fa(x)=6af'''_a(x) = 6a fa(0)=6a0f'''_a(0) = 6a \neq 0
2. fk(x)=kx+cos(x)f_k(x) = kx + cos(x) mit kRk \in \mathbb{R} 2.1 k=0.25,k=0.5,k=1k = 0.25, k = 0.5, k = 1 2.2 P(01)P(0|1) 2.3 π2πfk(x)dx\int_{\pi}^{2\pi} f_k(x) \, dx Rückseite beachten!

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Problem 1423

Exercice 1 Les figures ci-dessous représentent les coordonnées polaires, r(t)r(t) et θ(t)\theta(t), d'un mobile m :
1. Représenter la trajectoire du mobile dans le plan (Oxy)(Oxy) pour t[0;4]t \in [0; 4].
2. Donner les expressions de r(t)r(t) et θ(t)\theta(t) dans chaque phase.
3. Déterminer la vitesse radiale et transversale dans chaque phase. En déduire le module du vecteur vitesse.
4. Calculer l'accélération tangentielle ata_t en déduisant la nature du mouvement dans chaque phase.
5. Le vecteur position est donné, en coordonnées cartésiennes, par : OMundefined=r(t)=x(t)i^+y(t)j^\overrightarrow{OM} = r(t) = x(t)\hat{i} + y(t)\hat{j} Retrouver les équations des abscisses et des ordonnées, x(t)x(t) et y(t)y(t), dans chaque phase.
6. Tracer les vecteurs position et vitesse, sur la trajectoire, à l'instant t=3t = 3 s.

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Problem 1424

The circumference of a circle is increasing at a rate of π2\frac{\pi}{2} meters per hour.
At a certain instant, the circumference is 12π12\pi meters.
What is the rate of change of the area of the circle at that instant (in square meters per hour)?
Choose 1 answer: A 6 B π4\frac{\pi}{4} C 3π3\pi D 36π36\pi

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Problem 1425

The radius of the base of a cylinder is increasing at a rate of 1 meter per hour and the height of the cylinder is decreasing at a rate of 4 meters per hour.
At a certain instant, the base radius is 5 meters and the height is 8 meters.
What is the rate of change of the volume of the cylinder at that instant (in cubic meters per hour)?
Choose 1 answer: A 20π20\pi B 180π180\pi C 180π-180\pi D 20π-20\pi

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Problem 1426

Given that y=2w4+3y=2 w^{4}+3, find ddw(y33sinw)\frac{d}{d w}\left(y^{3}-3 \sin w\right) in terms of only ww.

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Problem 1427

Question 4 (7 points) Find the slope of the line tangent to the curve given parametrically by x=2e6tx = 2e^{6t}, y=(t4)2y = (t-4)^2 at the point (x,y)=(2,16)(x,y) = (2,16). Your score will be based entirely on the work you submit - no need to fill the box below.

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Problem 1428

Use Newton's method to estimate the two zeros of the following function. Start with x0=0x_0 = 0 for the xx2+9x - x^2 + 9
If x0=0x_0 = 0, then x2=4.73684x_2 = -4.73684. (Do not round until the final answer. Then round to five decimal places as needed.)
If x0=4x_0 = 4, then x2=3.57218x_2 = 3.57218. (Do not round until the final answer. Then round to five decimal places as needed.)

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Problem 1429

Pearson Session Ended lab.pearson.com/Student/PlayerTest.aspx?testId=265408904 Ilege Algebra ce Final Question 20 of 45 This quize: This ques
A rancher needs to enclose two adjacent rectangular corrals, one for cattle and one for sheep. If the river forms one side of the corrals and 180 yd of fencing is available, find the largest total area that can be enclosed. \qquad
What is the largest total area that can be enclosed? \square yd2y d^{2}

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Problem 1430

\begin{tabular}{|c|c|c|} \hlinexx & 4 & 8 \\ \hlinef(x)f(x) & 11 & 6 \\ \hlinef(x)f^{\prime}(x) & -4 & -3 \\ \hline \end{tabular}
The table above gives selected values for a differentiable and decreasing function ff and its derivatives. Let gg be the decreasing function given by g(x)=f(4x)f(2x)g(x)=f(4 x)-f(2 x), where g(2)=f(8)g(2)=f(8)- f(4)=5f(4)=-5. What is (g1)(5)\left(g^{-1}\right)^{\prime}(-5) ?

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Problem 1431

A rectangle has one side of 10 cm. How fast is the area of the rectangle changing at the instant when the other side is 13 cm and increasing at 2 cm per minute? (Give units.) Answer:

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Problem 1432

The following situation involves a rate of change that is constant. Write a statement that describes how one variable changes with respect to the other, give the rate of change numerically (with units), and use the rate of change rule to answer any questions. You run along a path at a constant speed of 2.8 miles per hour. How far do you travel in 1.5 hours? in 3.1 hours?
Which statement describes this situation? A. The distance traveled varies with respect to time with a rate of change of 2.8 h/mi2.8 \mathrm{~h} / \mathrm{mi}. B. Time varies with respect to distance traveled with a rate of change of 2.8mi/h2.8 \mathrm{mi} / \mathrm{h}. C. Time varies with respect to distance traveled with a rate of change of 2.8 h/mi2.8 \mathrm{~h} / \mathrm{mi}. D. The distance traveled varies with respect to time with a rate of change of 2.8mi/h2.8 \mathrm{mi} / \mathrm{h}.
What is the distance traveled after 1.5 hours? \square miles (Type an integer or a decimal.)

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Problem 1433

The top and bottom margins of a poster are 8 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 388 square centimeters, find the dimensions of the poster with the smallest area.
Width = _______ (include units) Height = _______ (include units)
Note: You can earn partial credit on this problem. printed material

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Problem 1434

Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius rr. List the dimensions in non-decreasing order (the answer may depend on rr).

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Problem 1435

Find the point P on the graph of the function y=xy = \sqrt{x} closest to the point (2,0)(2, 0) The xx coordinate of P is:

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Problem 1436

For some positive constant CC, a patient's temperature change, TT, due to a dose, DD, of a drug is given by T=(C2D3)D2T = (\frac{C}{2} - \frac{D}{3})D^2.
What dosage maximizes the temperature change?
D=D =
The sensitivity of the body to the drug is defined as dT/dDdT/dD. What dosage maximizes sensitivity?
D=D =

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Problem 1437

Given f(x,y)=3x5+2xy3+4y2f(x, y) = 3x^5 + 2xy^3 + 4y^2, find
fx(x,y)=f_x(x, y) =
fy(x,y)=f_y(x, y) =

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Problem 1438

Find the average rate of change of g(x)=2x22xg(x)=-2 x^{2}-2 x from x=4x=-4 to x=1x=1. Simplify your answer as much as possible.

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Problem 1439

x(t)=at+bx(t) = at + b, where aa and bb are both nonzero.
Find the acceleration of the bug at any given time tt.

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Problem 1440

0t100 \le t \le 10 xP(t)x_P(t) xQ(t)x_Q(t) vQ(t)v_Q(t) aQ(t)a_Q(t)
d) For 0t100 \le t \le 10, particles P and Q move along the x axis. The position of particle P can be modeled by xP(t)x_P(t) as shown in the figure above. The position of particle Q is defined by xQ(t)x_Q(t). Selected values of xQ(t)x_Q(t), vQ(t)v_Q(t), and aQ(t)a_Q(t) are given in the table above. At what time tt are particles P and Q moving towards each other?

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Problem 1441

Question 5 4 pts The business manager of a company that produces in-ground outdoor spas determines that the cost (in dollars) of producing xx spas in thousands of dollars is given by C(x)=0.04x2+4.1x+100C(x) = 0.04x^2 + 4.1x + 100 Find the average rate of change in cost if production is changed from 10 in-ground outdoor spas to 11 in-ground outdoor spas. \$4,994 \$149,940 \$4,940 \$145,000

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Problem 1442

Find and simplify the difference quotient f(x+h)f(x)h\frac{f(x+h)-f(x)}{h} for the functions: 71. f(x)=4xf(x)=4x, 72. f(x)=7xf(x)=7x.

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Problem 1443

Simplify the difference quotient f(x+h)f(x)h\frac{f(x+h)-f(x)}{h} for f(x)=x24x+3f(x) = x^2 - 4x + 3, with h0h \neq 0.

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Problem 1444

Find the inflection point of g(x)=1x2e2tdtg(x)=\int_{1}^{x^{2}} e^{2 \sqrt{t}} d t and give the exact xx-coordinate.

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Problem 1445

Find the tangent line equation for the curve at the specified parameter value for each case.
1. x=t4+1,  y=t3+t;  t=1x=t^{4}+1, \; y=t^{3}+t; \; t=-1
2. x=tt1,  y=1+t2;  t=1x=t-t^{-1}, \; y=1+t^{2}; \; t=1
3. x=et,  y=tlnt2;  t=1x=e^{\sqrt{t}}, \; y=t-\ln t^{2}; \; t=1
4. x=cosθ+sin2θ,  y=sinθ+cos2θ;  θ=0x=\cos \theta+\sin 2 \theta, \; y=\sin \theta+\cos 2 \theta; \; \theta=0

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Problem 1446

Find the derivatives of these functions: 1) y=sin2x+(2x5)2y=\sin 2x + (2x-5)^{2} 2) y=cotx+sec2x+5y=\cot x + \sec^{2} x + 5

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Problem 1447

Find the derivative of these functions:
1. y=cot4xy=\cot ^{4} x
2. y=sec3xy=\sec ^{3} x
3. y=cosx2y=\cos x^{2}
4. y=secx+tanxsecxtanxy=\sqrt{\frac{\sec x+\tan x}{\sec x-\tan x}}
5. y=sin2x+(2x5)2y=\sin 2 x+(2 x-5)^{2}
6. y=cotx+sec2x+5y=\cot x+\sec ^{2} x+5
7. y=sec1secx+1y=\frac{\sec -1}{\sec x+1}
8. y=cosec3x3y=\operatorname{cosec} 3 x^{3}
9. y=sin3xx+1y=\frac{\sin 3 x}{x+1}
10. y=sin2x2x+5y=\frac{\sin 2 x}{2 x+5}
11. y=cot2xy=\cot 2 x
12. y=(3x+1)cos2xexy=\frac{(3 x+1) \cos 2 x}{e^{x}}
13. y=secxy=\sec x

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Problem 1448

Find the derivatives of these functions: 1. y=sin3xx+1y=\frac{\sin 3 x}{x+1}, 2. y=sin2x2x+5y=\frac{\sin 2 x}{2 x+5}, 3. y=cot2xy=\cot 2 x, 4. y=(3x+1)cos2xexy=\frac{(3 x+1) \cos 2 x}{e^{x}}, 5. y=secxy=\sec x.

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Problem 1449

Is the statement true or false? The derivative of f(x)f(x) is the instantaneous rate of change of y=f(x)y=f(x) with respect to xx. Choose A, B, C, or D.

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Problem 1450

Find the average velocity from t=2t=2 to t=3t=3 and t=2t=2 to t=4t=4, and the instantaneous velocity at t=2t=2.

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Problem 1451

Is it true or false that a derivative can exist where a function does not? Explain your reasoning.

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Problem 1452

Find the derivatives of these functions:
1. y=cotx+sec2x+5y=\cot x+\sec ^{2} x+5
2. y=1secx+1y=\frac{1}{\sec x+1} or y=sec(x1)secx+1y=\frac{\sec (x-1)}{\sec x+1}.

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Problem 1453

Find the derivative of f(x)=x2+9x6f(x)=-x^{2}+9x-6 using the limit definition: limh0f(x+h)f(x)h\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}.

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Problem 1454

6. [6 marks] Find the values of xx at which the function f(x)=(x23x)5f(x) = (x^2 - 3x)^5 has horizontal tangent lines.

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Problem 1455

11. The horizontal position of the pendulum of a grandfather clock can be modelled by h(t)=Acos(2πtT)h(t) = A\cos(\frac{2\pi t}{T}) where AA is the amplitude of the pendulum, tt is time, in seconds, and TT is the period of the pendulum. Determine the formula for the velocity of the pendulum at time tt.

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Problem 1456

Derivative Test 23 Remaining Time: 13:35:48
The following is a graph of the first derivative, f(x)f^{\prime}(x), of a function f(x)f(x).
Using the graph, answer the following questions about f(x)f(x). (Remember "increasing/decreasing at x=c\mathrm{x}=\mathrm{c} " means "increasing/decreasing on an open interval containing x=c\mathrm{x}=\mathrm{c} ") a) List all critical points of f(x)f(x) that lie in the interval 7x6-7 \leq x \leq 6. x=x= \square (If there is more than one xx-value, then separate with commas. For example, x=2,3x=2,3.) b) Which statement is true of f(x)f(x) at x=1x=-1 ? f(x)f(x) has a local maximum at x=1x=-1. f(x)f(x) has a local minimum at x=1x=-1. f(x)f(x) is increasing at x=1x=-1 f(x)f(x) is decreasing at x=1x=-1 c) Which statement is true of f(x)f(x) at x=6x=6 ? f(x)f(x) has a local maximum at x=6x=6. f(x)f(x) has a local minimum at x=6x=6. f(x)f(x) is undefined at x=6x=6. f(x)f(x) has a horizontal tangent at x=6x=6.

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Problem 1457

Question The height of a bottle rocket, in meters, is given by the function s(t)=15t2+64t+554s(t) = -15t^2 + 64t + 554. Find the average velocity of the bottle rocket over the intervals [4.999,5.0][4.999, 5.0] and [5.0,5.001][5.0, 5.001]. Use this information to approximate the instantaneous velocity of the bottle rocket at time t=5.0t = 5.0. (Round your answer to the nearest integer if necessary.) Provide your answer below: vinst=v_{\text{inst}} = m/s

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Problem 1458

Find drdθ\frac{d r}{d \theta}. r=5secθcscθdrdθ=5sec2θ5csc2θ\begin{array}{c} r=5 \sec \theta \csc \theta \\ \frac{d r}{d \theta}=5 \sec ^{2} \theta-5 \csc ^{2} \theta \end{array}

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Problem 1459

Given the function f(x)=x321xf^{\prime}(x)=x^{3}-21 x, determine all intervals on which ff is increasing.

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Problem 1460

The equations for free fall near the surfaces of two celestial bodies in outer space (ss in meters, tt in seconds) are: Celestial Body A, s=2.15t2s = 2.15t^2; Celestial Body B, s=13.89t2s = 13.89t^2. How long would it take a rock falling from rest to reach a velocity of 27.1msec27.1 \frac{m}{sec} on each celestial body? It would take ___ sec on Celestial Body A and ___ sec on Celestial Body B. (Round to three decimal places as needed.)

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Problem 1461

Question 22
A local office supply store has an annual demand for 20,000 cases of photocopier paper per year. It costs \$3 per year to store a case of paper, and it costs \$80 to place an order. How many cases of paper should be ordered at a time so that inventory costs are minimized?
1033 cases 1,066,667 cases 327 cases 730 cases

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Problem 1462

The function f(x)f(x) is twice differentiable on the closed interval [3,4][-3, 4]. Selected values of f(x)f(x), f(x)f'(x), and f(x)f''(x) are given in the table above.
| f(x)f''(x) | f(x)f'(x) | f(x)f(x) | xx | |---|---|---|---| | 5-5 | 22 | 1919 | 3-3 | | 33 | 00 | 2323 | 1-1 | | 2-2 | 1-1 | 77 | 00 | | 66 | 33 | 2-2 | 11 | | 4-4 | 44 | 00 | 22 | | 88 | 1-1 | 55 | 44 |
Name:
Section 2: Unit 5 Review
Name:
maximum value of the functions below on the given intervals.

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Problem 1463

Problems 9-12 use the graph of the derivative of // above.
9. On what interval(s) is the graph of ff increasing? a. (3,2)(-3,-2) b. (1,1)(-1,1) and (3,4)(3,4) c. (1,1)(-1,-1) d. (3,4)(3,4)

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Problem 1464

C(x)=4000+55x+0.1x2C(x) = 4000 + 55x + 0.1x^2 C(x)=totalCx=4000x+55+0.1x\overline{C}(x) = \frac{total C}{x} = \frac{4000}{x} + 55 + 0.1x a)a) instantaneous rate of change of C(x)\overline{C}(x) C(x)=4000x2+0.1\overline{C}(x)' = \frac{-4000}{x^2} + 0.1
b)b) level of production C(x)=0C'(x) = 0 4000x2+0.1=0\frac{-4000}{x^2} + 0.1 = 0 4000=0.1x24000 = 0.1x^2 x2=40000.1x^2 = \frac{4000}{0.1} x=200x = 200 unit
c)c) C(x)=55+0.2xC'(x) = 55 + 0.2x C(200)=95C'(200) = 95 C(200)=$95\overline{C}(200) = \$95

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Problem 1465

NYA Module 6: Problem 2 (1 point)
The function f(x)=2x318x2+48x11f(x)=2 x^{3}-18 x^{2}+48 x-11 has two critical values. The smaller one equals \square and the larger one equals \square
Note: You can earn partial credit on this problem. Preview My Answers Submit Answers
You have attempted this problem 0 times. You have unlimited attempts remaining. Email Instructor

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Problem 1466

Find derivative: f(x)=sin(sin(ex))f(x) = \sin(\sin(e^x))

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Problem 1467

15. f(x)=sinx21f(x) = \sin\sqrt{x^2 - 1}

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Problem 1468

The position of a body at time tsec\mathrm{t} \sec is s=t39t2+24tm\mathrm{s}=\mathrm{t}^{3}-9 \mathrm{t}^{2}+24 \mathrm{t} \mathrm{m}. Find the body's acceleration each time the velocity is zero.
The body's acceleration each time the velocity is zero is \square m/s2\mathrm{m} / \mathrm{s}^{2} (Simplify your answer. Use a comma to separate answers as needed.)

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Problem 1469

Determine the location of each local extremum of the function. f(x)=x3+7x2+8x+3f(x)=x^{3}+7 x^{2}+8 x+3 A. Local maximum at 43-\frac{4}{3}; local minimum at -2 B. Local maximum at 2 ; local minimum at 43\frac{4}{3} C. Local maximum at 23\frac{2}{3}; local minimum at 4 D. Local maximum at -4 ; local minimum at 23\frac{-2}{3}

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Problem 1470

Find dydx\frac{dy}{dx} for y=2x+3sinxy = \frac{2}{x} + 3\sin{x}.
ddx(2x+3sinx)=2x2+3\frac{d}{dx} \left( \frac{2}{x} + 3\sin{x} \right) = -\frac{2}{x^{-2}} + 3

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Problem 1471

f(x)=5x28x2+6x3f(x) = 5x^2 - 8x^{-2} + 6x^{-3}
Find f(x)f'(x).
f(x)=f'(x) =

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Problem 1472

a=a = v=17costv = 17 \cos{t} s=17sints = 17 \sin{t}
A weight hanging from a spring bobs up and down with a position function s=f(t)s = f(t) (ss in meters, tt in seconds). What are its velocity and acceleration at time tt? Describe its motion.
Question 5, 2.5.11 Part 2 of 3
HW Score: 91.07%, 12.75 of 14 points Points: 0.25 of 1

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Problem 1473

Find the derivative of the implicit function: y2+siny+x+y+ysinx+ysiny+cos(y2+1)+siny+x2+4y=cosxy^2 + \sin y + x + y + y \sin x + y \sin y + \cos(y^2 + 1) + \sin y + x^2 + 4y = \cos x and 3xy2+cosy2=2x3+53xy^2 + \cos y^2 = 2x^3 + 5 and tan(5y)ysinx+3xy2=9\tan(5y) - y \sin x + 3xy^2 = 9.

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Problem 1474

Find the derivative f(x)f^{\prime}(x) of the function f(x)=9x2f(x)=9x-2.

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Problem 1475

Find the derivative f(x)f'(x) of the function f(x)=x25f(x) = x^2 - 5.

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Problem 1476

Find the derivative f(x)f^{\prime}(x) for the function f(x)=7x2+xf(x)=7 x^{2}+x.

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Problem 1477

Find the derivative R(t)R'(t) for R(t)=0.1t2R(t) = -0.1 t^{2} and calculate R(1)R'(1).

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Problem 1478

Find the derivative of R(t)=0.1t2R(t)=-0.1 t^{2} and evaluate it at t=1t=1. What is R(1)R'(1)?

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Problem 1479

Find the equation of the tangent line to the curve y=4xy=\frac{4}{x} at the point (3,43)\left(3, \frac{4}{3}\right).

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Problem 1480

Estimate the growth rate of Whooping cranes in 2007 using rr and given values. Choose all correct answers. dN/dt=rNtr=ln( lambda )dN/dt=0.086258r=ln(1.09) d N / d t=r^{*} N_{t} \\ r=\ln (\text { lambda }) \\ d N / d t=0.086^{*} 258 \\ r=\ln (1.09)

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Problem 1481

Find the derivative U(t)U'(t) of U(t)=5.1t21.2tU(t)=5.1 t^{2}-1.2 t and evaluate it at t=8t=8.

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Problem 1482

Find the tangent line equation to the curve y=x2+5x4y=-x^{2}+5x-4 at x=1x=1.

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Problem 1483

Find dydx\frac{d y}{d x} for y=2x4+9x24xy=\frac{2 x^{4}+9 x^{2}}{4 x}. Choices include options A to E.

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Problem 1484

Find dydx\frac{d y}{d x} for y=2x4+9x24xy=\frac{2 x^{4}+9 x^{2}}{4 x}. Options: A, B, C, D, E.

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Problem 1485

Find dydx\frac{d y}{d x} for y=2x4+9x24xy=\frac{2 x^{4}+9 x^{2}}{4 x}. Choices are A, B, C, D, E.

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Problem 1486

Find the derivative of f(x)=3x+54xf(x)=\frac{3x+5}{4-x}.

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Problem 1487

Find the derivative of f(x)=(2xx3)2x2f(x)=(2x-x^{3})\sqrt{2-x^{2}}.

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Problem 1488

Find the derivative of f(x)=25xcos10xf(x)=\frac{2-5 x}{\cos 10 x}.

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Problem 1489

y=y' =
y=lnx2+2xy = \ln{\frac{x^2 + 2}{x}}
Find the derivative of the function.

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Problem 1490

Using the Rules for Finding the Derivative of a Function.
For the following exercise, assume that f(x)f(x) and g(x)g(x) are both differentiable functions of xx with values as given in the following table. Use the table to calculate the following derivatives. \begin{tabular}{|l|c|ccc|} \hlinexx & 1 & 2 & 3 & 4 \\ \hlinef(x)f(x) & 19 & 9 & 3 & 1 \\ \hlineg(x)g(x) & -2 & 0 & 2 & 16 \\ \hlinedfdx\frac{d f}{d x} & -12 & -8 & -4 & 0 \\ \hlinedgdx\frac{d g}{d x} & 6 & 0 & 6 & 24 \\ \hline \end{tabular}
1. If h(x)=xf(x)+4g(x)h(x)=x f(x)+4 g(x) then, h(1)=h^{\prime}(1)= \square
2. If h(x)=f(x)g(x)h(x)=\frac{f(x)}{g(x)} then, h(2)=h^{\prime}(2)= \square
3. If h(x)=2x+f(x)g(x)h(x)=2 x+f(x) g(x) then, h(3)=h^{\prime}(3)= \square
4. If h(x)=1x+g(x)f(x)h(x)=\frac{1}{x}+\frac{g(x)}{f(x)} then, h(4)=h^{\prime}(4)= \square If you believe that a value is not defined, type "ND"

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Problem 1491

The Differential of a Function.
Find the differential of the given function. Then, evaluate the differential at the indicated values.
If y=xcos(x)y=x \cos (x) then the differential of yy is dy=d y=\square
Note: Type dxd x for the differential of xx Evaluate the differential of yy at x0=4.71239,dx=0.3x_{0}=4.71239, d x=0.3 dyx=x0dx=0.3=\left.d y\right|_{\substack{x=x_{0} \\ d x=0.3}}=\square
Note: Enter your answer accurate to 4 decimal places if it is not an Integrer.

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Problem 1492

1. [-/1 Points] DETAILS MY NOTES TANAPCALCBR10 3.
Find dydx\frac{d y}{d x} by implicit differentiation. dydx=(4x+3y)1/3=x2\frac{d y}{d x}=\square(4 x+3 y)^{1 / 3}=x^{2} Need Help? Read It Watch It

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Problem 1493

For the function f(x)=8x26x1f(x)=-8 x^{2}-6 x-1, find the equation of the tangent line at x=12x=-12.

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Problem 1494

b. y=114x+744y=\frac{1 \frac{1}{4} x+\frac{7}{4}}{4}
12. Find the derivative of the following function. f(x)=6xx+2f(x)=\begin{array}{r} f(x)=\frac{6 x}{x+2} \\ f^{\prime}(x)= \end{array}

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Problem 1495

Let u(x)=sin(x)u(x) = \sin(x) and v(x)=x12v(x) = x^{12} and f(x)=u(x)v(x)f(x) = \frac{u(x)}{v(x)}. Find u(x)=u'(x) = v(x)=v'(x) = f=f' =
!!! The challenge is that the Quotient Differentiation Rule on Earth, f=uvuvv2f' = \frac{u'v - uv'}{v^2}, is "twisted" on Z Planet as the following: f=uvuvv2f' = \frac{u'v' - uv}{v^2} (all the other rules might be also alternated but they are not given, so don't use them).
Here Is The Story that happened to you earlier... but now you're on board the spaceship #1573343500, and the captain is asking to solve tricky "ZP" (Z Planet) problem (you know what it means when captain is "asking"... that's an order): Next Question

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Problem 1496

Points: 0 of 1 Save
Determine where the function is (a) increasing; (b) decreasing; and (c) determine where relative extrema occur. Do not sketch the graph. y=x332x2+12x3y=-\frac{x^{3}}{3}-2 x^{2}+12 x-3 (a) For which interval(s) is the function increasing? Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The function is increasing on (6,2)(-6,2). (Type your answer in interval notation. Use a comma to separate answers as needed.) B. The function is never increasing. (b) For which interval(s) is the function decreasing? Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The function is decreasing on \square . (Type your answer in interval notation. Use a comma to separate answers as needed.) B. The function is never decreasing.

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Problem 1497

The graphs of y=x+1y = x + 1 intersects the graph of y=5cos(xπ3)y = 5\cos\left(x - \frac{\pi}{3}\right) at the point AA in the first quadrant. By using Newton-Raphson method, find the coordinates of AA correct to four decimal places.

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Problem 1498

f(x)={2x+3si x<212x2si x2f(x) = \begin{cases} -2x+3 & \text{si } x < 2 \\ \frac{1}{2}x - 2 & \text{si } x \ge 2 \end{cases}
1. ff est-elle continue sur R\mathbb{R}?
2. ff est dérivable sur R\mathbb{R}?

Exercice 2 (8 pts) :
Partie A : Étude d'une fonction auxiliaire Soit gg la fonction définie sur R\mathbb{R} par g(x)=(x+2)ex42g(x) = (x+2)e^{x-4} - 2
1. Déterminer la limite de gg en ++\infty.
2. Démontrer que la limite de gg en -\infty vaut 2-2.
3. On admet que la fonction gg est dérivable sur R\mathbb{R} et on note gg' sa dérivée. Calculer g(x)g'(x) pour tout réel xx puis dresser le tableau de variations de gg.
4. Démontrer que l'équation g(x)=0g(x) = 0 admet une unique solution α\alpha sur R\mathbb{R}.
5. En déduire le signe de la fonction gg sur R\mathbb{R}.
6. À l'aide de la calculatrice, donner un encadrement d'amplitude 10310^{-3} de α\alpha.

Partie B : Étude de la fonction Soit ff la fonction définie sur R\mathbb{R} par f(x)=x2x2ex4f(x) = x^2 - x^2e^{x-4}
1. Résoudre l'équation f(x)=0f(x) = 0 sur R\mathbb{R}.
2. On admet que la fonction ff est dérivable sur R\mathbb{R} et on note ff' sa fonction dérivée. Montrer que, pour tout réel xx, f(x)=xg(x)f'(x) = -xg(x) où la fonction gg est celle définie à la partie A.
3. Étudier les variations de la fonction ff sur R\mathbb{R}.
4. Démontrer que le maximum de la fonction ff sur [0;+[[0 ; +\infty[ est égal à α3α+2\frac{\alpha^3}{\alpha+2}.

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Problem 1499

Determining Concavity In Exercises 5-16, determine the open intervals on which the graph of the function is concave upward or concave downward.
5. f(x)=x24x+8f(x)=x^{2}-4 x+8
6. g(x)=3x2x3g(x)=3 x^{2}-x^{3}
7. f(x)=x43x3f(x)=x^{4}-3 x^{3}
8. h(x)=x55x+2h(x)=x^{5}-5 x+2
9. f(x)=24x2+12f(x)=\frac{24}{x^{2}+12}
10. f(x)=2x23x2+1f(x)=\frac{2 x^{2}}{3 x^{2}+1}

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Problem 1500

2. ddxx2sin(x)\frac{d}{dx} x^2 \sin(x).

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