Math 2413 - Thomas - Practice Exam 3

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SHOW YOUR WORK.

Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.1) y = cos3 x

2) y = tan ! - 8x

Given y = f(u) and u = g(x), find dy/dx = f′ (g(x))g ′ (x).3) y = sin u, u = 9x + 16

Suppose that the functions f and g and their derivatives with respect to x have the following values at the given valuesof x. Find the derivative with respect to x of the given combination at the given value of x.

4)x f(x) g(x) f ′ (x) g ′ (x)3 1 9 8 34 3 3 2 -5

g(x), x = 3

5)x f(x) g(x) f ′ (x) g ′ (x)3 1 9 6 34 3 3 5 -5

f(x) + g(x), x = 3

Find the derivative of the function.6) y = x7cos x - 6x sin x - 6 cos x

7) y = (x + 1)2(x2 + 1)-3

Find dy/dt.8) y = cos( 8t + 11)

Find y ′ ′ .9) y = 7x + 4

10) y = 3x4(2x + 7)2

Use implicit differentiation to find dy/dx.11) x4 = cot y

12) x = sec(5y)

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Solve the problem.13) The position (in centimeters) of an object oscillating up and down at the end of a spring is given by

s = A sin km t at time t (in seconds). The value of A is the amplitude of the motion, k is a measure of the

stiffness of the spring, and m is the mass of the object. How fast is the object moving when it is movingfastest?

At the given point, find the slope of the curve or the line that is tangent to the curve, as requested.14) 3x2y - ! cos y = 4!, slope at (1, !)

15) x3y3 = 8, tangent at (2, 1)

Use implicit differentiation to find dy/dx and d2y/dx2.16) y2 - x2 = 4

17) x3/5 + y3/5 = 3

Find dy/dx by implicit differentiation.18) x1/3 - y1/3 = 1

At the given point, find the line that is normal to the curve at the given point.19) x3y3 = 8, normal at (2, 1)

Find the derivative of y with respect to x, t, or !, as appropriate.20) y = ln 8x2

21) y = ln 1 + xx3

Find the derivative of y with respect to the independent variable.22) y = 9cos !"

Find the indicated tangent line.23) Find the tangent line to the graph of f(x) = 2e-8x at the point (0, 2).

Find the derivative of the function.24) y = log |-7x|

25) f(x) = log8 (x4 + 1)

Use logarithmic differentiation to find the derivative of y.26) y = sin x 5x + 8

2

27) y = x x5 + 2(x + 9)1/3

Use logarithmic differentiation to find the derivative of y with respect to the independent variable.28) y = (ln x)ln x

Find the derivative of y with respect to x.

29) y = -csc-1 6x + 119

30) y = 3x3 sin-1 x

Find the value of df-1/dx at x = f(a).31) f(x) = 2x2, x ≥ 0, a = 4

32) f(x) = x2 - 4x + 7; a = 5

Find the formula for df-1/dx.

33) f(x) = 12 x +

74

Provide an appropriate response.34) Consider the graphs of y = cos-1 x and y = sin-1 x. Does it make sense that the derivatives of these functions

are opposites? Explain.

Solve the problem.35) Water is falling on a surface, wetting a circular area that is expanding at a rate of 4 mm2/s. How fast is the

radius of the wetted area expanding when the radius is 163 mm? (Round your answer to four decimalplaces.)

36) A piece of land is shaped like a right triangle. Two people start at the right angle of the triangle at the sametime, and walk at the same speed along different legs of the triangle. If the area formed by the positions ofthe two people and their starting point (the right angle) is changing at 2 m2/s, then how fast are the peoplemoving when they are 3 m from the right angle? (Round your answer to two decimal places.)

Solve the problem. Round your answer, if appropriate.37) Water is being drained from a container which has the shape of an inverted right circular cone. The container

has a radius of 3.00 inches at the top and a height of 5.00 inches. At the instant when the water in thecontainer is 2.00 inches deep, the surface level is falling at a rate of 1.6 in./sec. Find the rate at which water isbeing drained from the container.

38) Electrical systems are governed by Ohm's law, which states that V = IR, where V = voltage, I = current, and R= resistance. If the current in an electrical system is decreasing at a rate of 9 A/s while the voltage remainsconstant at 30 V, at what rate is the resistance increasing (in #/sec) when the current is 40 A? (Do not roundyour answer.)

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Answer KeyTestname: 2413,PRACTICEEXAM3,FA14

1) y = u3; u = cos x; dydx

= - 3 cos2 x sin x

2) y = tan u; u = ! - 8x; dydx

= 8x2

sec2 ! - 8x

3) 9 cos (9x + 16)

4) 12

5) 92 10

6) -x7 sin x + 7x6 cos x - 6x cos x

7) -2(x + 1)(2x2 + 3x - 1)(x2 + 1)4

8) 48t + 11

sin( 8t + 11)

9) - 494(7x + 4)3/2

10) 360x4 + 1680x3 + 1764x2

11) - 4x3

csc2 y

12) 15 cos(5y) cot(5y)

13) A km cm/sec

14) -2!

15) y = - 12x + 2

16) dydx = xy; d2ydx2

= y2 - x2

y3

17) dydx = - y

2/5

x2/5; d2ydx2

= 2x3/5 + 2y3/5

5x7/5y1/5

18) yx2/3

19) y = 2x - 3

20) 2x

21) -6 - 5 x2x(1 + x )

22) -!9cos !" ln 9 sin !"23) y = -16x + 2

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Answer KeyTestname: 2413,PRACTICEEXAM3,FA14

24) 1x(ln 10)

25) 4x3

(ln 8) (x4 + 1)

26) sin x 5x + 8 cot x + 52(5x + 8)

27) x x5 + 2(x + 9)1/3

1x

+ 5x4

2x5 + 4 - 13x + 27

28) ln (ln x) + 1x (ln x)

ln x

29) 54

(6x + 11) (6x + 11)2 - 81

30) 3x3

1 - x2 + 9x2 sin-1 x

31) 116

32) 1633) 234) Yes, They both have domains -1 ≤ x ≤ 1. They have the same basic shape with opposite slopes. Since the slopes are

opposites the derivatives will be opposites.35) 0.0039 mm/s36) 0.67 m/s37) 7.24 in.3/s

38) 27160 #/sec

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