Homework Homework Assignment #14 Read Section 3.6 Page 165, Exercises: 1 – 49 (EOO) Rogawski...

Homework

Homework Assignment #14 Read Section 3.6 Page 165, Exercises: 1 – 49 (EOO)

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Homework, Page 165Calculate the second and third derivatives.


21. 14y x

28

28

0

y x

y

y



345.

3y r

2 4

8

8

y r

y r

y



19. y z

z

1

1 1 2

3

4

1

1 1

2

6

y z z zz

y z z

y z

y z



13. 2 xy x e

2 x

x

x

y e

y e

y e

Homework, Page 165Calculate the derivative indicated.


23 2

2

1

17. , 4 3t

d yy t t

dt

4

2 255

2 2

1

2

2

1

12 6

48 6 48 1 6 54

54

t

t

dyt t

dt

d y d yt

dt dt

d y

dt



43

44

16

21. , t

d xx t

dt

27 11

4 42

3 415 19

4 43 4

419 54

4

16

45

4

16

3 21

4 16

231 3465

64 256

3465 346516 2.581 10

256 134,217,728

34652.581 10

134,217,728

t

t

dx d xt t

dt dt

d x d xt t

dt dt

d x

dt

d x

dt



125. 1 ,

1h h x

x

12

2 2

2 3 12 2

4

2 32

4

1 11 0 12 2

1 1

1 1 11 2 1

4 2 2

1

1 11 1

4 2

1

x xx

h xx x

x x x xx

h xx

x x xx

x


25. Cont’d


2 32

4

1 11 1 1 1 1

4 2 1 1 1 11

16 81 1

11

8

h

h

Homework, Page 16529. Calculate y (k) (0) where y = x4 + ax3 + bx2 + cx + d , with a, b, c, and d constant for 0≤ k ≤ 5.


0 4 3 2

1 3 2

2 2

3

4

5

4 3 2

12 6 2

24 6

24

0

y x ax bx cx d

y x ax bx c

y x ax b

y x a

y

y

Homework, Page 165Find a general formula for f (n) (x).


133. 1y x

2

3

4

1

1

2 1

6 1

1 ! 1n nn

y x

y x

y x

y n x

Homework, Page 16537. (a) Find the acceleration at t = 5 min of a helicopter whose height (in ft) is h (t) = – 3t3 + 400t.

(b) Plot the acceleration h″ (t) for 0 ≤ t ≤ 6. How does the graph show that the helicopter is slowing during this interval?

The negative values of the function indicate decreasing velocity and, therefore, decreasing speed.


3 23 400 9 400 18

5 18 5 90 ft/min 5 90 ft/min

h t t t h t t h t t

h h

Homework, Page 16541. Figure 7 shows the graph of the position of an object as a function of time. Determine the intervals on which the acceleration is positive.

Acceleration is positive where the

slope of the position curve is

increasing. This appears to be

on (10, 20) and (30,40).


Homework, Page 16545. Which of the following descriptions could not apply to Figure 8? Explain

(a) Graph of acceleration when velocity is constant.

(b) Graph of velocity when acceleration is constant.

(c) Graph of position when acceleration is zero.

The slope of the curve would be

units of distance divided by units

of time which yields units of

velocity, so neither (a) nor (b) apply,

but (c) does, as the slope of the curve is not changing.


Homework, Page 16549. According to one model that attempts to account for air resistance, the distance s(t) (in feet) travel by a falling raindrop satisfies:

where D is the raindrop diameter and g = 32 ft/s2. Terminal velocity vterm is defined as the velocity at which the drop has zero acceleration.

(a) Show that


22

2

0.0005d s dsg

dt D dt

term 2000v gD2 22

term 2

2

term

0.0005 0.0005When , 0

2000 20000.0005

ds d s ds dsv g g

dt dt D dt D dt

ds gDgD v gD

dt

Homework, Page 16549. Continued

(b) Find vterm for drops of diameter 0.003 and 0.0003ft.

(c) In this model do rain drops accelerate more rapidly at higher or lower velocities?

Rain drops accelerate more rapidly at lower velocities.


term

term

0.003 2000 32 0.003 13.856 ft/s

0.0003 2000 32 0.0003 4.382 ft/s

v

v


Chapter 3: DifferentiationSection 3.6: Trigonometric Functions

Jon Rogawski

Calculus, ET

First Edition


Recalling that all angles will be measured in radians, unless otherwisestated, the derivatives of the sine and cosine functions are given below.



Compare the slope of the graph of y = cos (x) to the values of itsderivative y = – sin (x) for any x.

y = - sin (x)

y = cos (x)

Example, Page 170 Find an equation of the tangent line at the point indicated.

2.


cos , 3

y x x

Example, Page 170 Find the derivative of each function.

6.


2 cosf x x x

Example, Page 170 Find the derivative of each function.

10.


sin xf x

x


The derivatives of the other four trig functions may be found by using the definitions of the functions in terms of sine and cosineand differentiating using the quotient rule. The results are given inTheorem 2.


Example, Page 170 Calculate the second derivative.

28.


tanf x x


Example, Page 170 Find an equation of the tangent line at the point specified.

32.


tan , 4

y

Example, Page 170 44. Find y(157), where y = sin x.


Homework

Homework Assignment #15 Read Section 3.7 Page 170, Exercises: 1 – 49 (EOO), 43


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Homework Homework Assignment #14 Read Section 3.6 Page 165, Exercises: 1 – 49 (EOO) Rogawski...

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