homework homework assignment #15 read section 3.7 page 170, exercises: 1 – 49 (eoo), 43 rogawski...

Homework

Homework Assignment #15 Read Section 3.7 70, Exercises: 1 – 49 (EOO), 43

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

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

1.


sin , 4

y x x

2 2 2sin , cos cos

4 4 2 4 2 4 4 2

2 2

2 2 4

y y x y

y x

Homework, Page 170 Find the derivative of each function.

5.


sin cosf x x x

2 2

2 2

sin cos sin , cos ; cos , sin

sin sin cos cos cos sin

cos sin

f x x x u x u x v x v x

f x uv vu x x x x x x

f x x x


9.


3 sinf x x x

3 3 2

3 2 3 2

3 2

sin , 3 ; sin , cos

cos sin 3 cos 3 sin

cos 3 sin

f x x x u x u x v x v x

f x uv vu x x x x x x x x

f x x x x x


13.


2cosh e

2 2

2

cos , ; cos , 2cos sin

2cos sin cos cos 2sin cos

cos 2sin cos

h e u e u e v x v x x

h uv vu e x x x e e x x x

h x e x x x


17.


2

sec xf x

x

22

2

22 2

3

secsec , sec tan ; , 2

sec tan sec 2

sec tan 2sec

x uf x u x u x x v x v x

x v

x x x x xvu uvf x

v x

x x x xf x

x


21.


sin 2

xf x

x

2 22

2

, 1; sin 2, cossin 2

sin 2 1 cos sin cos 2

sin 2 sin 2

sin cos 2

sin 2

x uf x u x u v x v x

x vx x xvu uv x x x

f xv x x

x x xf x

x


25.


sec xg x

x

2 2

2

secsec , sec tan ; , 1

sec tan sec 1

sec tan sec

x ug x u x u x x v x v

x vx x x xvu uv

g xv x

x x x xg x

x

Homework, Page 170 Calculate the second derivative.

29.


sing

sin , 1; sin , cos

cos sin 1 , 1; cos , sin

sin cos 1 cos 2cos sin

2cos sin

g u u v v

g uv vu u u v v

g

g

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

33.


2sin 3cos , 3

y x x x

3 1 32sin 3cos 2 3 3

3 3 3 2 2 2

2cos 3 sin 2cos 3sin

1 3 3 32cos 3sin 2 3 1

3 3 3 2 2 2

3 3 3 3 33 1 1

2 2 3 2

y

y x x x x

y

y x y x

3 33

3 2 2

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

37.


cos , 0xy e x x

0

0

cos , ; cos , sin , 0

0 cos 0 1 1 1

sin cos 0 cos 0 sin 0 1 1 0 1

1 1 0 1

x x x

x x

y e x u e u e v x v x x

y e

y e x x e y e

y x y x

Homework, Page 170 Verify the formula.

41.


csc csc cotd

x x xdx

2 2

1 1csc 1, 0; sin , cos

sin sinsin 0 1 cos cos 1 cos

csc csc cotsin sin sinsin

csc csc cot

d d ux u u v x v x

dx dx x x vx xd x x

x x xdx x x xx

dx x x

dx

Homework, Page 170 43. Calculate the first five derivatives of f (x) = cos x. Then determine f (8) and f (37).


4 5

8 37 36

8 37

cos sin cos sin

cos sin

cos cos sin

cos sin

f x x f x x f x x f x x

f x x f x x

d df x x f x f x x x

dx dx

f x x f x x

Homework, Page 170 45. Calculate f ′(x) and f ′″(x) where f (x) = tan x


2

2

2 4 2 2

tan sec sec sec

sec sec tan sec sec tan 2sec sec tan

2 sec sec sec sec sec tan tan sec tan sec tan

sec 2sec 4sec tan

f x x f x x x x

f x x x x x x x x x x

f x x x x x x x x x x x x

f x x f x x x x

Homework, Page 170 49. The height at time t (s) of a weight, oscillating up and down at the end of a spring, is s(t) = 300 + 40 sin t cm. Find the velocity and acceleration at t = π/3 s.


2

300 40sin 40cos

40 sin 40sin

140cos 40 20

3 3 2

340sin 40 20 3

3 3 2

20 cm/s3

20 3 cm/s3

s t t v t s t t

a t v t s t t t

v

a

v

a


Jon Rogawski

Calculus, ET

First Edition

Chapter 3: DifferentiationSection 3.7: The Chain Rule

Composite Functions

Composite functions are combinations of simpler functions, for instance f (x) = sin ex. None of the rules of differentiation we learned thus far permit us to differentiate a composite function.

Remembering that the composition of f (x) and g (x) is written as f ○ g (x) or f (g (x)), we use the Chain Rule to differentiate composites.



If we choose, we may also represent a composite function of f (x) and u (x) as f (u). The chain rule then becomes:

which may also be written as:

Example, Page 1788. Calculate d/dx f (x2 + 1) for the following choices of f (u ):


3 22a sin b 3 c f u u f u u f u u u

Example, Page 178Find the derivative of f ○ g.

16.


2 1 sinf u u g x x

homework homework assignment #15 read section 3.7 page 170, exercises: 1 – 49 (eoo), 43 rogawski...

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