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Question 4

Let f be a function defined on the closed interval -3 ~ x ~ 4 with

f (O) - 3. The graph of f', the derivative of f, consists of one line

segment and a semicircle, as shown above.

(a) On what intervals, if any, is f increasing? Justify your answer.

(b) Find the ~coordinate of each point of inflection of the graph of f

on the open interval -3 ~ 4. Justify your answer.

(c) Find an equation for the line tangent to the graph of f at the

point (O,3).

(d) Find f ( -3) and f ( 4 ) . Show the work that leads to your answers.

Graph of f'

2{

| 1 : x - O and x - 2 only

2: ~

I I : justification

| I : equation

(b) x - O and x - 2

f' changes from decreasing to increasing at

x O and from increasing to decreasing at

_

_

(c) f'(0) = -2

Tangent line is y - -2x + 3.

(d) f (0) - f(-3) = fo_3f '(t)dt

I 1 3

= 2- (I)(I) - ~ (2)(2) = -`2

3 9

f (-3) - f(0) + ~ = ~

f (4) - f (0) - fo4 f '(t) dt

- -(8 - ~ (2)2 7r) - -8 + 27r

f(4) - f(O) - 8 + 2P - -5 + 2P

I : ~(~- 2)

(difference of areas

of triangles)

1 : answer for f(-3) using FTC

I : ~(8 - ~(2)27r)

(area of rectangle

area of semicircle)

1 : answer for f(4) using FTC

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