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'
| 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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