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Exercise 2. In parts (a)-(j) below, find each max/min point (using the second derivative test to be sure what type of point it is), point of inflection, and asymptote (vertical, horizontal, or slanted), if there are any. Also indicate where the curve is concave up and concave down. Sketch the graph.

(a)

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(b) ,

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(c) ,

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(d) ,

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(e) ,

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(f) ,

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(g) ,

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(h) ,

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(i) ,

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(j) .

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Exercise 3: Suppose:

> 0 for | x | > 2 ;
for | x | < 2 ;
< 0 for x < 0 ; and
> 0 for x > 0

From the given information, sketch a possible graph of f(x) . How could your answer vary and still be correct?

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Exercise 4. Suppose <0 for x<1 and >0 for . Suppose also that f(1) = 1 . Sketch a possible graph of f(x) , assuming that

i) = 0 , ii) > 0 and iii) < 0 .

Exercise 5 : Below is a sketch of .

(a) Find the exact coordinates (x,y) of the local maxima and local minima.
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(b) For what values of x is the function concave upward?
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(c) Find the exact x -coordinates of all points of inflection. Hint: Color the part of the curve that is concave up blue and color the part that is concave down red. Points of inflection occur only where the curve changes color!)

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(d) Explain how you know from your calculations (not from the sketch you were given) which of your answers to part (a) are minima and which are maxima.

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