1. (25 pt) Find the following limits

 

a) b) c) d)

 

e)

2. (20 pt) Find the absolute maximum and minimum values of on [2,4].

 

3. (15 pt) a) If you have two functions and such that then what can you say about the relationship between and ?

b) Find where and is a constant.

c) Use the results from a) and b) to show that .(Hint, plug in 1).

 

4. (20 pt) For the following functions sketch the graphs. You should make clear where local extrema, points of inflection, intervals of increase/decrease, concavity, and asymptotes are.

5. (20 pt) You are given the following information about the function

, is continuous, when and when , when and when , when and when and when and when You may assume that –1, 0, and 1 are critical numbers.

1. Where (if anywhere) do the local extrema occur (and what kind of extrema are they)?
2. Where (if anywhere) do points of inflection occur?
3. At which of the critical points is it not possible to have derivative 0?(Justify)
4. In which direction is it possible to have a horizontal asymptote? (Justify)
5. Sketch the graph of a function that satisfies the above conditions.

6. Bonus (5 pt) Sketch the graph of a function that has a point on its graph that is both a local minimum and a point of inflection. Circle the point.
7. Bonus (5 pt) is continuous on [-1,1] and . But there is no place on the function where