MATH 160
EXAM 3
- (25 pt) Find the following limits
a) b) c) d)
e)
- (20 pt) Find the absolute maximum and minimum values of on [2,4].
- (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).
- (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.
a) b)
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.
- Where (if anywhere) do the local extrema occur (and what kind of extrema are they)?
- Where (if anywhere) do points of inflection occur?
- At which of the critical points is it not possible to have derivative 0?(Justify)
- In which direction is it possible to have a horizontal asymptote? (Justify)
- Sketch the graph of a function that satisfies the above conditions.
- 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.
- Bonus (5 pt) is continuous on [-1,1] and . But there is no place on the function where
Does this contradict Rolle’s Theorem (why)?