Math 151 Review for final.

1. If the graph of is given together with its first and second derivatives below. Label the graphs for and

Ans: is red, is thin and is black thick.

2. If the graph of is given below:

1. Compare in increasing order. [ Ans:

2. Suppose the function (graph of , is given above) represent the distance that a car travels.

1. When do you think the car has zero velocity between and ? [Ans: look for the point where so it is about  x=4 (max of f) or x=0.8 (min of  f ].

2. Approximate the time when the car reaches its highest speed for and . [Note: speed = ].

[Ans: We look for the slope of to be the largest, so it is the inflection point of .]

3. Find the tangent line equations and the at a given point for the following functions:

1. at [Ans: y=0.6708 x+1.565]

2. at [y=-3*x-4]

4. Find [Hint: Identify a function and a point and use the derivative at one point.]

[Ans. This limit is the same as when we pick and so it is ]

5. Find the following limits with explanations:

1. [ans. ]

2. [

3. [

4. [=0

5. [=0

6. [=0

6. Let the graph of a velocity function (x = time, y = feet/sec) be given below: Assuming the x-intercepts for the following graph is at and

1. What is the initial velocity (when time=0)? [Ans: Since when , is not found, so we do not have enough info to find the initial velocity.]

2. Explain how velocity function can be negative sometimes. [Ans. When we travel in opposite direction, the velocity is negative.]

3. Estimate the inflection point(s) for the distance function[Ans, they are at the local min and the local max of the velocity, so it about and ].

4. Estimate the maximum and minimum for the distance function[Ans, max is at and minimum at

5. Find the interval(s) where the acceleration is negative. [Ans. This is where the velocity has negative slopes].

7. If the graphs of and are given below: Then

1. identify the graph for and respectively. [Ans. is the one always positive and the last one is for ]

2. find the interval(s) where is increasing or decreasing, [ans. is always increasing]

3. find the maximum and minimum for in the interval [since is increasing in the minimum is at and the maximum is at

4. find the interval(s) where is concave upward and concave downward. [Since in , is concave in this interval;similarly, in is concave down this interval.]

8. Find the first derivative for the following functions:

1. [

2. [

3. [

4. [done in class]

5. [

6. [

9. If the derivative of a function is Then

1. find the interval(s) where original function is increasing and decreasing, [hint: draw

2. find the interval(s) where the original function is concave upward and concave downward. [hint: find and use the signs of to find the intervals where is concave upward or downward]

3. graph one possible function which has the derivate function **this is a typo, ignore this one]

10. Find if [Use implicit differentiation to find

11. Find if [ ].

12. Find [

13. Find the first derivative for the following functions:

1. [

2. [

14. Use the product or quotient rule to find

15. Find the followings:

1. [

2. [

16. If Find by hand.

17. Suppose the cost, in dollars, for a company to produce pairs of a new line of jeans is

1. Find the marginal cost function. [

2. Find and explain its meaning. What does it predict? [This will predict the cost for the 101st unit]

3. Estimate the cost of making the 99 th pair of jeans. [Use use calculator to find this value]

18. If

1. use the signs of to find the intervals where is increasing or decreasing, [hint: plot , which is shown above. We need find the zero of which is at so is increasing in and is decreasing in ]

2. find the relative maximum and minimum for [ has a relative minimum at

3. find [hint:

4. use the signs of to find the intervals where is concave upward or downward. [hint: after finding you need to plot