AB Calculus
Topics for Midterms
Ch 2: Limits
Continuity (What does it mean to be a continuous function?)
Ch 3: Derivatives
Tangent Lines
Rules for derivatives
When is a function differentiable?
Velocity and Acceleration
Chain Rule
Implicit Differentiation
Trig Functions
Inverse Functions (including inverse trig functions)
Logs and Exponents
Piece-wise functions
Ch 4: Applications of Derivatives
Extreme Value Theorem
Mean Value Theorem
Connecting graphs of functions and their derivatives
Modeling and Optimization
Related Rates

AB Calculus
Midterm Review
1. Let f

1

20
3 e
 x
. Find a formula for f

1
.
2. Solve the equation algebraically. 1 .
08 x 
5
3. Solve the equation sin x

0 .
2 on the interval 0
 x

2

.
4. Which of the following statements are true about the function f whose graph is shown?
I.
II.
III. f x lim
 
1

  x lim

1
 f
 f
  lim x

1
 

 f
1

1
A) I and II B) I and III C) II and III D) I only E) I, II, and III
5. Determine lim x

0 sin
2 x x

3 x

4
6. Let a) b) c) d) x x x x f


2


2
 




5 x x



 x
2
3
2
,
, x x

2

2
find the limit of f
 
as:
7. Let f

3 x
5
2
 x
2
2 x
2

5

3
. Find a power function end behavior model for f .

8. Let f


 x
3 x

4
, x

3 , x x


4 , x
4
 
4
Find the point(s) of discontinuity of the function f . Identify each type of discontinuity.
9. Let f
 x x
2
2


3
4 x x

4

3
.
Give a formula for the extended function that is continuous at x = 1.
10. An object is dropped from a 75ft cliff. Its height in feet above the beach after t seconds is given by h
 

75

16 t
2
. a) Find the average velocity during the interval from t = 1 to t = 2. b) Find the instantaneous velocity at t = 2.
11. Let f


3

 4
 x x
2
,

9 , x x


2
2
Determine whether the graph of y
 f
12. The graph of y
 g
has a tangent at x = 2. If not, explain why not. is shown. Graph the function’s derivative.

13. The graph shows the derivative of a continuous function, f , where possible graph of y
 f
 
. f
 

1 . Sketch a
14. Find all values of x for which the function y
 x
5 x

2
is differentiable. dy
15. Find dx
, where y

 x

2
 x

4
 x

1
16. A particle moves along a line so that its position at any time s
 

2 t
3 
1 .
5 t

6 t

0 is given by the function
where s is measured in meters and t is measured in minutes. a) Find the displacement during the first 5 minutes. b) Find the average velocity during the first 5 minutes. c) Find the instantaneous velocity when t = 5 minutes. d) Find the acceleration of the particle when t = 5 minutes. e) At what value(s) of t does the particle change direction?
17. The values of the coordinate s of a moving body for various values of t are given. a) Plot s versus t, and sketch a smooth curve through the given points. b) Assuming this smooth curve represents the motion of the body, estimate the velocity at t = 10 sec and at t = 25 sec.

dy
18. Find dx
if y

4
3 tan
 cos x x dy
19. Find dx
if y
 sec
 
20. Find
 f  g
 
at x = 4 if f
 u
3 
5 and u
 g

3 x dy
21. Use implicit differentiation to find dx
if cos xy

2 x
2 
3 y . dy
22. Find dx
if y
 tan

1
 x

4

.
23. Let y
 x cos x
, find dy dx
. dy
24. Find dx
if y
 e
2 x  ln x
2
.
25. Find the global extreme values and where they occur.
Min: _____ at x = _____
Max: _____ at x = _____

26. For y

1
4 x
4 
2 x
3 
1
2 x
2 
3 find the exact intervals on which the function is:
3 a) Increasing b) Decreasing c) Concave up d) Concave down e) Find any local extremes f) Find any inflection points
27. A car’s odometer(mileage counter) reads exactly 59,024 miles at 8:00am and 59, 094 miles at 10:00am. Assuming the car’s position and velocity functions are differentiable, what theorem(s) can be used to show that the car was traveling at exactly 35 mph at some time between 8:00am and 10:00am?
I)
II)
Intermediate Value Theorem for Derivatives
Extreme Value Theorem
III) Mean Value Theorem a) I only b) II only c) III only d) I and III e) II and III
28. Sketch a possible graph of a continuous function f that has domain [-3, 3], where f
 
 
2 and the graph of y
 f
  
is shown below.
29. A piece of cardboard measures 22- by 35-in. Two equal squares are removed from the corners of a 22-in side as shown. Two equal rectangles are removed from the other corners so that the tabs can be folded to from a rectangular box with a lid. a) Write a formula V
 
for the volume of the box. b) Find the domain of V . c) Find the maximum volume and the value of x that gives it.
30. The function f
 x
3  ax
2  bx has a local minimum at x = 3 and a point of inflection at x = -1. Find the values of a and b .

31. Find the linearization L
 
of f

3 x
4 
5 x
3
at x = 2.
32. Suppose that the edge lengths x, y, and z of a closed rectangular box are changing at the following rates: dt
Find the rates at which the box’s a) Volume b) Surface area dx

4 ft/sec, dy dt

0 ft/sec, dz dt
 
3 ft/sec. c) Diagonal length s
 x 2  y 2  z 2
Are changing at the instant when x = 10 ft, y = 8 ft, and z = 5 ft.