AB Calculus
1st Semester Exam Review
I. FUNCTIONS
 Be able to solve any type of equation within contexts of different problems either analytically or graphically.
 Be able to note the domain and range of given functions within contexts of problems involving derivatives.
 Composites
 Be able to apply properties of logarithms to solve equations.
 Examples:
1.
If
f ( x)  x 2  4 x and g ( x)  x  1 , what is the domain of
2.
If
f ( x)  4 x 2  1 and g ( x)  x ,


f ( x)
?
g ( x)
What is f(g(x))?
What is g(f(x))?
3.
What is/are the solutions to
x3  2x 2  3 ?
4.
What is/are the solutions to
3 x 3  sin x  2
5.
Solve
4 x  3 x2
II. Limits and Continuity
 Be able to apply properties to evaluate limits
 Find horizontal asymptotes
 Evaluate limits as x approaches ±infinity
 Be able to use one-sided limits
 Use the definition of continuity and find when functions are discontinuous
 Use Intermediate Value Theorem
 Examples:
1.
Find the limit: lim 3 x sin x
2.
Find the limit:
3.
Given
x 
t2  2
t
lim
t 0
x 2  2x  3
, find
x 3
lim f ( x )
lim f ( x)
f ( x) 

lim
25  x
x 5
x3
2
4.
Find
5.
Evaluate the limit
6.
Find the horizontal and vertical asymptotes of
x5
lim f ( x )
x 3
x 3
lim
x
2x  1
x2  3
.
f ( x) 
3x  5
.
x2
x5
.
x x2
 x 2  2 x, x  6
8. For what value of k is the function f ( x)  
continuous at x = 6?
2
x

k
,
x

6

7.
Find the points of discontinuity of
9.
Show that
f ( x) 
f ( x) 
2
x2  5
has a root between x = 2 and x = 3 using the IVT.
x 1
10. Describe what discontinuities, if any, the function
k ( x) 
x2  4
. If the function has a removable
x 2  3x  2
discontinuity, write a new function that removes it.
III. Derivatives
 Be able to notice the definition of a derivative.
 Know and apply all rules for differentiation (power rule, product rule, quotient rule, chain rule, trig derivatives, exponential
derivatives, logarithmic derivatives, inverse trig derivatives)
 Find second derivatives.
 Be able to find displacement, average velocity, instantaneous velocity, speed, and acceleration given a position function.
 Know the difference between average rates of change and instantaneous rates of change.
 Find derivatives using implicit differentiation.
 Find derivatives of inverses (using tables of values).
 Write equations of tangent and normal lines
 Examples:
1.
Find s’(t) and s’’(t) if
2.
If
3.
If y 
s(t )  (t 2  1)(1  t ) 2 .
f (t )  e 2t sin 3t , find f’(0).
5
(1  x 2 ) 3
, find
dy
.
dx
2v
, find f’(v).
1  2v 2
2
5. If y  tan( 2 x  1) , find y’
4.
If
f (v ) 
6.
If
4
 
g ( x)  1  sin 2 3x , find g '  
2
7.
If
y  sin 1 x  ln x 2  2 x  1 , find y’.
8.
Find
9.
A function f and its derivative take on the values shown in the table. If g is the inverse of f, find g’(6).
x
f(x)
f'(x)
2
6
1


dy
d2y
2
2
and
if x  y  1 .
dx
dx 2
6
8
3
3
2
10. Find the equations of the tangent and normal lines to the curve of
11. Find the equation of the tangent line to the curve of
12. Find the coordinates of any point on the curve of
13. Find the x-coordinate of any point on the curve of
f ( x)  x 3  3x 2 at the point (1, –2).
x 2 y  x  y 3  8 at the point where x = 0.
y 2  4 xy  x 2  5 for which the tangent is horizontal.
y  sin 2 ( x  1) for which the tangent is parallel to the line
3x – 3y – 5 = 0.
14. From a 400-foot tower, a bowling ball is dropped. The position function of the bowling ball is
s(t )  16t 2  400 , t ≥ 0 is in seconds.
a.
b.
c.
Find the instantaneous velocity of the ball at t = 2 sec.
Find the average velocity for the first 3 seconds
Find when the ball hits the ground.
15. The position function of a particle moving on a straight line is
a.
b.
c.
d.
s(t )  2t 3  10t 2  5 . Find
The position at t = 1.
The instantaneous velocity at t = 1.
The acceleration at t = 1.
The speed at t = 1.
 3

sin 
 h 1
 2
 .
16. Evaluate lim
h 0
h
2
17. Let G  400(15  t ) be the number of gallons of water in a cistern t minutes after an outlet pipe is opened.
a.
b.
Find the average rate of drainage during the first 5 minutes.
Find the rate at which the water is running out at the end of 5 minutes.
IV. Curve Sketching
 Be able to find critical points.
 Know the importance of the Extreme Value Theorem.
 Be able to find absolute and relative extrema.
 Be able to find points of inflection.
 Write an explanation of the number line illustration of the First Derivative Test. (Tell what the First Derivative Test shows
you about a function.)
 Sketch a curve using f’ and f’’.
 Connect the graph of a function to the graph of its derivative.
 Know the hypotheses of the Mean value Theorem and be able to find values that satisfy the MVT.
 Examples:
1. You left home one morning and drove to a cousin’s house 300 miles away, arriving 6 hours later. What does the
Mean Value Theorem say about your speed along the way?
2.
Let f be the function
f ( x)  ( x  1) 3 . Are the hypotheses of the MVT satisfied on [3, 6]? If so, find all values of
2
c that satisfy the conclusion to this theorem.
3.
Find any maximum, minimum, or inflection points on the graph of
f ( x)  x 3  5x 2  3x  6 and sketch the
curve.
4.
Find any maximum, minimum, or inflection points on the graph of
5.
Find the absolute extrema of
6.
f ( x)  x 4  4 x 3 and sketch the curve.
f ( x)  2 x 3  3x 2  12 x on [–2, 3].
A particle moves along a line according to the law s  2t
a. Find all t for which the distance s is increasing.
b. Find all t for which the velocity is increasing..
t
3
 9t 2  12t  4 where t ≥ 0.
3
.
2
c.
Find the speed when
d.
Find the total distance traveled between t = 0 and t = 4.

f ( x)  x 2  9
7.
Find the open intervals on which
8.
Given the graph of f(x) shown below, sketch f’(x).

2
3
is increasing or decreasing.
c3
c1
c2
c4
c5
9.
Given the graph of f’ to the right, find where the function f
a. Has relative maximum(s) or relative minimum(s)
b. Is increasing or decreasing
c. Has its point(s) of inflection
d. Is concave up or down
e. If f(–2) = f(2) = 1 and f(0) = –3, draw a sketch of f.
–4
–2
0
2
4
V. Optimization
 Be able to set up an optimization problem and find the values that maximize/minimize a situation.
 Examples:
1. You need to fence in a rectangular play zone for children, to fit into a right-triangular plot with sides measuring 4
meters and 12 meters. What is the maximum area for this play zone?
2. A carpenter wants to make an open-topped box out of a rectangular sheet of tin 24 inches wide and 45 inches long.
The carpenter plans to cut congruent squares out of each corner of the sheet and then bend the edges of the sheet
upward to form the sides of the box. If the box is to have the greatest possible volume (V = Lwh), what should its
dimensions be?
3. A dune buggy is on the desert at a point A located 40 km from a point B, which lies on a long, straight road. The
driver can travel at 45 km/h on the desert and 75 km/h on the road. The driver will win a prize if she arrives at the
finish line at point D, which is 50 km from point B, in 85 minutes or less. Help her decide on a route that will
minimize the time of travel. Does she win the prize?
4. A farmer wishes to fence in a rectangular pasture on a 3750 square-foot piece of riverfront property. He also plans
to separate the pasture into four regions, as shown in the figure below. What is the least amount of fencing (in feet)
he will need to purchase, assuming that he will not erect a fence along the river?
5. An open box can be created from a rectangular sheet of cardboard by cutting congruent squares from each of its
corners. What is the largest volume of such a box given cardboard that measures 20 by 30 inches?
6. The highway department is planning to build a rectangular picnic area for motorists along a major highway. It is to
have an area of 5000 square yards and is to be fenced off on three sides not adjacent to the highway. What is the
least amount of fencing that will be needed to complete the job?
7. A poster is to contain 108 cm2 of printed matter, with margins 6 cm each at top and bottom and 2 cm on the sides.
What is the minimum cost of the poster if it to be made of material costing $0.20 per square cm?
8. An isosceles trapezoid has a small base of 14 centimeters and slant sides of 6 centimeters. What is the largest area
of such a trapezoid?