Name: ____________________
Calculus Test #3 (Sectinos 4.4 – 5.5)
1. (10 pts.) Sketch the graph of ONE SINGLE function that satisfies all of the following properties: f (x)
is odd, has a vertical asymptote at x  3 , lim f ( x)  2 , f ( x)  0 on 0, 3  3,  , f ( x)  0 on
0, 3 and f ( x)  0 on 3,  .
2. (15 pts.) Find the following limits:
a.
x2  2 x
x 0 sin x
lim
x 
1
b.
lim x x
x 
3. (10 pts.) Estimate
are doing.

2
1
e x dx using right-hand endpoints with 3 subdivisions. Draw a picture of what you
4. (15 pts.) Suppose the graph shown is the graph of f (x ) , the first derivative of f. Find the following for
f (x) : (you need only write the x-value(s).)
When is f (x) concave up? (STILL ASSUME THE GRAPH IS OF f (x ) )
When is f ( x)  0 ? (STILL ASSUME THE GRAPH IS OF f (x ) )
When is f (x) increasing? (STILL ASSUME THE GRAPH IS OF f (x ) )
When is f ( x)  0 ? (STILL ASSUME THE GRAPH IS OF f (x ) )
When does f (x) have any maximum values? (STILL ASSUME THE GRAPH IS OF f (x ) )
When does f (x) have any minimum values? (STILL ASSUME THE GRAPH IS OF f (x ) )
When does f (x) have any inflection points? (STILL ASSUME THE GRAPH IS OF f (x ) )
Integrate the following:
a. (5 pts)
b. (5 pts)

x
dx
4x  9

(t  1) 2
t2
dt
c. (5 pts)

2 x
3 dx
0
d. (5 pts)  sec 3  tan d
5. a) (5 pts.) Explain in words what

2004
f (t )dt represents if f (t ) is the rate at which the world’s
2000
population is growing in year t , in billions of people per year.
b) (5 pts.) If w(t ) is the growth rate of a child in pounds per year, write a definite integral that
expresses the total amount of weight a child gains from the time the child is 2 years old to 5 years
old? What units will the integral have?
6.
(15 pts.)
a.
Change the following integral using the definition of the integral (make into a limit of sums),
using right hand endpoints
3
1
x2  1dx . You may need one or more of the formulas written
on the board to evaluate.
b) Check your answer from part a by evaluating the integral
Theorem of Calculus
3
1
x2  1dx using the Fundamental
7.
(10 pts.)
Let
g ( x) 
 f (t )dt
x
, where
f
is the function whose graph is shown.
0
a. At what values of x do the local maximum and minimum values of g occur?
b. Where does g attain its absolute maximum?
c. On what intervals is g concave down?
d. On what intervals is g concave up?
Extra Credit: Which of the following areas are equal? Why? (You must explain your answer,
guessing will NOT give you any credit)