MATH 160 Final Exam, Spring 2008
1.
page 1
Name: _______________________
Evaluate the following limits algebraically (without using l’Hôpital’s Rule or a calculator). Show the details of
the algebra that leads to the limit or shows that the limit does not exist. (Of course, you may use l’Hôpital’s
Rule or a calculator to check your answer.)
x2  x  2
(a) xlim
=
2
1
2x  x 1
(b) x lim
0
(c)
2x
x9  3
t 3
lim
t  3  1 1 =
  
3 t 
f( x)
(d) x lim
4
2.
=
1

 3  4 x
where f( x) =  x  4

 x  2
when x  4
when x  4
Two function y = f( x) and y = g ( x) have the following properties:
• f( 2 ) = 3
• g(2) = 4
f( x) = 5
• g is continuous at x = 2.
• x lim
2
EITHER find x lim
 2 f( x ) g ( x) and explain how you know this is the limit
O R give an example of two functions that have the properties listed but x lim
 2 f( x) g ( x) does not exist.
MATH 160 Final Exam, Spring 2008
3.
Name: _______________________
Calculate the following derivatives by using the differentiation formulas/theorems.
Show clearly how you calculated each derivative without your calculator. Simplification is not required.
4 x 2  3x  2 x
x
(a)
f( x) =
(b)
g ( t) = ( t 2 + 3 t – 1 ) cos t
(e)
y = (1 + ( x 2 + 2 ) 3 )
(f)
F(x) =

0
4.
page 2
x
f′( x) =
dy
=
dx
5
 1
tan
2
1 u
g ′( t) =

 du

F ′( x) =
Find an exact equation in point-slope form for the line tangent to the graph of f( x) =
the graph where x = 2. Find the equation without a calculator. Show the details.
2x 2  1
at the point on
x5
MATH 160 Final Exam, Spring 2008
5.
page 3
Name: _______________________
Sand falls from a conveyor belt onto a conical pile as suggested by the figure. The base of the pile of sand is
always a circle with diameter equal to the height of the pile. Sand is being added to the pile at the rate of 5
ft3/min. How fast is the height of the pile increasing when the pile of sand is 6 feet high?
Reminder: When we grade this problem we will look for:
(i) Constants and/or variables (quantities that change with time) correctly labeled in the diagram.
(ii) Clear identification of the variable(s) whose rates of change are known and the variable whose rate of change
is to be found.
(iii) An equation that relates the variable whose rate of change is to be found with the variable or variables whose
rates of change are known and that holds over an interval of time (not just at one instant).
(iv) Clear presentation of using calculus methods correctly to find the unknown rate of change.
6.
(a) Sketch the graph of a continuous function that has all the following properties:
(i)
the function is defined at every x
between – 5 and 5
(ii) f ′ ( x) > 0 for – 5 < x < 0
(iii) the graph is concave down
on the interval – 5 < x < – 2
(iv) f ′ ′ ( x) > 0 for – 2 < x < 0
(v) f ′ ( x) < 0 for 0 < x < 5
(vi) f ′ ′ ( x) > 0 for 0 < x < 2
(vii) the graph is concave down
on the interval 2 < x < 5 .
_____________________________________________________________________________________________________________________________________________________
-5
-4
(b) Are there any points where the
function you graphed is not differentiable?
What do you see in the graph you drew
that tells you?
-3
-2
-1
0
1
2
3
4
5
MATH 160 Final Exam, Spring 2008
7.
page 4
Name: _______________________
Evaluate the following definite and indefinite integrals without a calculator. Show details of algebra,
u-substitutions, etc.
(a)

( x 4  2 x )( 2 x 3  3 ) dx =
(b)

tan x sec3 x dx =
(c)

8
0
(d)
x4
( x  8 x  17 ) 3
2

dx =

2
[ sin(2 x)  cos(2 x)] dx =
0
8.
(a) Without using a calculator, show that

x cos x dx  cos x  x sin x  C .

(b) Without using a calculator, evaluate

0
x cos x dx . Show the details.
MATH 160 Final Exam, Spring 2008
9.
page 5
Name: _______________________
(a) Write a single integral that represents the area of the region enclosed by the graphs of y = 1 – |x |
and
y=–
2
1 x .
(b) Use familiar area formulas to find the value of the integral you found in (a).
Explain clearly how you used these formulas to find the value of the integral.
10. A parabolic tank is formed by revolving the region enclosed by the parabola y = x2 and the line y = 4 around
the y-axis. Thus, the tank is 4 feet deep and its top is a circle with radius 2 feet.
(a) What is the volume of the tank?
(b) The tank is full of olive oil weighing 57 lbs/ft3. How much work is required to pump the contents of the
tank to a point 2 feet above the top of the tank?