MATH 106, Fall 2005
Final Exam
Name:
____
____
Sufficient work must be shown for problems that require it. Each problem is worth 3 points except where noted
otherwise.
1. Use the graph of f (x) to answer the following questions:
(1.5 pts. each)
a) State the domain of f ____________________________________________________________
b) State the range of f _____________________________________________________________
c) Evaluate f ( f (-4)): _____________________________________________________________
d) State the equations of all vertical & horizontal asymptotes: ______________________________
e)
f)
g)
lim f ( x)  ___________________________________________________________________
x - 7
lim f ( x)  ____________________________________________________________________
x  3
lim f ( x)  ____________________________________________________________________
x 1
h) State the value(s) of x for which f has a removable discontinuity: _________________________
i) For what value(s) of x is f not differentiable?___________________________________________
j) What are the critical values of f? ____________________________________________________
k) For what value(s) of x does f have a local minimum? ____________________________________
l) f ′(-4) =________________________________________________________________________
m) For what interval(s) of x is f increasing? ______________________________________________
n) For what interval(s) of x is f ' increasing? _____________________________________________
o) For what interval(s) of x is f ′(x) < 0?_________________________________________________
p) For what interval(s) of x is f ″(x) < 0?________________________________________________
2. Find the following limits or state that they do not exist.
a)
lim t 3  1 
t 2
b) lim
sin 2 (2 x)

x 2 cos(3x)
c)
x 2  6x

36  x 2
x 0
lim
x 6
x 2  6x
d) lim

x    36  x 2
e)
lim
x - 
x3
9x 2  9
=
 x  4
if x  1
3. Find a value of k that makes h(x) continuous at x = 1 if h( x)   2
2 x  k if x  1
4. Use the definition of the derivative (either form) to find f '(2) for f ( x)  x 2  3x .
5. Find the derivatives of the following. Do NOT simplify.
9
a) f ( x)  6 x 3  x 8  5  csc x  10
x
b) g ( x)  (3  10 x 7 )(9 x  x 2  5)
c) f (t )  tan(sin( t 3  4t ))
x2 1
6. Find the equation of the tangent line to f ( x)  2
at x = 1.
x 1
7. A particle has the position function y  x 3  5 x 2  8x where x is measured in seconds and y in feet.
(worth 2 pts. Each)
a) Find the velocity at time x.
b) What is the velocity after 3 seconds?
c) When is the particle at rest?
8. Find
dy
if x 2  4 xy  y 2  13 .
dx
9. At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/hr and ship B is sailing
north at 25 km/hr. How fast is the distance between the ships changing at 4 pm? (worth 6 pts.)
10. A farmer that wants to test 3 different kinds of feed for a herd wishes to fence off 3 identical adjoining
rectangular pens (see below), using a total of 540 square feet of area. The outer boundary of the pens
requires heavy fence that costs $3 per foot, but the two internal partitions require fence costing only
$2 per foot. What dimensions will produce the least expensive cost for the pens? Use calculus to find
and show that this is a minimum. (worth 6 pts.)
x2  9
18  x 2


and f ( x) 
:
2x 4
x5
a) Find the intervals on which f is increasing and decreasing.
11. Suppose there is a function f such that f ( x) 
(worth 6 pts. Total)
b) Label each critical number as a local max., min., or neither.
c) Find the intervals on which f is concave up and concave down.
If the original function f that has the derivatives above is a function whose domain is x ≠ 0, and
lim - f ( x)   ; lim  f ( x)   with lim f ( x)  0 and lim f ( x)  0 , and if f (  3 2 ) = 5,
x 0
x 0
x 
x - 
f(-3) = 7, f( 3 2 ) = -5, f(3) = -7 :
d) State the equations of all asymptotes of f.
e) Sketch a graph of f(x).
12. Suppose that f ''(x) = -12x + 8, f '(1) = -3 and f(-2) = 18. Find f(x). (worth 4 pts.)
13. Find the following antiderivatives:
4
a)  ( x 3  2  sin x   )dx
x
b)
(1  x ) 4

x
dx
14. Evaluate the following definite integrals:

a)
 2 cos x dx =
2
5
b)

3 x  1 dx =
1
Hope you have a safe, happy, and peaceful holiday!
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