Math
165
Fall 2002
Exam 1
1.
(25
pt) Evaluate the following limits.
a) 
b)
c) 
d)
e) 
2.
(32
pt) Find the derivatives of the following functions.
a) 
b) 
c) 
d) 
3.
(10
pt) Use the definition of the derivative to find the derivative of the
function 
4.
(20
pt) On a planet with no atmosphere, the height above ground of a projectile
that is shot straight up is given by 
a)
What
does the 
represent?
b)
Find
a formula for the velocity, 
and explain why 
must be negative (you
may assume that the projectile returns to the ground eventually).
c)
What
is the initial velocity 
?
d)
Now
use your formula for velocity and find a formula for the acceleration of the
projectile. Use this to explain what 
represents.
5.
(7
pt) Show that the equation 
has at least one real
solution.
6.
(6
pt) For this problem we will assume that 
are positive integers.
a)
Given
the fact that 
(extra credit: verify
this), show that 
b)
Now
show that 
(Hint: let 
and note that
; take the derivatives of both sides and solve for 
)
7.
(10
pt) Consider the functions pictured below.
a)
Find
b)
For
what value(s) of b is the function 
continuous
everywhere?
g(x) 2 f(x)
b
M 1
L L -1 M