Math 131 Exam 2, November 16, 2011
Read each problem carefully and show ALL your work. No calculators.
1. (16 pts) A ball is thrown upward into the air at time
0t
seconds. Its height in feet above
the ground is given by
2
( ) 16 64 80h t t t
.
a. Find the velocity of the ball after t seconds.
b. After how many seconds will the ball reach its highest point?
c. How high does the ball rise?
d. When will the ball strike the ground?
2. (12 pts) Differentiate the following functions: (simplify your answers)
a)
22
( ) ln ln x
f x x x x
b)
2
( ) 2 2 x
f x x x e
c)
2
2
( ) ln 2
x
f x x x
3. (18 pts) Differentiate the following functions: (simplify answer in part b)
a)
sin
( ) tan( )
xx
f x e e
b)
ln(sin ) cos
( ) ln( )
xx
f x e e
c)
2
( ) cot secy x x
4. (12 pts) Find
dy
dx
where: a)
24x
xy y e
b)
sin 1
x
yy
5. (12 pts) Find the equation of the:
a. Tangent line to
33
7xy
at (-1, 2).
b. Normal line to
33
7xy
at (-1, 2).
6. (14 pts) Use logarithmic differentiation to find
dy
dx
where: a)
3
2
21
5
x
yx
; b)
2x
yx
7. (8 pts) Suppose that the differentiable function
()y f x
has an inverse
()y g x
and that
the graph of
()fx
passes through the point (3, 2) and has slope 4 there. Find the equation of
a line that is tangent to
()y g x
.
8. (8 pts) Find a so that
()fx
is continuous for all
x
, where
2
2
1 where 1
1
() 3 where 1
2
xx
x
fx
ax
x
