Final Review Math 131
1
Day 1:
1. If
( )
3x
fx x
−
=
, find
( )( )
ffx
. What is the domain of
( )( )
ffx
?
2. Are the lines
21xy+=
and
21xy−=
perpendicular?
3. True or False: If
lim ( )
xa
fx L
→
=
then
( )
fa L=
.
4. If
( ) ( )
and fx gx are differentiable, then
( ) ( )
( )
df xgx
dx =
_____________________.
5. If
1
y
x
=, use the definition of derivative to find
dy
dx
.
6. Sketch a function
f
where
f
is continuous at
xa=
but
f
is not differentiable at
xa=
.
7. If
( ) [ ( )], (-3) 5, '(-3) 2, (5) 3, and '(5) -3gx f ux u u f f= = = = =
, find an equation of the tangent
line to the graph of
( )
gx
at
3x= −
.
8. Find the maximum and minimum values of
( )
f x Ax B= +
where
0 and AB>
are constants on
[a, b].
9.
0
sin 2
lim ____
x
x
x
→
=
.
10. Does the function
( )
2gx x= −
satisfy the hypotheses of the Mean Value Theorem on [1,4].
11. By the ______________________theorem, if f is continuous on the interval [a, b] and K is
between
( ) ( )
and fa fb
then
( )
K fc=
for some c in (a, b).
12. Find the equation of the tangent line to the graph of the equation
2
tan( )xy y=
at (π/4, 1).
13. Evaluate the following limits
a.
23
0
1 tan
lim
x
x
xx
→
−
b.
2
1
lim ln
x
x
xx
→∞
+
