Practice Problems for Test 2
Note:
These are not sample test questions. However, your understanding of the topics below will help you on your
second test. You should also review the derivatives on Worksheet 5 and the concepts presented in Worksheets
3, 4, and 6. Be sure to review all class notes and assigned homework problems.
1) Use the definition:
¢ f (x)=lim
hÆ0
f(x+h)-f(x)
h
to show that
¢ f (x)=4x-3
if
f(x)=2x2-3x
2) Use
†
g(x)=x+1 if x£2
2x-1 if x>2
Ï
Ì
Ó
a) Is
g(x)
continuous for all x?
b) Is
g(x)
differentiable for all x? How did you decide?
c) Sketch the graphs of
g(x)
and
¢ g (x)
.
3) Find
dy
dt
for each of the following:
a)
y=t3cos3(t3)
b)
sin2(5t)+cos2(3y)=2t
c)
y=10t-5
d)
y=(t2+5)10 (1 -t)
e)
y=sin(
p
2t2)
f)
y=t3tan(t-1)
g)
y=sin(cost)
h)
y=cos(
p
)
i)
y=tan-1(cos t)
j)
†
y=e5t
1+3t
k)
†
y=54t3
l)
sin y=sin-1t
4) Find the equation for the line that is tangent to the graph of the given function,
y=f(x)
, at the given point:
a)
f(x)=sin(5x) at the point (
p
,f(
p
))
b)
x2y3-4y=7x-4 at the point (1,-1)
c)
y=x
(x2+1)5 at the point 1, 1
32
( )
5)
If x=r2-2 and r=cos(
q
)
find
dx
d
q
