Answer:
(a) Assume that f(x) is a real-valued function defined for all real numbers x
on an open interval whose center is a certain real number a. What does
it mean to say that f(x) has a derivative f
0
(a) at x = a, and what is the
value of f
0
(a)? (Give the definition of f
0
(a).)
(b) Use the definition of f
0
(a) you have just given in part (a) to show that if
f(x) = 1
2x − 1
then f
0
(3) = −0.08.
(c) Find lim
h→0
sin7
π
6 +
h
2
−
1
2
7
h
.
2. Let I be a bounded function on R and define f by f(x) = x
2
I(x). Show that
f is differentiable at x = 0.
3. Use the definition of derivative to find f
0
(2) for f(x) = x +
1
x
.
4. If g is continuous (but not differentiable) at x = 0, g(0) = 8, and f(x) = xg(x),
find f
0
(0).
5. (a) State the definition of the derivative of f(x) at x = a.
(b) Using the definition of the derivative of f(x) at x = 4, find the value of
f
0
(4) if f(x) = √
5 − x.
6. Let f be a function that is continuous everywhere and let
F(x) = f(x) sin2 x
x
if x 6= 0,
0 if x =