Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 12
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41Worksheet # 9: The Derivative
1. Comprehension check:
(a) What is the definition of the derivative f (a) at a point a?
(b) What is the geometric meaning of the derivative f (a) at a point a?
(c) True or false: If f (1) = g(1), then f (1) = g (1)?
2. Consider the graph below of the function f (x) on the interval [0, 5].
(a) For which x values would the derivative f (x) not be defined?
(b) Sketch the graph of the derivative function f .
3. Find a function f and a number a so that the following limit represents a derivative f (a).
3
(4 + h)
64
lim
h
0
4. Let f (x) = x . Find f (1), f (0) and f ( 1) or explain why the derivative does not exist.
5. Find the specified derivative for each of the following.
(a) If f (x) = 1/x, find f (2).
(b) If g(x) =
x, find g (2).
2
(c) If h(x) = x
, find h (s).
3
(d) If f (x) = x
, find f ( 2).
(e) If g(x) = 1/(2
x), find f (t).
6. Let
2
at
+ bt + c if t
0
g(t) =
.
2
t
+ 1
if t > 0
Find all values of a, b, and c so that g is differentiable at t = 0.
7. Let f (x) = e and estimate the derivative f (0) by considering difference quotients (f (h)
f (0))/h for
small values of h.
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