Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 16
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41Worksheet # 11: Rates of Change
2
1. Given that
( ) = 4
3 + 42 find the instantaneous rate of change when = 3.
3
2. A particle moves along a line so that its position at time is ( ) = 3
12 where ( ) represents the
distance to the right of the origin.
(a) Find the velocity and speed at time = 1.
(b) Find the acceleration at time = 1.
(c) Is the velocity increasing or decreasing when = 1?
(d) Is the speed increasing or decreasing when = 1?
2
3. An object is thrown upward so that its height at time seconds after being thrown is ( ) =
4 9
+
20 + 25 meters. Give the position, velocity, and acceleration at time .
4. An object is thrown upward with an initial velocity of 5 m/s from an initial height of 40 m. Find the
2
velocity of the object when it hits the ground. Assume that the acceleration of gravity is 9.8 m/s
.
5. An object is thrown upward so that it returns to the ground after 4 seconds. What is the initial
2
velocity? Assume that the acceleration of gravity is 9.8 m/s
.
6. Suppose that height of a triangle is equal to its base . Find the instantaneous rate of change in the
area respect to the base
when the base is 7.
2
7. The cost in dollars of producing
bicycles is
( ) = 4000 + 210
1000.
(a) Explain why
(40) is a good approximation for the cost of the 41st bicycle.
(b) How can you use the values of
(40) and
(40) to approximate the cost of 42 bicycles?
(c) Explain why the model for
( ) is not a good model for cost. What happens if
is very large?
8. Suppose that
is a function with
(7) = 34 and
(7) = 4. Explain why we can use 42 as an
approximation for (9). What is an approximate value for (6)?
9. (Review) Suppose that the tangent line to the graph of
at (2 (2)) is
= 3 + 4. Find the tangent
line to the graph of
( ).
10. (Review) Differentiate the following functions.
(a)
( ) = ( + 1)(
2)
1
(b) ( ) =
+1
(c)
( ) = sin( )
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