Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 34
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41Worksheet # 26: The Fundamental Theorems of Calculus and the Net
Change Theorem
1. (a) State both parts of the Fundamental Theorem of Calculus using complete sentences.
5
(b) Consider the function ( ) on [1
) defined by ( ) =
1 . Find the derivative of .
1
Explain why the function
is increasing.
3
5
(c) Find the derivative of the function ( ) =
1
on (1
).
1
2. Use Part I of the Fundamental Theorem of Calculus to evaluate the following integrals or explain why
the theorem does not apply:
5
4
2
(a)
6
(d)
sec
( )
2
0
7
1
sin(2 )
6
(b)
(e)
5
sin( )
2
3
1
+1
(c)
1
3. Use Part II of the Fundamental Theorem of Calculus to find the derivative of the following functions:
0
4 5
3
(a) ( ) =
2 +
(d) ( ) =
sin
( )
1
1
2
4
2
5
(b)
( ) =
cos
(e)
( ) =
sin( )
2
3
3
(c)
( ) =
1 +
0
2
4. A population of rabbits at time
increases at a rate of 40
12 + 3
rabbits per year where
is
measured in years. Find the population after 8 years if there are 10 rabbits at = 0.
2
5. Suppose the velocity of a particle traveling along the -axis is given by ( ) = 3
+ 8 + 15 m/s at time
seconds. The particle is initially located 5 meters left of the origin. How far does the particle travel
from = 2 seconds to = 3 seconds? After 3 seconds, where is the particle with respect to the origin?
2
6. Suppose an object traveling in a straight line has a velocity function given by ( ) =
8 + 15 km/hr.
Find the displacement and distance traveled by the object from = 2 to = 4 hours.
0 01
7. An oil storage tank ruptures and oil leaks from the tank at a rate of ( ) = 100
liters per minute.
How much oil leaks out during the first hour?
8. Recognize each of the sums as a Riemann sum, express the limit as an integral and use the Fundamental
Theorem to evaluate the limit.
3 +
(a) lim
=1
2
2
(2 +
)
(b) lim
2
=1
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