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AN APPLICATION OF THE
FUNDAMENTAL THEOREM OF
CALCULUS: RATE GRAPHS
Section 4-L
.
1) The graph of a function f consists of a
quarter circle and line segments.
Let g be the function given by
x
g  x    f  t  dt
0
a) Find g  0 , g  1 , g  2 , g 5
Graph of f
.
b) Find all values of x on the open interval  1, 5
at which g has a relative maximum
c) Find the absolute minimum
value of g on  1, 5
And the value of x at which
it occurs
Graph of f
.
d) Find the x-coordinate of each point of
inflection of the graph of g on  1, 5
Graph of f
2)
x
d sin t
dt

dx 3 t
3) The graph of the velocity , in ft/sec, of a car traveling
on a straight road, for is shown in the figure.
a) Find the average acceleration of the car,
over the interval
b) Find an approximation for the acceleration
of the car at t = 20.
c) Approximate with a Riemann sum, using
the midpoints of three subintervals of equal
length. Explain the meaning of this integral.
4) (modification of 2006 BC 4)
Rocket A has positive velocity v(t) after being launched
upward from an initial height of 0 feet at time t = 0
seconds. The velocity of the rocket is recorded for
selected values of t over the interval 0  t  80 seconds
as shown in the table below
t (seconds)
0
10 20 30 40 50 60 70 80
(ft per sec)
5
14 22 29 35 40 44 47 49
4) (modification of 2006 BC)
70
a) Explain the meaning of 10 v  t  dt in terms
of the rocket’s flight. Use a midpoint Riemann
sum with 3 subintervals of equal length to
approximate 70 v  t  dt
10
t (seconds)
0
10 20 30 40 50 60 70 80
(ft per sec)
5
14 22 29 35 40 44 47 49
4) (modification of 2006 BC)
b) Rocket B is launched upward with an acceleration
of
3
a t  
t 1
feet per second per second. At time t=0 seconds, the
initial height of the rocket is 0 feet, and the initial
velocity is 2 feet per second. Which of the two rockets
is traveling faster
at t = 80 seconds?
HOME WORK
Worksheet 4-L and FRQ 4