An Application of the fundamental theorem of calculus: Rate graphs

AN APPLICATION OF THE FUNDAMENTAL THEOREM OF CALCULUS: RATE GRAPHS

Section 4-L

0

xg x f t dt

0 , 1 , 2 , 5g g g g

1) The graph of a function f consists of aquarter circle and line segments. Let g be the function given by

.

Graph of f

a) Find

.

1, 5

1, 5

Graph of f

b) Find all values of x on the open intervalat which g has a relative maximum

c) Find the absolute minimum value of g on

And the value of x at which it occurs

. 1, 5

Graph of f

d) Find the x-coordinate of each point of inflection of the graph of g on

2) x

dtt

t

dx

d

3

sin

3) The graph of the velocity , in ft/sec, of a car travelingon 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 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

800 t

4) (modification of 2006 BC)a) Explain the meaning of in terms of the rocket’s flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate

70

10v t dt

70

10v t dt

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

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?

3

1a t

t

HOME WORKWorksheet 4-L and FRQ 4