Review for the MVT, Rolle’s Theorem, and Approximating The Definite Integral Quiz
Name ______________________________________________
1) What does the MVT guarantee?
2) Given f x   x 3  4 x 2  5 x , find all the numbers c in the open interval (2, 4) where the
instantaneous rate equals the average rate of change.
3)
4) A policeman clocks a commuter’s speed at 50 mph as he enters a tunnel whose length is
exactly 0.75 miles. A second officer measures the commuter’s speed at 45 mph as he
exits the tunnel 43 seconds later and tickets the driver for exceeding the posted speed
limit of 50 mph. Use the MVT to justify the speeding charge levied by the officer.
5) How is Rolle’s Theorem different than the MVT? Explain.
6) Find the area bounded by f x   2 x 2  1 and the x-axis on the interval [0, 6] using 6
intervals and:
a) left-Riemann sum
b) right-Riemann sum
c) midpoint Riemann sum
d) Trapezoidal Rule
7) How are the left & right-Riemann sums related to the Trapezoidal Rule?
8) If
5
5
2
2
 2 f x   3dx  17 , find  f x dx .
2
2
2
0
9) If f x  is even and
10) If f x  is odd and
  f x   3dx  8, find  f xdx .
3
3
2
2
 f x dx  30 , find  f x dx .
Answers to MVT, Definite Integral Approximation, and Properties of Definite Integrals Quiz Review
1) That for a continuous function on [a, b] and differentiable on (a, b) that there is a point c,
f b   f a 
 f ' c  . In other words, there must be at least one point c in the
a < c < b, such that
ba
open interval (a, b) where the instantaneous rate of change at c will be equal to the average rate of
change of the function over the interval from a to b.
2) x  3.097
3) There are four places at approximately x = -5,-3, 2, 4,
4) By the MVT the driver’s speed was 62.791 at least once inside the tunnel.
5) Rolle’s Theorem is a special case of the M.V.T. where f a  0, f b  0, or f a f b  0 so that
f ' c = 0.
6)
a)
b)
c)
d)
116
188
149
152
7) The Trapezoidal Rule is the average of the left and right Riemann sums.
8) 4
9) 10
10) 30