Precalculus
Difference Quotient Quiz Review
Average Slope
m([p, q]) =
f (q) − f (p)
q−p
Difference Quotient
f (p + h) − f (p)
h
The Mean Value Theorem
DQ(p, h) =
If f (x) is a quadratic function, then for every interval
[a, b], there exists r ∈ [a, b] such that the slope of the
tangent line to the graph of y = f (x) at the point (r, f (r))
is m([a, b]).
1. Let f (x) = x2 + x + 2. Find the average slope of
the graph of y = f (x) for x ∈ [1, 5]
2. Let f (x) = 2x2 − x − 1. Find the average slope of
the graph of y = f (x) for x ∈ [−1, 3]
3. Find the difference quotient for f (x) = x2 −2x. Use
your answer to find the average slope of the graph of
y = f (x) for (a) x ∈ [−2, 4] and (b) x ∈ [2, 7]
4. Find the difference quotient for f (x) = 2x2 + x +
1. Use your answer to find the average slope of the
graph of y = f (x) for (a) x ∈ [−1, 3] and (b) x ∈
[1, 6]
5. Let f (x) = x2 + 3x. Find r ∈ [0, 5] so that the slope
of the graph of y = f (x) at (r, f (r)) is 5.
6. A particle travels on the real line so that its position
is given by r(t) = −t2 + 14t − 13 for t ∈ [0, 12],
where t is the number of seconds after the initial observation and r(t) is in meters.
t=0
∅
r = -13
r=0
(a) What is the position of the particle after 3 seconds?
(b) For which time(s) is the particle at the origin?
(c) What is the maximum distance of particle from
the origin?
(d) What is the average velocity of the particle for
t ∈ [5, 11]
(e) For which instant in time is the velocity of the
particle 2 m/s?
7. A water tank is being filled through a faucet and simultaneously drained throug a spigot in such a way
that the volume of water in the tank, in gallons, is
given by v(t) = 2t2 − 24t + 150, where t is in minutes and t ∈ [0, 10].
(a) What is the volume of water in the tank after 3
minutes?
(b) What are the maximum and minimum volumes
of water in the tank over the 10-minute inteval?
(c) What is the average rate at which water is being
added to the tank for the interval 1 ≤ t ≤ 4?
(d) At what instant of time is water being added to
the tank at a rate of 4 gal/s?
6. a) r(3) = 20
1. m ([1, 5]) = 7
b) t = 1
2. m ([−1, 3]) = 3
c) At t = 7 s, r(5) = 36
3. DQ(p, h) = 2p − 2 + h
d) m ([5, 11]) = −2 m/s
a) DQ(−2, 6) = 0
b) DQ(2, 5) = 7
4. DQ(p, h) = 4p + 2h + 1
a) DQ(−1, 4) = 9
e) t = 6s
7.
a) v(3) = 96 gal
b) Max: 150 gal; Min 78 gal
c) m ([1, 4]) = −14 gal/min
b) DQ(1, 5) = 29
5. r = 1
d) t = 7s