Download Quiz Three Calculus A Professor D. Olles Name - RIT

Quiz Three
Calculus A
Professor D. Olles
Name:
There is partial credit awarded for incorrect answes on the multiple choice questions. No formula
sheets or calculators are allowed. You have 10 minutes to complete this quiz.
Question:
1
2
3
Total
Points:
6
2
12
20
Score:
1. Use the given graph of f to state the value of each quantity, if it exists. If it does not, explain why.
(a) limx→1 f (x) = 2
(b) limx→3 f (x) DNE
The graph of the function does not approach any ONE y-value, in particular, as x approaches
3. From the left, the function approaches 1 and from the right, the function approaches 4.
(c) f (3) = 3
2. Is it possible for f (2) = 3 if limx→2 f (x) = 5?
Yes. The function could be approaching y = 5 as x tends toward 2, but if there is a hole at this
x-value and the function is piecewise defined to give the point (2, 3), the limit would not be equal to
the function’s evaluation at that x-value.
3. If a ball is thrown into the air with a velocity of 40 ft/s, its height in feet t seconds later is given by
y = 40t − 16t2 .
(a) Find the average velocity for the time period beginning when t = 2 and lasting 1 second.
s(b) − s(a)
s(3) − s(2)
=
b−a
3−2
40(3) − 16(3)2 − 40(2) − 16(2)2
=
1
= 120 − 16(9) − 80 + 16(4)
ms =
= 40 − 16(9 − 4)
= 40 − 16(5)
= 40 − 80
= −40 ft/s
(b) Estimate the instantaneous velocity when t = 2.
s(t) − s(2)
t−2
40t − 16t2 − 40(2) − 16(2)2
= lim
t→2
t−2
mt = lim
t→2
40t − 16t2 − 80 + 64
t→2
t−2
= lim
−16t2 + 40t − 16
t→2
t−2
−8 2t2 − 5t + 2
= lim
t→2
t−2
= lim
= lim
t→2
−8(2t − 1)(t − 2)
t−2
= lim [−8(2t − 1)]
t→2
= −8(4 − 1)
= −24 ft/s