Math 125 Exam 2 Fall 2010
3x2 − 13x + 4
1. (a) Let f (x) = . Identify any discontinuities in the graph of f (x). Classify each as
16 − x2
a removable disconuity (or hole), a jump discontinuity, or a vertical asymptote. Justify your
answer with the appropriate limits.
x3 + 8x x2 − 2x
(b) Evaluate each limit: lim lim
x→∞ 4 − x − 2x3 x→−2 x + 1
2. The capacity of Dave’s lung x minutes after it begins to collapse is given by V (x) = litres. Find
6 + ex
the instantaneous rate at which Dave’s lung capacity is changing when 2 minutes have elapsed. Round
your answer to the nearest 0.0001.
3. Dave is being pumped full of the drug codeine in an eﬀort to manage the excruciating pain of a collapsed
lung. Shortly after the drug was ﬁrst administered, Dave’s bloodstream had absorbed 180 mg of codeine.
While his lungs might not be at capacity, Dave’s liver is working overtime– one-third of the drug is
removed from his bloodstream every 4 hours.
(a) Find an exponential model of the form y = Aekt for the amount of codeine in Dave’s bloodstream
after t hours.
(b) Determine the average rate of change of the codeine level in Dave’s bloodstream over the interval
[0, 12]. Round your answer to the nearest 0.0001.
4. The number of chest x-rays that Dave must have on the x-th day of his hospital stay is modeled
by f (x) = . Use the limit deﬁnition of the derivative (also known as the four-step process) to
determine f (1). Interpret your answer in terms relevant to Dave’s hospital stay.
5. Find the derivative of each function. You must show algebraic simpliﬁcations and appropriate
work to indicate the rules that you are using.
x3 − 5x2 1
f (x) = h(x) = 4ex − 3x4 + √ − 2 log4 x + π 4
2x + 1 √ 34x
g(x) = x ln 3
6. The line y = 2x − 9 is tangent to the graph of f (x) = x2 + ax + b at the point (4, −1). Determine the
values of a and b.
7. Identify each statement as True (T) or False (F) by circling the best response.
T F If lim f (x) = 5 then the graph of f (x) must contain the point (3, 5).
T F If f (x) is continuous at x = 4, then f (x) is diﬀerentiable at x = 4.
T F The line y = 4x is tangent to the graph of f (x) = x2 + 3x at x = 1.
T F If lim f (x) does not exist, then f (x) has a vertical asymptote at x = 2.
T F If lim f (x) = lim f (x), then f (x) is continuous at x = 3.
x3 − 2x if x ≤ 2
T F The function f (x) = x2 − 4 is continuous at x = 2.
2 if x > 2
x − 3x − 2