MATHEMATICS 184 (Section 922) - TERM EXAM 2
NAME:
STUDENT ID NUMBER:
SIGNATURE:
INSTRUCTIONS: No notes or books are to be used. Calculators
are allowed. No credit will be given for the correct answer without
the (correct) accompanying work. Use the back of the pages if you
need extra space.
1. For the function
f (x) = ln(x)
(a) Estimate f 0 (1). Use the numbers 0.1, 0.01, 0.001, −0.001,
−0.01, −0.1 as your test values.
[3]
(b) Estimate f 0 (2). Use the numbers 0.1, 0.01, 0.001, −0.001,
−0.01, −0.1 as your test values.
[3]
(c) Estimate f 0 (3). Use the numbers 0.1, 0.01, 0.001, −0.001,
−0.01, −0.1 as your test values.
[3]
(d) Does this suggest a general formula for the derivative function
f 0 (x)? If so, what is it?
[2]
2. Let W be the amount of water, in gallons, in a bathtub at time
t, in minutes.
(a) What are the meaning and units of
dW
?
dt
(b) Suppose the bathtub is full of water at time to , so that W (to ) > 0
Subsequently, at time tp > t0 the plug is pulled. Is dW
positive,
dt
negative, or zero:
(i) For to < t < tp ?
(ii) After the plug is pulled?
(iii) When all the water has drained from the tub?
[4]
[6]
3. Using the definition of the derivative (ie. no shortcuts), find the
derivative function f 0 (x) for the function
f (x) =
p
2x2 + x
[10]
4. A 500 gram sample of radioactive lead decays at a continuous
rate of 1% per year.
(a) Write an equation that gives the amount Q of lead remaining
after t years.
[2]
(b) Find the time t (to the nearest year) when the lead decays to
one quarter of its initial amount.
[6]
5. Suppose the number of cubic metres of water that have passed
through a water treatment system after t seconds is given by a
function y = f (t) which has the graph:
(a) In the table below, indicate whether f , f 0 , f 00 at each point is
positive, negative, or zero.
[5]
(b) What is happening (physically) at each point? Hint: What do
the signs of f 0 and f 00 tell you in each case?
[6]
A:
B:
C: