MATH 1106, Spring 2017
Tutorial 9
02/21/17
REVIEW FOR PRELIM 1
Tutorial 9.1. Linear approximation
(a) Let C(x) = 3x2 + 200.
(i) Use C 0 (x) to approximate ∆C when ∆x = 1 and x = 3. Is
this a good approximation?
(ii) Use C 0 (x) to approximate ∆C when ∆x = 1 and x = 50. Is
this a good approximation?
(b) (Example 5 Chapter 6.6) In a precision manufacturing process,
ball bearings must be made with a radius of 0.6mm, with a maximum error in the radius of ±0.015mm. Estimate the maximum
error in the volume of the ball bearing.
Hint: The volume of a sphere is given by V = 43 πr3 where r is the
radius.
(c) (Exercise 34 Chapter 6.6) A cubical crystal is growing in size.
Find the approximate change in the length of a side when the
volume increases from 27 cubic mm to 27.1 cubic mm.
(d) (Exercise 29 Chapter 6.6) A sphere has a radius of 5.81 inches
with a possible error of ±0.003 inches. Estimate the maximum
error in the volume of the sphere.
Tutorial 9.2. Derivatives of Exponentials
(a) (Exercise 52 chapter 4.4) The age/ weight relationship of female
Arctic foxes caught in Svalbard, Norway, can be estimated by the
funtion
−0.022(t−56)
M (t) = 3102e−e
where t is the age of the fox in days and M (t) is the weight of the
fox in grams.
(i) Estimate the weight of a female fox that is 200 days old.
(ii) Use M (t) to estimate the largest size that a female fox can
attain (Hint: Find limt→∞ M (t)).
(iii) Estimate the age of a female fox when it has reached 80% of
its maximum weight.
(iv) Estimate the rate of change in weight of an Arctic fox that is
200 days old.
(b) Differentiate √the following functions:
t
−t
t
(i)f (t) = 2 · 3 t (ii) f (t) = e −et
(iii) f (t) = ete2t+2
+1
1
2
REVIEW FOR PRELIM 1
(c) The value of a particular investment changes over time according to the function
S(t) = 5000e0.1(e
0.25t )
,
where S(t) is the value after t years. Which of the following is
the rate at which the value of the investment is changing after 8
years? (i) 618 (ii) 1934 (iii) 2011 (iv) 7735 (v) 10 468
(d) It has been observed that there has been an increase in the proportion of medical research papers that use the word “novel” in
the title or abstract, and that this proportion can be accurately
modeled by the function
p(x) = 0.001131e0.1268x
where x is the number of years since 1970.
(i) Find p(40).
(ii) If this phenomenon continues, estimate the year in which
every medical article will contain the word “novel” in its
title or abstract.
(iii) Estimate the rate of increase in the proportion of medical
papers using this word in the year 2014.
Tutorial 9.3. Derivatives of logarithmic functions
(a) Differentiate
p the following functions:
√
(i) f (x) = ln |x − 3| (ii) f (x) = 10x log x (iii) f (t) = e−t + ln 2t
(b) (Exercise 65 chapter 4.5) Suppose that the population of a certain
collection of rare Brazilian ants is given by
P (t) = (t + 100) ln(t + 2)
where t represents the time in days. Find the rates of change of
the population on the second day and on the eigth day.
(c) (Exercise 67 chapter 4.5) The following formula shows a relationship between the amount of energy released and the Richter
number.
2
E
M = log
3
0.007
where E is measured in kilowatt-hours.
(i) For the 1933 earthquake in Japan, what value of E gives a
Richter number M = 8.9?
(ii) If the average household uses 247 kWh per month, how
many months would the energy released by an earthquake
of this magnitude power 10 million households?
(iii) Find the rate of change of the Richter number M with respect to energy when E = 70, 000 kWh.
REVIEW FOR PRELIM 1
(iv) What happens to
dM
dE
3
as E increases?
Tutorial 9.4. (Exercise 55 Chapter 5.2)
The mathematical relationsip between the age of a captive female
moose and its mass can be described by the function
M (t) = 369(0.93)t t0.36 ,
t ≤ 12
where M (t) is the mass of the moose (in kilograms) and t is the age
(in years) of the moose. Find the age at which the mass of a female
moose is maximised. What is the maximum mass?
Tutorial 9.5. Consider the function
f (x) = 0.008x3 − 0.288x2 + 2.304x + 7.
Find the domain of f , the critical points of f , the intervals on which
f is increasing and relative minima and maxima (use the second derivative test then the first derivative test).
Tutorial 9.6. (Exercise 58 Chapter 5.2)
Researchers determined that the ratings people gave for a film could
be approximated by
20t
R(t) = 2
t + 100
where t is the length of the film (in minutes). find the film length that
received the highest rating.