Elementary Calculus with Applications

135 Pages·2017·14.726 MB·English

by  Elena Devdariani

Elementary Calculus with Applications
Elena Devdariani
Carleton University, Ottawa, Canada
Fourth Edition
Copyright (c) 2017 Bukar
Contents
1 Elementary Functions 1
1.1 Definition, Domain and Range of a Function....................................................... 1
1.2 Algebra of Functions................ •............................................................................... 2
1.3 Transformations of Graphs....................................................................................... 5
1.4 Polynomial, Rational and Power Functions.......................................................... 8
1.5 Exponential Functions.............................................................................................. 11
1.6 Logarithmic Functions.............................................................................................. 13
2 Limits 22
2.1 The Limit of a Function at a Point....................................................................... 22
2.2 Properties of Limits..................................................................................................... 26
2.3 Limits at Infinity........................................................................................................ 28
2.4 Continuous Functions............................................................................................... 31
2.4.1 The Intermediate Value Theorem............................................................... 34
3 The Derivative and Rules of Differentiation 38
3.1 The Derivative ........................................................................................................... 38
3.1.1 The rate of change of a function and the slope of its graph................... 38
3.1.2 The derivative as the slope of the tangent line........................................ 39
3.1.3 The derivative as a rate of change............................................................... 41
3.2 Rules of Differentiation.............................................................................................. 43
3.2.1 Basic rules of differentiation......................................................................... 43
3.2.2 The Product and Quotient Rules............................................................... 45
3.2.3 The Chain Rule............................................................................................... 45
3.2.4 Implicit Differentiation.................................................................................. 48
3.3 Higher Order Derivatives........................................................................................... 51
4 Applications of the Derivative 53
4.1 Determining the Intervals Where a Function is Increasing or Decreasing. . . 53
4.2 Marginal Concepts in Economics ........................................................................ 55
4.3 Elasticity of Demand.................................................................................................. 58
4.4 Related Rates............................................................................................................... 60
4.5 Maximum and Minimum Values............................................................................... 63
4.6 Applications of the Second Derivative..................................................................... 70
4.7 Curve Sketching.............................. 75
4.8 Some Optimization Problems .................................................................................. 80
4.9 Exponential Models...................................................................................................... 85
4.9.1 Continuously Compounded Interest............................................................ 85
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11
4.9.2 Exponential Growth and Decay................................................................... 86
4.9.3 Learning Curves................................................................................................. 88
5 Functions of Several Variables 91
5.1 Examples of Functions of Two and Three Variables........................................... 91
5.2 Partial Derivatives..................................................................................................... 94
5.3 Maxima and Minima of Functions of Two Variables........................................... 98
5.4 Lagrange Multipliers. Constrained Optimization................................................. 102
6 Integration 109
6.1 Antiderivatives and the Rules of Integration........................................................ 109
6.2 Integration by Substitution........................................................................................ 114
6.3 The Definite Integral.................................................................................................. 117
6.4 The Fundamental Theorem of Calculus.................................................................. 120
6.5 Evaluating Definite Integrals..................................................................................... 122
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111
Chapter 1
Elementary Functions
These notes set out an introductory course in calculus for students in business, economics
and the life sciences.
Mathematics in general may be regarded as a special ’’language” that allows us to formalize,
analyze and solve many problems in the real world. Calculus in particular deals with quanti­
ties that change with respect to one another. As any language has its vocabulary and rules,
calculus has its own concepts and methods. We begin the study with the main concept in
calculus, namely, the notion of a function.
1.1 Definition, Domain and Range of a Function
A function is a basic tool for describing relations among quantities. We begin by looking
at relations between two quantities which we call x and y. For example, x may denote the
time and y the amount of money in someone’s savings account, or x may be the speed of a
vehicle and y the distance traveled by the vehicle. In this context, x and y are real numbers,
which we call variables. One can say that, in both examples, the variable y depends on
the variable x, as the amount of money depends on the time, and the distance depends on
the speed. Moreover, y depends on x in a certain way, as the amount of money will be
determined by the interest rate offered by the bank, and the distance will depend on the way
the driver regulates the speed. A particular way, or rule, by which the quantity y depends
on a?, is called a “function”:
Definition 1. A function / of a variable a; is a rule that assigns to each value of a; a
unique number y = f(x). The number x is called the independent variable, or the argument,
and the number y is called the value of the function at x, or the dependent variable.
1. fix'} = x2 — 5. a(x h(x) = 5X, P(x) = y7 — x are functions of x.
x — 1
Each function in the Example above is defined by a formula. A function may also be defined
by a table of values for x and y, by diagram, or by graph. We are going to work mostly with
functions defined by formulas.
Next, we define the domain and the range of a function. The set of all real numbers will be
denoted by R.
1
CHAPTER 1. ELEMENTARY FUNCTIONS 2
Definition 2. The set of all values that the independent variable is allowed to assume is
called the domain of the function.
Definition 3. The set of all values that the function assumes as the independent variable
varies throughout the domain is called the range of the function.
Example 2. If f(x) = x2 — 5, then x can be any real number. We then say that the
domain of f is the set of all real numbers, and write Dom{f] — (—oo, oo), or Dom{f] = R.
then x cannot take the value of 1, as division by zero is not permitted.
Thus, we write Dom{g} = (-oo, 1) U (1, oo), or Dom{g} = {x G R : x ± 1}.
If h(a?) — 51, then x can be any real number: Dom{h] = R.
If P(x) = \/7 — x, then, for an even root to be defined, the expression under the root must be
nonnegative: 7—x >0, 7 > x. Thus, Dom{P} = (—oo, 7], or Dom{P} = {z € R : x < 7}.
Example 3. Finding the range of a function may not be as straightforward as finding the
domain. In some cases, the graph of a function can be useful.
If f(x) = x2 — 5, then the graph of the function lies on and above the line y = —5, so we
write Range{f} = [—5, oo), or Range{f} = {y G R : y > —5}.
If P(x) = V7 — x, then, since the square root assumes only nonnegative values,
Range{P} — [0, oo), or Range{P} = {y G R : y > 0}.
Exercises 1.1
In Exercises 1 — 10, find the domain of each function.
2. f(x) = 3. f(x) =
5. f(x) — \/x + 3. 6. f(x) = y/4 — x
\[x — 2
9. f(x) = 10. /(a;) =
{x — 5)(x + 1)'
In Exercises 11 — 14, find the range of each function.
14. /(a;) =
1.2 Algebra of Functions
Let / and g be functions with domains F and G, respectively, We define the sum, the
difference, the product and the quotient of f and g as follows:
(/±p)(a;) = /(ar) ±g(x),
= f(x)-g(x),
CHAPTER 1. ELEMENTARY FUNCTIONS 3
The domain of the sum f + g, difference f — g and product fg is the set F Cl G (read “F
f
intersect G”). The quotient — has the domain F Cl G, excluding x such that g(x) = 0.
g
9(l) = ih
Example 4. Given fix) = find the sum, the product and the
quotient of f and g.
Example 5. (Cost and Profit Functions)
The cost of operating a business consists of two major parts:
- Fixed costs (rental fees, management salaries, etc.)
- Variable costs (cost of raw materials, wages, etc.)
Suppose that a company manufactures x units of a product per month. If we denote the
fixed cost as F, the variable cost as V(z), and the total cost of operation per month as C(x),
then
C(z) = V(x) + F.
The profit function P(x) is the difference between the total revenue R(x) and the cost C(x):
P(x) = R(x) - C(x).
Example 6. The small company “DomoCo” produces chairs. The weekly fixed cost is
$500, and the variable cost in dollars for x chairs per week is
V(x) = 2 • 10~4z3 - O.lz2 + 25z.
It was estimated that the weekly revenue in dollars from the sale of x chairs is given by
R(x) = — O.lz2 + 95z (0 < x < 950).
(a) Find the total cost function.
(b) Find the profit of the company if 60 chairs are produced and sold every week.
CHAPTER 1. ELEMENTARY FUNCTIONS 4
Solution:
(a) C(x) = V(x) + 500 = 2 • 10-4a;3 - 0.1a;2 + 25a; + 500.
(b) P( ) = R(x) - C(x) = —0.1a;2 + 95a; - 2 • 10"4x3 + 0.1a;2 - 25a; - 500 = -0.0002a;3 +
t
70a; - 500.
P(60) = -43.2 + 4200 - 500 = 3,656.80 (dollars).
Composition of Functions
Definition 4. Given two functions f and g. the composition of f and g is denoted by
fog and is defined by
The domain of f o g is the set of all x in the domain of g for which g{x) lies in the domain
of/.
Example 7. If f(x) = V# + 1 and g(x} = 2a;2 — 3, then
/(<?($)) = y/g(x) + 1 = x/2a;2 — 3 + 1 = -\/2a;2 — 2,
g(J(xf) = 2(/(a;))2 - 3 = 2(\/^TT)2 - 3 = 2(x + 1) - 3 = 2a; - 1.
Here, the domain of / is the set of x for which x + 1 > 0, that is, x > —1, and the domain
of g is the set of all real numbers. Thus, the domain of f(g(x)') is the set of all x for which
g(x) = 2a?2 — 3 > —1, that is, 2a;2 > 2, so that x 6 (—00, —1] U [l,oo). Next, the domain
of ff(/(a?)) is the set of all real x for which /(a;) is defined, as g imposes no restrictions on
the values of /. Since / is defined for x > —1, then the domain of g(f{xf) is the interval
1, oo).
Example 8. Suppose that the factory’s cost of manufacturing x machine parts is
C (ar) = 2,500 + 60a;. An assembly line produces N(f) = lOt — -t2 parts, 1 < t
Express the factory cost as a function of hours of operation.
Solution: C(AT(t)) = 2,500 + 60 (lOt - |t2 ) = 2,500 + 600t - 20t2.
\ O /
Exercises 1.2
In Exercises 1 — 2, find the functions f + g, f — g, fg, and f /g.
1. /(a;) = x2 — 3, g(x) — 2 — y/x + 1.
In Exercises 3 — 4, find the functions f(g(x)) and g(f(xf).
g(x) =
In Exercises 5 — 6, find functions / and g such that h — f o g. (Note: The answer is not
unique.)
CHAPTER 1. ELEMENTARY FUNCTIONS 5
5. h(x) — (rr3 — x2 + 5)4. 6. hfx) =
7. A manufacturer has a monthly fixed cost of $24,000, and a production cost of $5 per
unit. The product sells for $9 per unit. Find the cost, the revenue and the profit function.
Compute the profit (loss) in producing 5,000 and 8,000 units.
8. The revenue of a travel agency is given by R(x) = — O.lz2 + 100x dollars, where x is
the dollar amount spent on advertising. The amount of dollars spent at time t is given by
/(t) = 3t + 100, where t is measured in months. Find the function What does it
represent?
1.3 Transformations of Graphs
Definition 5. The graph of a function f is the collection of all points (x, y) in the zy-plane
such that x G Dom{f} and y = f(x).
The Vertical Line Test determines whether a curve in the rry-plane is the graph of a function.
Vertical Line Test. A curve in the zy-plane is the graph of a function y = f(x) if and
only if each vertical line intersects it in at most one point.
Thus, a circle in the ary-plane fails the Vertical Line Test, and therefore is not the graph of
a function.
Much information about a function can be gained from its graph. By applying certain
transformations to the graph of a given function, we can obtain the graphs of certain related
functions.
Vertical and Horizontal Shifts. Let the graph of f(x) be given and let c be a positive
number. To obtain the graph of
y = f(x) + c, shift the graph of f(x) up by c units;
y = f(x) — c, shift the graph of f(x) down by c units;
y = f[x — c), shift the graph of f(x) to the right by c units;
y = f(% + c), shift the graph of f(x) to the left by c units.
Vertical and Horizontal Stretching. Let c > 1. To obtain the graph of
y = cf(x), stretch the graph of /(x) vertically by a factor of c;
y = — f(x), compress the graph of f(x) vertically by a factor of c;
c
y = f(cx), compress the graph of f(x) horizontally by a factor of c;
X
y — /(-), stretch the graph of f(x) horizontally by a factor of c.
c
CHAPTER 1. ELEMENTARY FUNCTIONS 6
Reflections. To obtain the graph of
y = — f(x), reflect the graph of /(x) about the a?-axis;
y = /(—x), reflect the graph of /(x) about the y-axis;
Example 9. Given the graph of y = a;2, sketch the graph of
(a) y — x2 — 2 (6) y = (x + 2)2 (c) y = — (x + 2)2 (d~) y — -x2
3
Solution:
We begin with the graph of y = x2.
-5 -4 -3 -2 -1 0 1 2 3 4 5
Figure 1.1: y = x2.
(a) The graph of y = x2 — 2 is obtained from the graph of y = x2 by shifting it 2 units
downwards along the t/-axis.
T"“ " I--------------- 1--------------- 1--------------- 1-------------- Illi
-5 -4 -3 -2 -1 0 1 2 3 4 5
Figure 1.2: y = x2 — 2.