Module 1
Module 1
Module 1
Module 1 : Functions
Course Title : Differential Calculus
Course Number : Math 111
Course Description : An introductory course covering the core concepts of limit,
continuity and differentiability of functions involving one or
more variables. This also includes the application of
differential calculations in solving on optimization, rate of
change, related rates, tangents and normal, and
approximations; partial differentiation and transcendental
curve tracing. .
Total Learning Time : 4 units (4 hours lecture per week)
Pre-requisites : N/A
(if there’s any)
Overview:
In this chapter, our attention will be confined almost entirely to the study of function
defined by equations. Such function occur often in mathematics and in physical
applications, and they furnish illuminating examples of the power of the tools to be
developed throughout the present course of study.
Learning Outcomes:
At the end of this module, the student should be able to:
1. Work with functions represented in a variety of ways: graphical, numerical,
analytical, or verbal. They should understand the connections among these
representations.
2. Understand the meaning of the derivative in terms of a rate of change and local linear
approximation and should be able to use derivatives to solve a variety of problems.
3. Calculate problems involving relationships between changing quantities, (e.g. related
rates problems)
Indicative Content:
This module discusses at least the following topics: Functions, graph of functions,
dependent and independent variables, Algebraic and Transcendental Functions, Single-
valued and many-valued Functions, notation of function, inverse function Implicit and
Explicit.
Pre-Assessment:
1. y = -2x
2. y = x2 + 2x -2
3. y =x3 – 1
1
4. y = 𝑥2
5. y = √𝑥 + 2
For the given functions, sketch the graph of the curves and determine whether the
functions are even, odd or neither.
1. 𝑦 = 𝑥 2 + 1
2. 𝑦 = 𝑥 3 + 5
3. 𝑦 = 𝑒 𝑥
Functions
x2 + y2 = a2
x and y are variables, but a is a constant. For as the point whose coordinates are x,
y, moves along the curve, the values of x and y are continually changing, while the value
of the radius a remains unchanged.
Constants are usually denoted by the first letters of the alphabet, a,b,c, α, β, γ, etc.
Variables are usually denoted by the last letters of the alphabet, x, y, z, ϕ, ψ
Definition of a Function
When one variable quantity so depends upon another that the value of the latter
determines that of the former, the former is said to be a function of the latter.
For example, the area of a square is a function of its side ; the volume of a sphere
is a function of its radius ; the sine, cosine, and tangent are functions of the angle ; the
expressions,
are functions of x.
𝑥 2 + 𝑥𝑦 + 𝑦 2 , log(𝑥 2 + 𝑦 2 ) 𝑎 𝑥+𝑦 ,
The expressions,
𝑥+𝑦
𝑥𝑦 + 𝑦𝑧 + 𝑧𝑥, √ , log(𝑥 2 + 𝑦 − 𝑧)
𝑧
are functions of x, y, and z.
𝑦 = 𝑥 2 , y = tan 4 x, y = 𝑒𝑥 = 1
𝑧 + 𝑥 − 𝑦 = 0, 𝑤 + 𝑤𝑧 + 𝑧𝑥 + 𝑦 = 0
one must be regarded as the dependent variable, and the others as independent variables.
An algebraic function is one that involves only a finite number of the operations
of addition, subtraction, multiplication, division, involution and evolution with constant
exponents. All other functions are called transcendental functions. Included in this class
are exponential, logarithmic, trigonometric or circular, and inverse trigonometric,
functions.
Note. —The term "hyperbolic functions" is applied to certain forms of exponential
functions.
A more general definition of Algebraic Function is, a function whose relation to
the variable is expressed by an algebraic equation.
In the equation,
𝑦 = 𝑥 2 − 2𝑥
𝑥 = 1± √𝑦 + 1
Here each value of y determines two values of x. In the former case, y is a single-
valued function of x. In the latter case, x is a two-valued function of y. A n -valued
function of a variable x is a function that has n values corresponding to each value of x.
The inverse trigonometric function, tan-1 x, has an unlimited number of values for each
value of x.
Notation of Functions
The symbols F(x), f(x),𝜙(𝑥), ψ(𝑥) and the like, are used to denote functions of x.
Thus, instead of “y is a function of x," we may write
𝑦 = f(x) or 𝑦 = 𝜙(𝑥)
A functional symbol occurring more than once in the same problem or discussion is
understood to denote the same function or operation, although applied to different
quantities. Thus if,
𝑓(𝑥) = 𝑥 2 + 5
Then
𝑓(𝑦) = 𝑦 2 + 5 𝑓(𝑎) = 𝑎2 + 5
𝑓(𝑎 + 1) = (𝑎 + 1)2 + 5 = 𝑎2 + 2𝑎 + 6
𝑓(2) = 22 + 5 = 9, 𝑓(1) = 6
Inverse Function.
𝑦 = 𝜙(𝑥)
𝑥 = 𝜓(𝑦)
When one quantity is expressed directly in terms of another, the former is said to
be an explicit function of the latter. For example, y is an explicit function of x in the
equations,
𝑦 = 𝑥 2 + 2𝑥 𝑦 = √𝑥 2 + 1
2𝑥𝑦 + 𝑦 2 = 𝑥 2 + 1 𝑦 + log 𝑦 = 𝑥
Sometimes, as in the first of these equations, we can solve the equation with
reference to y, and thus change the function from implicit to explicit. Thus, we find from
this equation,
𝑦 = −𝑥 ± √2𝑥 2 + 1
Solution:
To determine if an equation defines a functional relationship between its
variables, isolate the dependent variable on the left side and the independent variable on
the right side.
Considering y as the dependent variable and x as the independent variable, we
have,
For ( a),
𝑦 = ±√3 − 𝑥, this is not a function since there are two values of y for some
values of x.
For (b),
𝑦 = 2 − 𝑥, this is a function since there is only one value for y for every values of
x.
For (c),
𝑦 = 1 − 𝑥 2 , this is a function.
For (d),
𝑦 = ±√5 − 𝑥 2 , this is not a function.
1
2. Determine the domain and range for the function defined by 𝑦 = 𝑥−3.
Solution:
The domain is all real numbers x ≠ 3.
The function has a zero denominator when x = 3, thus, the domain is restricted to all
real numbers except x = 3. The range is all real numbers, y ≠ 0.
Solution:
The domain is all real numbers x ≥ 1.
5. Express the altitude of a right circular cone as a function of the volume with a
fixed radius of 3 inches. Graph the function.
Solution:
Solution:
For 𝑓(−2)
𝑓(−2) = (−2)2 + 3(−2) − 7 = −9
For 𝑓(0)
𝑓(0) = (0)2 + 3(0) − 7 = −7
For 𝑓(4)
𝑓(4) = (4)2 + 3(4) − 7 = 21
For 𝑓(3𝑥)
𝑓(−3𝑥) = (−3𝑥)2 + 3(−3𝑥) − 7 = 9𝑥 2 − 9𝑥 − 7
For 𝑓(2𝑦)
𝑓(2𝑦) = (2𝑦)2 + 3(2𝑦) − 7 = 4𝑦 2 + 6𝑦 − 7
The values, 𝑓(−2), 𝑓(0), 𝑓(4), 𝑓(−3𝑥) and 𝑓(2𝑦) are called functional values, and they
lie in the range of f. This means that the values, 𝑓(−2), 𝑓(0), 𝑓(4), 𝑓(−3𝑥) and
𝑓(2𝑦)are y-values and thus the points,
𝜋
8. If 𝑔(𝑥) = sin 2𝑥 − cos 𝑥, 𝑓𝑖𝑛𝑑 𝑔(𝜋), 𝑔 ( 2 ) , 𝑔(𝜋 + 𝑥), 𝑔(−𝑥) 𝑎𝑛𝑑 𝑔(0)
Solution:
For 𝑔(𝜋)
𝑔(𝜋) = sin 2𝜋 − cos 𝜋 = 0 − (−1) = 1
𝜋
For 𝑔 ( 2 )
𝜋 𝜋 𝜋
𝑔 ( ) = sin 2 ( ) − cos ( ) = 0 − 0 = 0
2 2 2
For 𝑔(𝜋 + 𝑥)
𝑔(𝜋 + 𝑥) = sin 2(𝜋 + 𝑥) − cos(𝜋 + 𝑥)
= 𝑠𝑖𝑛2𝜋𝑐𝑜𝑠2𝑥 + 𝑐𝑜𝑠2𝜋𝑠𝑖𝑛2𝑥 − cos 𝜋 cos 𝑥 − sin 𝜋 sin 𝑥
= 𝑠𝑖𝑛2𝜋𝑐𝑜𝑠2𝑥 + 𝑐𝑜𝑠2𝜋𝑠𝑖𝑛2𝑥 − cos 𝜋 cos 𝑥 + sin 𝜋 sin 𝑥
= sin 2𝑥 + cos 𝑥
For 𝑔(−𝑥)
𝑔(−𝑥) = sin 2(−𝑥) − cos(−𝑥)
= −sin 2𝑥 + cos 𝑥
For 𝑔(0)
𝑔(0) = sin 2(0) − cos(0) = 0 − 1 = −1
Solution:
The composition of 𝑓(𝑥) and 𝑔(𝑥) is (𝑓 ∘ 𝑔)(𝑥) =𝑓(𝑔(𝑥)). This is called as
function composition.
For (𝑓 ∘ 𝑔)(2)
(𝑓 ∘ 𝑔)(2) = 𝑓(𝑔(2))
But, 𝑔(2) = 5(2) − 1 = 9, then
𝑓(𝑔(2)) = 𝑓(9) = 2𝑥 2 − 𝑥 + 5
=2(9)2 − 9 + 5
= 158
For (𝑓 ∘ 𝑔)(𝑥)
(𝑓 ∘ 𝑔)(𝑥) = 𝑓(𝑔(𝑥))
But, 𝑔(𝑥) = 5(𝑥) − 1 = 5𝑥 − 1, then
𝑓(𝑔(𝑥)) = 𝑓(5𝑥 − 1) = 2𝑥 2 − 𝑥 + 5
= 2(5𝑥 − 1)2 − (5𝑥 − 1) + 5
= 2(25𝑥 2 − 10𝑥 + 1) − 5𝑥 + 1 + 5
= 50𝑥 2 − 20𝑥 + 2 − 5𝑥 + 1 + 5
= 50𝑥 2 − 25𝑥 + 8
For (𝑔 ∘ 𝑓)(𝑥)
(𝑔 ∘ 𝑓)(𝑥) = 𝑔(𝑓(𝑥))
Evaluation:
Solve the following problems. Write your solutions on the space provided.
3. If 𝑓(𝑦) = 𝑦(𝑦 − 3)2 , find 𝑓(𝑐), 𝑓(0), 𝑓(3), 𝑓(−1) 𝑎𝑛𝑑 𝑓(𝑥 + 3).
𝑏−𝑏 2 1
4. If 𝑓(𝑏) = , find 𝑓(0), 𝑓(1), 𝑓 (2) 𝑎𝑛𝑑 𝑓(tan 𝑥).
1+𝑏2
1
6. If 𝑔(𝑥) = 4𝑥 4 − 3𝑥 2 + 2𝑥 − 2, find 𝑔(2), 𝑔(1 − 2), 𝑔 (2) 𝑎𝑛𝑑 𝑔(− 𝑥).
8. Express the area of a triangle as a function of its altitude having a base of 4 units.
Graph the function.
10. Express the height of a right circular cylinder as a function of its volume having a
radius of 4 cm. Graph the function.
11. A parabola has an altitude of 4 units. Express the length of its base as a function
of its area. Graph the function.
References:
C.E. Love and E.D. Rainville. (1981). Differential and Integral Calculus, Sixth Edition.
New York: The Macmillan Company.
H.J. Terano. (2015). A Simplified Text in Differential Calculus. Camarines Sur Plytechnic
Colleges
E.D. Rainville. (1958). Elementary Differential Equations, Second Edition. Collier
Macmillan Canada, Ltd., Toronto Ontario