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Table of Contents
iii CHAPTER IV Applications of Derivatives of Algebraic and
- Definition and Notation of Functions CHAPTER I Functions, Limits and Continuity
- Domain and Range
- Graph of Functions
- Types of Functions
- Operations on Functions
- Limit of a Function
- One – sided Limits
- Continuity
- Chapter Test
- Differentition CHAPTER II Derivatives of Algebraic Functions
- The Increment Method
- Theories on Differentiation of Algebraic Functions
- Higher order Derivatives
- Implicit Differentiation
- Chapter Test
- Trigonometric Functions CHAPTER III Derivatives of Transcendental Functions
- Chain Rule of Derivatives of Trigonometric Functions
- Implicit Differentiation of Trigonometric Functions
- Derivatives of Inverse Trigonometric Functions
- Chain Rule of Inverse Trigonometric Functions
- Higher – order Derivative of Inverse Trigonometric Functions
- Implicit Differentiation of Inverse Trigonometric Functions
- Logarithmic Functions
- Higher – order Derivatives of Logarithmic Functions
- Implicit Differentiation of Lograithmic Functions
- Logarithmic Differentiation
- Exponential Functions
- Chain Rule of the Derivatives of Exponential Functions
- Higher – order Derivaives of Exponential Functions
- Implicit Differentiation of Exponential Functions
- Hyperbolic Functions
- Chain Rule of the Derivatives of Hyperbolic Functions
- Higher – order Derivatives of Hyperbolic Functions
- Implicit Differentiation of Hyperbolic Functions
- Chapter Test
- The Differential Transcendental Functions
- Application of the Differential
- Approximation Formulas
- Error Propagation
- Tangent Line and Normal Line to a given Curve
- Relative Exrema
- Increasing and Decresing Functions and the
- Concavity, Points of Inflection and the Second First Derivative Test
- Curve Tracing Derivative Test
- Optimization Problems
- Optimization Problems Involving Algebraic Functions
- Optimization Problems Involving Transcedental Functions
- Number Problems
- Related Rates
- Related Rates ProblemsInvolving Algebraic Functions
- Related Rates Problems Involving Transcendental
- Motion Problems Functions
- Chapter Test
- Definition of Parital Derivatives of a Functions CHAPTER V Partial Differentiation
- Partial Derivatives by Formulas of Differentiation
- Higher – order Partial Derivatives
- Total Derivatives
- Chain Rule of Partial Differentiation
- Implicit Partial Derivatives
- Chapter Test
Symbols such as f , g , and h are used to denote functions unless stated
otherwise. If the function is expressed in terms of the variable x , f x , g x , and
h x are used to denote this function. For instance, in the finding the area of
circle, A r r^2 is used to describe the relationship between the area and the
radius of the given circle. Here, it can be observed that the area, A is expressed as a function of r and the value of A depends on the value of r. So, we say that A is the dependent variable and r is the independent variable.
Example 1.1. Let f be defined by f x 2 x 1. It can be observed that f is function
since when x is replaced by any real numbers there is exactly one value of y
obtained.
x 3 2 1 0 1 2 3
f x 7 5 3 1 1 3 5
Example 1.2. The circle x^2 y^2 9 cannot be a function since when x is 0, y
assumes two values such as 3 and – 3. Thus, we have the ordered pairs 0 , 3 and
0 , 3 . Similarly, the ordered pairs 3 , 0 , 3 , 0 , 1 , 2 2 and 1 , 2 2 can also be
obtained from the given circle. Observe that 1 is assigned to two values of y ,
that is 2 2.
Figure 1.1. The circle x^2 y^2 9
Example 1.3. Let g be defined by g x x^2 1. Refer to the next table for the
values of x and its corresponding values of g x. Although, several values of g x
appear similar in the table, g can still be thought as a function since no two or
more values of x are repeated.
x 3 2 1 0 1 2 3
g x 10 5 2 1 2 5 10
The relations presented in Example 1.1, 1.2 and 1.3 can be illustrated by
mapping. Here, the set of real numbers x is called the domain while the set of
real numbers f x , y and g x are the range.
Figure 1.2 shows the mapping of the domain to the range of the given relations in Example 1.1, 1.2 and 1.3. Figure 1.2 ( a ) shows a one–to–one mapping
since there is exactly one value of x mapped to exactly one value of y. In ( b ), a
mapping of one–to–many is observed since there is exactly one value of x , say 0
and 1 mapped to two values of y , i.e. -3 and 3; and 2 2 and 2 2 , respectively.
In ( c ), a mapping of many–to–one can be seen as there are two values of x
mapped to one value of y. The ordered pairs 3 , 10 and 3 , 10 ; 2 , 5 and 2 , 5
; and 1 , 2 and 1 , 2 show this relation.
Figure 1.2. Mapping of a f x 2 x 1 , b x^2 y^2 9 and c g x x^2 1
Based on the examples presented in Figure 1.2 and Definition 1.1, a function possesses a one–to–one correspondence (bijection) and many–to–one correspondence (surjection) but not one–to–many or many–to–many.
Another way to determine whether a relation is function or not is by using the vertical line test. Given the graph of a function, draw vertical lines overlaying the graph. If the vertical lines pass through exactly one point on the graph, then it is a function. If it passes through two or more points, then it is not a function.
To evaluate function, a straight forward substitution is used. For instance,
given a function, f x x^2 2 , if we wish to find f 3 we shall replace x by -
and perform the operation leading us with f 3 11.
Example 1.4. Let f be a function defined by f x x^2 2 x 3 , find a. f 2 ; b.
2
f^1 ;
c. f 1 ; d. f 2 x and e. f x h .
Solution :
a. f x^ ^ x^2 ^2 x ^3 d. f x^ ^ x^2 ^2 x ^3
Domain
x
Range
f x
Domain
x
Range y
0 1
3
2 2
0
2 2
3
1
2
5
10
Domain
x
Range
g x
a b c
Example 1.7. Let f be a function defined by f x x 3. Here, x 3 cannot be a negative number, so we can let x 3 0. Solving this inequality, we have (^) x 3. So, the domain of this function is the set of all numbers
greater than or equal to -3 or 3 ,. The range of this function is 0 ,.
Example 1.8. [2] Let f be a function defined by f x x^2 9. Since f
involves square root, then x^2 9 0. Here, we shall think of any number
that when replaced to x will give x^2 9 0. Solving this inequality, we
have x 3 or x 3. So the domain of f is given by , 3 3 , and
the range is 0 ,.
Example 1.9. Find the domain and range of the function y^ ^3 x. Recall that
the cube root of any negative number is defined on the set of negative real number, the cube root of 0 is 0 and the cube root of a positive real number is still defined on the same set of positive real numbers. Therefore, the
domain of y ^3 x is ,and the range is ,.
From this example, we can conclude that for any function f defined
by f x^ n^ x where n is any odd positive integer, then the domain and
range of f is ,.
Example 1.10. Find the domain and range of the following functions:
a.
x
f x ^2
b.
x
gx x
c. 12
x
f x
d.
f x (^) x (^2)
Solution:
a. Since x can be found on the denominator, then x cannot be replaced by
0. So the domain of
x
f x ^1 is the set of real numbers except 0 or
interval notation we have , 0 0 ,and the range is also the set
of real numbers except 0, , 0 0 , since the numerator is
constant.
b. For
x
gx x , the denominator x 2 should be equal to 0, i.e. x 2
. The domain of g is the set of real numbers except -2 or
, 2 2 ,. For the range of g , observed that the numerator is
no longer constant so g x assumes any number except 1. Thus, the
range of g is , 1 1 ,.
c. Similar with a, x appears on the denominator. So we have x^2 0 giving
us with the domain , 0 0 ,. For the range, take note that the
numerator is constant, 1 and the denominator is (^) x^2. Given this facts,
f x do not assumes values such as 0 and any negative numbers, thus
the range of this function is 0 ,.
d. The domain of
f x (^) x (^2) is the set of real numbers except 4
since when x is 4 or -4, the denominator becomes 0. The range is
^
(^) 0 , 16
,^1.
Example 1.11. Let y be function defined by y 2 x. The function y 2 x is
an exponential function whose domain is the set of real numbers, ,
and the range is is the set of all positive real numbers, 0 ,.
Example 1.12. The domain of the function f x sin 2 x is the set of real
numbers ,while the range is any numbers on the interval 1 , 1 .
Exercise 1.1. Find the domain and range of the following functions.
- 2 3
y 2 x
2. y x^2
3. y x^2 2 x 4
4. y ^ x^3 ^2
5. f x^ ^3 2 x ^3
6. f x 9 x^2
- 2
x
gx x
x
g x x
2 25
x
f x x
x y (^)
Figure1.4. Graph of f x 2 x 3 Figure1. 5. Graph of f x x 3
To facilitate the graphing of a function, the following steps are suggested:
- Identify the domain and range and the properties of the function.
- Choose suitable values of x from the domain of a function and solve for its corresponding value of y.
- Determine the behaviour of x and y.
4. Plot the points x , y on the plane.
- Smoothly trace the curve.
Example 1.13. Sketch the graph of the following functions: a. f x 2 x 3
b. f x x 3
c. f x 25 x^2
d. f x x^2 4
e.
x
f x
f. 12
x
f x
g. y 2 x
Solution :
a. b.
From Figure 1.4 it can be observed that the graph of f x 2 x 3 is a
line that passes through the points 0 , 3 and
, 0 2
3. The point 0 , 3 is the
x intercept and the point
, 0 2
3 is the y intercept of f x 2 x 3. We
can also verify that the domain and range of this function is any number on
the interval ,. From Example 1.7 the domain of f x x 3 is
3 , and the range is 0 ,. Also, it can be seen from Figure1.5 that as x increases, y also increases.
Figure1. 6. Graph of f x 25 x^2 Figure1. 7. Graph of f x x^2 4
Figure1. 8. Graph of f x x^2 Figure1. 9. Graph of f x x^12
Figure1. 10. Graph of f x 2 x
c. d.
The graph of f x 25 x^2 (see Figure 1.6) shows a semi – circle with
0 , 3 as y intercept and 3 , 0 and 3 , 0 as x intercepts. The graph of
f x x^2 4 in Figure 1.7 confirms that the domain of this function is , 2 2 ,^ and the range is^ 0 ,.
e. f.
g.
^
,^1. Finally, it can be seen from Figure 1.12 that domain of
x
f x x is
, 3 3 ,and the range is , 6 6 ,.
The lines x 5 and x 5 in Figure 1.11 and the lines x 4 and x 4 are called^ vertical asymptotes ; while the lines^ y 2 in Figure 1.
and y 0 in Figure 1.12 are called horizontal asymptotes. The asymptotes
of functions determine its discontinuity. For instance in the function
f x x 2 , when x is replaced by 4 or -4, you’ll have 0
f 4 ^1 and
f 4 ^1. We can also say that when x approaches 4 from the right, f x
gets arbitrarily large or approaching . Similarly, as (^) x approaches 4
from the left f x approaches . This idea gives the formal definition of
asymptotes.
From this definition, we set the rules to determine the asymptote of a function. Suppose that the rational function
1 1 1 0 ...
b x b x bx b
a x a x ax a qx
f x px m m
m m
n n n n
where q x 0
is in lowest terms.
If q a 0 , then x a is a vertical asymptote.
If n m , then the x axis is a horizontal asymptote.
If (^) n m , then the horizontal asymptote is the line m
n b
a (^) y .
If n m , then the graph of f has no horizontal asymptote.
Definition 1.4. Rees, R., (2003). Suppose that
q x
f x px is a rational
function in lowest terms and a is some real number where q x 0
i. The vertical line (^) x a is a vertical asymptote of the graph of f if
as x a , then f x .
ii. The horizontal line y a is a horizontal asymptote of the graph
of f if as x , then f x a.
For the function
x
f x x in Figure 1.12, observe that the f is
undefined at (^) x 3 , i.e., when (^) x is replaced by -3, the denominator becomes 0. However, based on Definition 1.4, the rational function should
be in lowest term leading us with f x x 3. So now, f 3 6. The
point 3 , 6 is called point discontinuity.
1.1.3. Types of Functions
In the study of Calculus we might encounter different forms of functions. Generally speaking functions can be classified into two such as algebraic functions and transcendental functions. Some forms of
algebraic functions are linear function, f x ax b ; quadratic function,
f x ax^2 bx c , for a , b , c and a 0 ; and polynomial functions,
f x an xn an 1 xn ^1 ... a 1 x a 0. We may also include the rational
functions of the forms q x
f x px where q x 0
(a) (b) (c) (d)
Figure 1.13. Some examples of (a) linear function; (b) quadratic function; (c) polynomial function, and (d) rational function
Transcendental functions are functions that transcend algebraic functions.
For instance, ... 1! 2! 3!
2 3 x x x transcends ex ; ... 3! 5! 7!
sin
3 5 7 x x x x x
and ... 2! 4! 6!
cos 1
2 4 6 x x x x . Some commonly used transcendental
functions are:
- Trigonometric functions ( Six Circular Functions )
- Inverse trigonometric functions
- Exponential functions
The domain and range of the trigonometric functions are:
Table Domain and Range of Trigonometric Functions Domain Range
sin ^ ^ , ^ ^1 ,^1
cos ^ ^ , ^ ^1 ,^1
tan
(^) x x n 2
where n Z ,
cot x^ ^ x n ^ where n^ Z ^ ,
sec
(^) x x n 2
where n Z , 1 1 ,
csc x^ ^ x n ^ where n^ Z ^ ^ ,^1 ^ ^1 ,
Some trigonometric identities are as follows:
x
x x
x x
x x
x
x x
x x x
x
x
x x x
x x
x x
x x
x x
x x
x x
x x
x x
1 cos 2
tan^1 cos^2
2
cos^1 cos^2
2
sin^1 cos^2
1 tan
tan 2 2 tan
sin 2 2 sin cos
2 cos 1
1 2 sin
cos sin cos 2
1 cot csc
1 tan sec
sin cos 1
tan cot 1
cos sec 1
sin csc 1
sin sin
cos cos
2
2
2
2
2
2
2 2
2 2
2 2
2 2
x y x y x y
x y x y x y
x y x y x y
x y x y x y
x y x y x y
x y x y x y
x y x y x y
x y
x y x y
x y
x y x y
x y x y x y
x y x y x y
x y x y x y
x y x y x y
2
sin^1 2
cos cos 2 sin^1
2
cos^1 2
cos cos 2 cos^1
2
sin^1 2
sin sin 2 cos^1
2
cos^1 2
sin sin 2 sin^1
cos 2
cos^1 2
cos cos^1
sin 2
sin^1 2
sin cos^1
cos 2
cos^1 2
sin sin^1
1 tan tan
tan tan tan
1 tan tan
tan tan tan
sin sin cos cos sin
sin sin cos cos sin
cos cos cos sin sin
cos cos cos sin sin
These identities will help students to simplify both trigonometric expressions and equations.
Let us consider the sine function r
sin y. If 2 2
^ , we see from
Figure 1.14 that the sine function attains the value on the interval 1 , 1
exactly once and so is one–to–one. For the cosine function, if we restrict
the value of inclusively between 0 and 2 2
^ for tangent
function, these gives cosine and tangent a one–t – one correspondence. On these intervals, we obtain their inverse functions as follows:
x y
x y
x y
1
1
1
tan
cos
sin
y x
y x
y x
tan
cos
sin
Table Domain and Range of the Inverse Trigonometric Functions Domain Range
sin^1 x ^1 ,^1
(^) 2
, 2
cos^1 x ^1 ,^1 ^0 ,
tan^1 x^ all real numbers
(^) 2
, 2
Exponential and Logarithmic Functions
A function y defined by the relation, y ax where a is a positive number except 1 is called an exponential function of x.
Example 1.15. Sketch the graph of (a) f x 2 x and (b)
x f x
2
1
Solution : The graphs of f x 2 x and
x f x
2
1 are shown in Figure 1.18.
Figure 1.1 6. Graphs of (a) y sin^1 x , (b) y cos^1 x and (c) y tan^1 x Reference: https://www.onlinemathlearning.com/inverse-sine-cosine-tangent.html
(a) y sin^1 x (b) y cos^1 x (c) y tan^1 x
Figure 1.21. Graph of f x log ax where a 1
Figure 1.22. Graph of f x log ax where 0 a 1
Logarithmic Functions
By definition of logarithm, the relation y ax , can be written as x log ay. Thus ax and log ax are inverse functions of one another, and it follows that a log a^ x x and log a a x x.
Example 1.16. Convert the following exponential to logarithmic functions
23 8 Ans. log 2 8 3
5
1 5
(^1) ^1
1 5
log^1 5
1
100 1 log 10 1 0
27
3 ^3 ^13 27
log 1 3
27
1 3
(^1) ^3
3 27
log^1 27
1
The logarithmic function takes a special when a e , that is called natural logarithm, given by log (^) e x ln x and when a 10 , called common logarithm and is written as log 10 x log x.
Properties of Logarithmic Function
For f x log ax where a 1
- Domain : 0 , Range : ,
- As x , f x increases and flattens
- As (^) x 0 from the right, f x decreases and gets steeper
- Asymptotic with respect to negative y – axis
For f x log ax where 0 a 1
- Domain : 0 , Range : ,
- As (^) x , f x decreases and flattens
- As x 0 from the right, f x increases and gets steeper
- Asymptotic with respect to positive y – axis
Some laws on logarithmic functions are as follows:
r x xr
x y y
x
xy x y
e
ln ln
ln ln ln
ln ln ln
ln 1
ln 1 0
a a r
a a a
a a a
a
a
r x x
x y y
x
xy x y
a
log log
log log log
log log log
log 1
log 1 0
Hyperbolic Functions
Hyperbolic functions are functions possessing similar characteristics with that of the six trigonometric functions derived from its relation to the equilateral hyperbola. The hyperbolic functions are defined as follows:
2
sinh ex^ e^ x x ^ x x e e
hx (^)
csc (^2) x x
x x e e
x e e
tanh
2
cosh ex^ e^ x x ^ x x e e
hx (^)
sec (^2) x x
x x e e
x e e
coth
Some properties and identities of hyperbolic functions and their domain and range are presented in Table 3 while their graphs are on Figure 1.23.
x
x x x e
x x e
x h x
x h x
x x
^
cosh sinh
cosh sinh
coth csc 1
tanh sec 1
cosh sinh 1
2 2
2 2
2 2
x x x
x x x
x y x y x y
x y x y x y
cosh 2 cosh^2 sinh^2
sinh 2 2 sinh cosh
cosh( ) cosh cosh sinh sinh
sinh( ) sinh cosh cosh sinh
Table 3 Domain and Range of Hyperbolic Functions Domain Range
cosh x , 1 ,
sinh x , , tanh x ^ ^ , ^1 ,^1
sec hx ^ , ^0 , 1
csc hx , 0 0 , , 0 0 , coth x , 0 0 , , 1 1 ,
