USTP
MATH 111A: Calculus 1
Name: _________________________________________________ Class number: ______
Section: ____________ Schedule: __________________________ Date: _____________
Lesson title: INTRODUCTION TO DIFFERENTIAL Materials: Ballpoint, Notebook, Calculator
CALCULUS Text book: Calculus by Ron Larson,11th Ed
Lesson Objectives: References:
At the end of the lesson, you should be able to: 1..The Calculus 7, by Louis Leithold
1. Define Differential Calculus and related terms 2. Calculus 7th Ed. By James Stewart
2. Classify Functions and their Graphs 3. Differential and Integral Calculus
3. Evaluate Functions by Clyde E. Love and Earl Rainville
Productivity Tip: Most things in life don’t come easy. You
have to make some tough decisions. Make some smart
choices. Being prepared is the only way to know that you
are going to win.
A. LESSON PREVIEW/REVIEW
Introduction
In order to pass the course, you shall be oriented on the following:
1. Course Outline
Differential Calculus is an introductory course covering the core concepts of limits, continuity and
differential of functions involving one or more variables. This also includes: the application of
differential calculations in solving problems on optimization, rates of change, related rates,
tangent and normal lines, partial differentiation and transcendental curve tracing
University of Science and Technology of Southern Philippines
1
USTP
MATH 111A: Calculus 1
Name: _________________________________________________ Class number: ______
Section: ____________ Schedule: __________________________ Date: _____________
3. Course Map
4. The first lesson will cover the following:
4.1 Definition of Differential Calculus and related terms
4.2 Classification of Functions and their Graphs
4.3 Evaluation of Functions
University of Science and Technology of Southern Philippines
2
USTP
MATH 111A: Calculus 1
Name: _________________________________________________ Class number: ______
Section: ____________ Schedule: __________________________ Date: _____________
Activity 1: What I Know Chart, part 1
What I Know Questions: What I Learned (Activity 4)
1 Do you have any knowledge of
differential calculus?
2 How about functions and their
graphs?
3 Can you evaluate a function?
B. MAIN LESSON
Activity 2: Content Notes
DEFINITIONS
Differential Calculus is the mathematics of the variation of a function with respect to changes in
independent variables; the study of slopes of curves, accelerations, maxima and minima, by
means of derivatives and differentials.
A function is a set of ordered pairs of numbers (𝑥, 𝑦) in which no two distinct ordered pairs have
the same first number. The set of all admissible values of 𝑥 is called the domain of the function,
and the set of all resulting values of 𝑦 is called the range of the function.
The symbols 𝑥 and 𝑦 denote variables. Because the value of 𝑦 is dependent on the choice of
x, 𝑥 denotes the independent variable and 𝑦 denotes the dependent variable.
A function may be denoted as 𝑦 = 𝑓(𝑥) which is read “y equals the function of x” or briefly as “y
equals f of x.” The other symbols used frequently are: 𝑦 = ℎ(𝑥), 𝑦 = 𝑔(𝑥), 𝑦 = 𝜃(𝑥). If we are
given a function 𝑓(𝑥), the value of the function when 𝑥 = 𝑎 is denoted by 𝑓(𝑎).
University of Science and Technology of Southern Philippines
3
USTP
MATH 111A: Calculus 1
Name: _________________________________________________ Class number: ______
Section: ____________ Schedule: __________________________ Date: _____________
GRAPHS OF SIX (6) BASIC FUNCTIONS
𝒇(𝒙) = 𝒙 𝒇(𝒙) = 𝒙𝟐
(1) Identity or Linear Function (2) Quadratic Function
𝒇(𝒙) = 𝒙𝟑 𝒇(𝒙) = +√𝒙
(3) Cubic Function (4) Radical or Square Root Function
University of Science and Technology of Southern Philippines
4
USTP
MATH 111A: Calculus 1
Name: _________________________________________________ Class number: ______
Section: ____________ Schedule: __________________________ Date: _____________
𝟏
𝒇(𝒙) = |𝒙| 𝒇(𝒙) =
𝒙
(5) Absolute Value Function (6) Rational Function
CLASSIFICATION OF FUNCTIONS
All functions are classified as either algebraic or transcendental. A function is algebraic if the operations
involved in the function are combinations of the six fundamental algebraic operations, namely, addition,
subtraction, multiplication, division, evolution (process of taking the nth roots of a number), and involution
(the operation of raising a number to a power).
The elementary transcendental functions are trigonometric functions (sine, cosine, tangent, cosecant,
secant, cotangent) and inverse trigonometric functions (arcsine, arccosine, arctangent, arc cotangent,
arc secant, arc cosecant) and the exponential and logarithmic functions.
University of Science and Technology of Southern Philippines
5
USTP
MATH 111A: Calculus 1
Name: _________________________________________________ Class number: ______
Section: ____________ Schedule: __________________________ Date: _____________
EVALUATION OF A FUNCTION
In evaluating a function, we will input or substitute a value of the independent variable to find the
value of the function. Examples are given for easy comprehension.
Example 1. If 𝑓(𝑥) = 3𝑥 − 5,
find: a) 𝑓(1)
b) 𝑓(−3)
c) 𝑓(𝑎)
d) 𝑓(4ℎ − 1)
Solution: Given, 𝑓(𝑥) = 3𝑥 − 5 Write the given function
a) 𝑓(1) = 3(1) − 5 Substitute 1 for x
= 3−5 Simplify
= −2
b) 𝑓(−3) = 3(−3) − 5 Substitute (-3) for x
= 3(9) − 5 Simplify
= 27 − 5
= 22
c) 𝑓(𝑎) = 3(𝑎) − 5 Substitute a for x
= 3𝑎 − 5 Simplify
d) 𝑓(4ℎ + 1) = 3(4ℎ + 1) − 5 Substitute (4h+1) for x
= 3(16ℎ + 8ℎ + 1) − 5 Expand the binomial
= 48ℎ + 24ℎ + 3 − 5 Simplify
= 48ℎ + 24ℎ − 2
Take note that different values of the
function will be obtained for different
input in the independent variable.
University of Science and Technology of Southern Philippines
6
USTP
MATH 111A: Calculus 1
Name: _________________________________________________ Class number: ______
Section: ____________ Schedule: __________________________ Date: _____________
Example 2. If 𝑓(𝑥) =
and 𝑔(𝑥) =
Find 𝑓[𝑔(𝑥)]
Solution:
Write
3𝑥 + 2
𝑓(𝑥) =
𝑥−1
Substitute 𝑔(𝑥) = to x of 𝑓(𝑥)
𝑥+2
3 +2
𝑓[𝑔(𝑥)] = 𝑥−3
𝑥+2
−1
𝑥−3
Simplify
3(𝑥 + 2) + 2(𝑥 − 3)
𝑓[𝑔(𝑥)] = 𝑥−3
(𝑥 + 2) − 1(𝑥 − 3)
𝑥−3
3(𝑥 + 2) + 2(𝑥 − 3) 𝑥−3
𝑓[𝑔(𝑥)] =
𝑥−3 (𝑥 + 2) − (𝑥 − 3)
3(𝑥 + 2) + 2(𝑥 − 3)
𝑓[𝑔(𝑥)] =
(𝑥 + 2) − (𝑥 − 3)
3𝑥 + 6 + 2𝑥 − 6
𝑓[𝑔(𝑥)] =
𝑥+2−𝑥+3
5𝑥
𝑓[𝑔(𝑥)] =
5
𝑓[𝑔(𝑥)] = 𝑥
University of Science and Technology of Southern Philippines
7
USTP
MATH 111A: Calculus 1
Name: _________________________________________________ Class number: ______
Section: ____________ Schedule: __________________________ Date: _____________
Example 3. If 𝑔(𝑦) = 𝑐𝑜𝑠 2𝑦 − 2𝑠𝑖𝑛 𝑦
find: a) 𝑔(𝜋),
b) 𝑔( ),
c) 𝑔(0),
d) [𝑔(𝑥) + 𝑔(−𝑥)]
Solution: From the definition of 𝑔(𝑦), it follows that,
a) 𝑔(𝜋) = 𝑐𝑜𝑠 2(𝜋) − 2𝑠𝑖𝑛(𝜋) Substitute 𝜋 for y
=1−0 Simplify
=1
𝜋
b) 𝑔 = 𝑐𝑜𝑠 2 − 2𝑠𝑖𝑛 ( ) Substitute for y
= −1 − 2 Simplify
= −3
c) 𝑔(0) = 𝑐𝑜𝑠 2(0) − 2𝑠𝑖𝑛 (0) Substitute 0 for y
=1−0 Simplify
=1
d) [𝑔(𝑥) + 𝑔(−𝑥)]
since: 𝑔(𝑥) = 𝑐𝑜𝑠 2(𝑥) − 2𝑠𝑖𝑛( 𝑥) ,
and 𝑔(−𝑥) = 𝑐𝑜𝑠 2(−𝑥) − 2sin (−𝑥) Note: s𝑖𝑛 (−𝑥) = −𝑠𝑖𝑛 𝑥
= 𝑐𝑜𝑠 (−2𝑥) − 2sin (−𝑥) 𝑐𝑜𝑠 (−𝑥) = 𝑐𝑜𝑠𝑥
= 𝑐𝑜𝑠 2𝑥 + 2𝑠𝑖𝑛 𝑥
Therefore:
[𝑔(𝑥) + 𝑔(−𝑥)] = [𝑐𝑜𝑠 2𝑥 − 2𝑠𝑖𝑛𝑥] + [𝑐𝑜𝑠 2𝑥 + 2𝑠𝑖𝑛𝑥] Substitution
= 2 𝑐𝑜𝑠2𝑥 Simplify
Take note: When dealing with trigonometric functions
with pi (𝜋);
Change the mode of your calculators into “radians
mode”
Or convert radians in degrees by multiplying it by,
Example 4. If ℎ(𝑏) = , find: a) ℎ(0)
University of Science and Technology of Southern Philippines
8
USTP
MATH 111A: Calculus 1
Name: _________________________________________________ Class number: ______
Section: ____________ Schedule: __________________________ Date: _____________
b) ℎ( )
c) ℎ(𝑡𝑎𝑛 𝑥)
Solution: a) ℎ(0) = Substitute 0 for b
= Simplify
=0
b) ℎ = Substitute for b
( )
= Simplify
𝑐) ℎ(t𝑎𝑛 𝑥) =
=
( )
𝑐𝑜𝑠 𝑥 𝑠𝑖𝑛 𝑥 − 𝑠𝑖𝑛 𝑥
= 𝑐𝑜𝑠 𝑥
𝑐𝑜𝑠 𝑥 + 𝑠𝑖𝑛 𝑥
𝑐𝑜𝑠 𝑥
= but: 𝑐𝑜𝑠 𝑥 + 𝑠𝑖𝑛 𝑥 = 1
=
1
h(tan x) = 𝑠𝑖𝑛 𝑥(𝑐𝑜𝑠 𝑥 − 𝑠𝑖𝑛 𝑥) factor out sin x
Example 5. If 𝑓(𝑣) = 𝑣 ,
University of Science and Technology of Southern Philippines
9
USTP
MATH 111A: Calculus 1
Name: _________________________________________________ Class number: ______
Section: ____________ Schedule: __________________________ Date: _____________
( ∆ ) ( )
𝑓𝑖𝑛𝑑: , ∆𝑥 ≠ 0
∆
𝑓(𝑥 + ∆𝑥) − 𝑓(𝑥)
∆𝑥
Separate evaluation for 𝑓(𝑥 + ∆𝑥) and 𝑓(𝑥)
Considering 𝑓(𝑥 + ∆𝑥)
Substitute: 𝑥 + ∆𝑥 for v
Write: 𝑓(𝑥 + ∆𝑥) = (𝑥 + ∆𝑥)
Expand 𝑓(𝑥 + ∆𝑥) = (𝑥) + 3 (𝑥) (∆𝑥) + 3(𝑥) (∆𝑥) + (∆𝑥)
Simplify 𝑓(𝑥 + ∆𝑥) = 𝑥 + 3𝑥 ∆𝑥 + 3𝑥∆𝑥 + ∆𝑥
Then, considering: 𝑓(𝑥)
Substitute: 𝑥 for v
Write 𝑓(𝑥) = (𝑥)
Simplify 𝑓(𝑥) = 𝑥
Substitute 𝑓(𝑥 + ∆𝑥) = 𝑥 + 3𝑥 ∆𝑥 + 3𝑥∆𝑥 + ∆𝑥 and 𝑓(𝑥) = (𝑥)
( ∆ ) ( )
To
∆
( ∆ ) ( ) ∆ ∆ ∆
=
∆ ∆
Since we have +x3 and -x3, therefore omit x3
( ∆ ) ( ) ∆ ∆ ∆
=
∆ ∆
Factor ∆x in the numerator
( ∆ ) ( ) ∆ ∆ ∆
=
∆ ∆
Cancel ∆x in the numerator and denominator
( ∆ ) ( )
= 3𝑥 + 3𝑥∆𝑥 + ∆𝑥
∆
University of Science and Technology of Southern Philippines
10
USTP
MATH 111A: Calculus 1
Name: _________________________________________________ Class number: ______
Section: ____________ Schedule: __________________________ Date: _____________
Activity 3: Skill-building Activities
Directions: Perform the indicated operations in each exercise. Simplify the results.
1. 𝐼𝑓 𝑓(𝑥) = 𝑥 − 4𝑥, 𝑓𝑖𝑛𝑑: 𝑎) 𝑓(−5)
𝑏) 𝑓(𝑦 + 1)
𝑐) 𝑓(𝑥 + ∆𝑥)
𝑑) 𝑓(𝑥 + 1) − 𝑓(𝑥 − 1)
2. 𝐼𝑓 ℎ(𝑦) = 𝑐𝑜𝑠 𝑦 − 𝑠𝑖𝑛 𝑦, 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 ℎ + 𝑥 = ℎ(𝜋 − 𝑥) = −ℎ(−𝑥)
Activity 4: What I Know Chart, part 2
What I Learned
1. ______________________________________________________________________________
______________________________________________________________________________
2. ______________________________________________________________________________
______________________________________________________________________________
3. ______________________________________________________________________________
______________________________________________________________________________
Activity 5: Check for Understanding
Directions: Perform the indicated operations in each exercise. Simplify the results.
1. 𝐼𝑓 𝑓(𝑥) = √𝑥 + 4, 𝑓𝑖𝑛𝑑: 𝑎) 𝑓(−2) DO YOUR
𝑏) 𝑓(2) BEST
𝑐) 𝑓(𝑥 + 𝑏𝑥)
2. 𝐼𝑓 𝑓(𝑥) = 𝑐𝑜𝑠 𝑥 𝑓𝑖𝑛𝑑: 𝑎) 𝑓(0)
𝑏) 𝑓(𝜋)
𝑐) 𝑓(𝜋 − 𝑦)
University of Science and Technology of Southern Philippines
11
USTP
MATH 111A: Calculus 1
Name: _________________________________________________ Class number: ______
Section: ____________ Schedule: __________________________ Date: _____________
B. LESSON WRAP-UP
Activity 6: Thinking about Learning
A. Work Tracker
You are done with this session! Let’s track your progress. Shade the session number you just
completed.
B. Think about your Learning
1. What motivated you to finish the lesson today?
_____________________________________________________________________________
_____________________________________________________________________________
2. What could you have done better to improve your learning today?
_____________________________________________________________________________
_____________________________________________________________________________
FAQs
1. What are the applications of differential calculus?
The applications of differential calculus are : (1)Calculation of profit and loss with respect to
business using graphs (2) calculation of the rate of change of temperature, (3) calculation of
speed or distance (4) to derive many Physics equations.
In Electrical Engineering, it used to determine the length of cable from one station to another.
2. What are the pre-requisites of calculus?
You must have prior knowledge in Algebra, Geometry and Trigonometry.
University of Science and Technology of Southern Philippines
12
