This document provides information about an Applied Calculus course taught by Imran Qasim at Mehran University of Engineering and Technology. The key points are: 1) The course covers topics in differential and integral calculus, including functions, limits, derivatives, integrals, and their applications. 2) Students are expected to have prior knowledge of functions, limits, and differentiation before taking the course. 3) The course contents will help students develop expertise in techniques for differentiation and integration, as well as apply calculus to solve real-world problems.

1. Course: Applied Calculus
Subject Teacher: Imran Qasim
Assistant Professor (BPS-19)
Department of Basic Science and Related Studies,
Mehran University of Engineering and Technology, Jamshoro.

2. Applied
Calculus

3. Skills Prior to Enrollment this Course
Before starting this course student must know how to:
• Apply concepts of functions and types of functions.
• To evaluate Limits.
• Apply the concepts of differentiations.
• To perform integrations.

4. Course Contents
• Functions and types of functions
• Physical applications of functions
• The concept of limit and theorems on limits
• Left and right hand limit
• Continuity of functions
• Case study problems of continuity and discontinuity
• Derivative and its rules
• Geometrical interpretation of derivative
• Physical meaning of derivative
• Case study problems of rates of change, related rates
• Marginal analysis
• Higher derivatives
• Leibniz theorem
• Partial derivatives

5. • Roll’s theorem and its geometrical meaning
• Mean value theorem and its geometrical meaning
• Maclaurin’s and Taylor’s Series
• L’Hopital’s rule and indeterminate Forms
• Tangent and normal
• Curvature and radius of curvature
• Circle of curvature
• Maxima and minima of a function of one variable.
• Applications of maxima and minima
• Differentials and their applications
• Homogenous functions
• Euler’s theorem on homogeneous functions
• Total differentials
• Maxima and minima of a function of two variables

6. • Introduction to integration
• Definite and indefinite integrals
• Methods of integration
• Integration by formulas
• Integration by substitution
• Integration by parts
• Integration of rational algebraic functions
• Integration by completing the squares method
• Integration of irrational algebraic functions
• The Definite integral as an area under a curve
• Fundamental properties of definite integrals
• Beta and Gamma function
• Applications of integration
• Volume of solid of revolution
• Area between the two curves

7. • Vectors, scalar and vector product of two vectors
• Scalar triple product or box product
• Differentiation and integration of vector functions
• DEL – the differential operator
• The gradient
• The divergence of a vector function
• The curl of a vector function
• Solenoidal and irrotational vectors
• Unit vector normal to the surface
• The directional derivative
• Some fundamental theorem/proofs

8. Text/Reference Books
Text Books
1.Calculus by Thomas/Finney
2. Calculus by Howard Anton
3. Calculus with Analytical Geometry by SM. Yousif
4. Applied Calculus by M. Urs shaikh
References Books
1.Calculus by Swokowaski
2.Calculus and Analytical Geometry by Sherman K.
Stein

9. Applied Calculus
What is Calculus: Calculus is the mathematics of motion and
change.

10. Applied Calculus
Branches of Calculus: Differential Calculus and Integral
Calculus.
Development of Calculus: The contributions of Sir Isaac
Newton and Gottfried Wilhelm Leibniz is appreciable.

11. Applied Calculus
Mathematical Scope of Calculus:
Learning calculus is not the same as learning
arithmetic, algebra and geometry.
Calculus involves those techniques and skills to
develop expertise and a deeper level in differentiations
and anti-differentiations (integrals).
Calculus introduces many new concepts and
computational operations, for instance, (see teaching
plan).

12. Applied Calculus
Function:
A function is a rule between two non-empty sets in which
every element of set (𝑋) has unique image with the element
of set (𝑌).
Function is denoted by 𝐹: 𝑋 → 𝑌 (read as) function from 𝑋 to
𝑌.

13. It is a function.
It is not a function
∵ 3 𝜖 𝑋 has no any
image of element of
𝑌
It is not a function.
∵ 3 𝜖 𝑋 has two images
of elements of 𝑌.

14. Applied Calculus
Additionally,

15. Applied Calculus
The concept of function 𝐹: 𝑋 → 𝑌 is given by Leibntiz, he
described that dependence of one quantity to an other quantity.
Mathematically,
𝑓 𝑥 = 𝑦
Independent variable (Domain) Dependent variable (Range)
Input Output
Elements of 𝑋 set (Domain) Elements of 𝑌 set (Range)

16. Applied Calculus
Types of function
The basic types of function are:
1. One-one function (Injective function)
2. Into function
3. Onto function (Surjective function)
4. Bijective function

17. One-One Function:
A function in which every element of X has distinct image of
elements of Y is called one-one function.

18. X Y
On-to Function:
A function in which range of f=Y is called on-to function.

19. X Y
In-to Function:
A function in which range of f≠Y is called in-to function.
OR
A function whose range is the proper subset of Y.
𝑎, 𝑏 ≠ 𝑎, 𝑏, 𝑐

20. Another types of Functions:
i. Polynomial Function:
A function whose exponent/Power is positive or zero is called
polynomial function.
𝑎0𝑥𝑛 + 𝑎1𝑥𝑛−1 + 𝑎2𝑥𝑛−2 +−−−−− −𝑎𝑛−1 𝑥 𝑎𝑛
Examples;
𝑓 𝑥 = 𝑎𝑥 + 𝑏 → Linear Function
𝑓 𝑥 = 𝑎2𝑥2 + 𝑏𝑥 + 𝑐 → Quadratic Function
𝑓 𝑥 = 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑 → Cubic Function

21. Applied Calculus
Linear Function
A function described by the equation: 𝑦 = 𝑚𝑥 + 𝑐 is called a
linear function.
In other words, a linear function of 𝑥 is one, which contains no
term in 𝑥 of degree higher than the first. The general form of a
linear function is 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0.
where 𝑥 and 𝑦 are variables, and 𝑎, 𝑏 and 𝑐 are constants. This
function is called linear because its graph in the Cartesian
coordinates is always a straight line.

22. Applied Calculus
This function is called linear because its graph in the Cartesian
coordinates is always a straight line.

23. Applied Calculus
Quadratic Function
A function defined by the equation 𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0, where
𝑎, 𝑏, 𝑐 are constants, is called a quadratic function. This
equation always represents a parabola. The graph of such a
parabola will have one of the shapes as shown below:

24. Applied Calculus
Cube Function
The cube function is defined as 𝑦 = 𝑓 𝑥 = 𝑎𝑥3
+ 𝑏𝑥2
+
𝑐𝑥 + 𝑑. The graph of such function is

25. Applied Calculus
Reciprocal Function
The reciprocal function is defined as 𝑦 = 𝑓 𝑥 =
1
𝑥
. The graph
of such function is

26. Real Valued Function:
A function whose domain and range both are real
numbers are called
real valued function. It is denoted by:
𝑓: 𝑅 ⟶ 𝑅

27. Applied Calculus
Constant Function
The constant function is defined as 𝑦 = 𝑓 𝑥 = 𝑐. The graph
of such function is

28. Applied Calculus
Algebra of Functions
Let two given functions 𝑓 and 𝑔 then, the sum 𝑓 + 𝑔, the
difference 𝑓 − 𝑔 , the product 𝑓𝑔 , the quotient
𝑓
𝑔
, the
reciprocal
1
𝑓
and the constant 𝑐 functions are described as:

29. Circular Function:
The trigonometric functions 𝑠𝑖𝑛 𝑥 and
𝑐𝑜𝑠 𝑥together are called circular
functions. Because these are very closely
related to the circle and also
satisfies equation of circle.
𝑥2 + 𝑦2 = 1
OR
𝑠𝑖𝑛2𝑥 + 𝑐𝑜𝑠2𝑥 = 1

30. Modulus Function or Absolute Valued Function:
𝑥 = 𝑥 if 𝑥 ≥ 0
𝑥 = −𝑥 if 𝑥 < 0
The modulus of every number is always positive.
Modulus is the distance of the real number from
origin.
OR
𝑥 = 𝑥2

31. Applied Calculus

32. Applied Calculus
Additionally, one can solve

33. Even and Odd Functions:
A function 𝑓 𝑥 is said to be even if 𝑓 −𝑥 = 𝑓 𝑥 .
A function 𝑓 𝑥 is said to be odd if 𝑓 −𝑥 = −𝑓 𝑥 .
Determine whether the given functions even, odd or neither.
Ex:1. 𝑓 𝑥 = 𝑥2
Solution:𝑓 −𝑥 = −𝑥 2
𝑓 −𝑥 = 𝑥2
𝑓 −𝑥 = 𝑓 𝑥 Even
Ex:2. 𝑓 𝑥 = 𝑥3
Solution:
𝑓 −𝑥 = −𝑥 3
𝑓 −𝑥 = − 𝑥 3
𝑓 −𝑥 = −𝑓 𝑥 Odd

34. EX.03 𝑓 𝑥 = 𝑥 − 𝑥2
Solution:
𝑓 −𝑥 = −𝑥— −𝑥 2
𝑓 −𝑥 = −𝑥 − 𝑥2
𝑓 −𝑥 = − 𝑥 + 𝑥2 Neither
Ex:4. 𝑓 𝑥 = 𝑠𝑖𝑛 𝑥
Solution:
𝑓 −𝑥 = 𝑠𝑖𝑛 −𝑥
𝑓 −𝑥 = −𝑠𝑖𝑛 𝑥 ∵ 𝑠𝑖𝑛 −𝑥 = −𝑠𝑖𝑛 𝑥
𝑓 −𝑥 = −𝑓 𝑥 Odd

35. Applied Calculus
Composite Function
Let two given functions 𝑓 and 𝑔 then, the composite functions
are written as 𝑓 ∘ 𝑔. The graph of composition functions is

36. Applied Calculus
Formulae for Composite Function
(i) (iii)
(ii) (iv)
It should be noted that

37. Applied Calculus
Solution:

38. Ex:2 If 𝑓 𝑥 = 𝑥 + 1 𝑎𝑛𝑑 𝑔 𝑥 = 𝑥2 + 2𝑥 + 1 then
find (fog)x and (gof)x
𝑓 𝑜 𝑔 𝑥 = 𝑓(𝑔(𝑥))
= 𝑔 𝑥 + 1
= 𝑥2 + 2𝑥 + 1 + 1
𝑔 𝑜 𝑓 𝑥 = 𝑥2 + 2𝑥 + 2
𝑔 𝑜 𝑓 𝑥 = 𝑔(𝑓(𝑥))
= 𝑓 𝑥 2 + 2 𝑓 𝑥 + 1
= (𝑥 + 1)2+2 𝑥 + 1 + 1
= 𝑥2 + 2𝑥 + 1 + 2𝑥 + 2 + 1
= 𝑥2 + 4𝑥 + 4 Ans.

39. Applied Calculus
Solution:

40. Applied Calculus
Piecewise function
A function on which different formula are used on different
parts of its domain is called piecewise function. In simple
word, it is two or more than two valued function. It is written
as
𝑓 𝑥 =
𝑝 𝑥 , 𝑎 < 𝑥 < 𝑏
𝑞 𝑥 , 𝑎 ≤ 𝑥 < 𝑏
𝑟 𝑥 , 𝑎 ≤ 𝑥 ≤ 𝑏
Example:

41. Identify Function:
The function 𝑓: 𝑋 ⟶ 𝑋 defined by 𝑓 𝑥 = 𝑥, ∀ 𝑥 ∈ 𝑋
is called the identify function. It is denoted by 𝐼 𝑥 .
e.g. Let 𝑋 = 3,4,6 and 𝑓: 𝑋 ⟶ 𝑋 be defined by
𝑓 3 = 3, 𝑓 4 and 𝑓 6 = 6.
Here 𝑓 is the identify function defined on 𝑋.

42. Exponential Function:
A function in which the variable appears as exponent
(power) is called exponential
functions, the functions 𝑦 = 𝑒𝑥, 𝑦 = 𝑒𝑎𝑥, 𝑦 = 2𝑥 =
𝑒𝑙𝑛2
, 𝑦 = 𝑎𝑥
= 𝑒𝑥𝑙𝑛𝑎
are exponential functions of 𝑥.

43. Hyperbolic Functions:
i. 𝑠𝑖𝑛ℎ𝑥 =
𝑒𝑥−𝑒−𝑥
2
is called hyperbolic sine function.
Its domain and Range are the set of all real numbers.
ii. 𝑐𝑜𝑠 ℎ𝑥 =
𝑒𝑥+𝑒−𝑥
2
is called hyperbolic 𝑐𝑜𝑠𝑖𝑛𝑒
function.
Its domain is the set of all real numbers and the
range is the set of all numbers in the interval. )
1, ∞ .
iii. The Remaining four hyperbolic functions are:
𝑡𝑎𝑛ℎ𝑥 =
𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥 , 𝑠𝑒𝑐ℎ𝑥 =
2
𝑒𝑥+𝑒−𝑥
𝑐𝑜𝑡ℎ𝑥 =
𝑒𝑥+𝑒−𝑥
𝑒𝑥−𝑒−𝑥 𝑐𝑜𝑠𝑒𝑐ℎ𝑥 =
2
𝑒𝑥−𝑒−𝑥

44. Explicit Function:
If 𝑦 is easily expressed in terms of the independent variables,
then 𝑦 is called an explicit function of 𝑥.
𝑓 𝑥 = 𝑦, 𝑦 = 𝑥2 + 2𝑥 − 1
𝑦 = 𝑥 − 1
xii. Implicit Function:
If 𝑥 and 𝑦 are so mixed up and 𝑦 cannot be expressed in terms
of independent variable 𝑥, then 𝑦 is called an implicit function of
𝑥.
𝑓 𝑥, 𝑦 = 0
𝑥2
𝑦 + 𝑦2
𝑥 = 4, , 𝑥2
+ 𝑥𝑦 + 𝑦3
= 4 etc…

45. Parametric Functions:
Sometime curve is described by expressing both 𝑥 and 𝑦 as
function of a third variable “t” and “𝜃” which is called parameter.
The equation of the type 𝑥 = 𝑓 𝑡 and 𝑦 = 𝑔 𝑡 are called
parametric equations of the curve.
1. 𝑥 = 𝑎𝑡2 2. 𝑥 = a cos 𝑡
𝑦 = 2𝑎𝑡 𝑦 = 𝑎 sin 𝑡
3. 𝑥 = a cos 𝜃 4. 𝑥 = 𝑎 sec 𝜃
𝑦 = 𝑏 sin 𝜃 𝑦 = 𝑎 tan 𝜃

46. Interval:
The sub set of real line contains two end points 𝑎 and 𝑏is called interval.
There are four types of interval.
i. Closed interval : 𝑎, 𝑏 or 𝑎 ≤ 𝑥 ≤ 𝑏
e.g. 1,5 = 1,2,3,4,5
contain 𝑎 and 𝑏
ii. Open interval : 𝑎, 𝑏 or 𝑎, 𝑏 or 𝑎 < 𝑥 < 𝑏
e.g. 1,5 = 2,3,4
Does not contain 𝑎 and 𝑏
iii. Half closed interval : )
𝑎, 𝑏 𝑜𝑟 𝑎 ≤ 𝑥 < 𝑏
e.g. 1,5) = 1,2,3,4
𝑎 contains,𝑏 does not contain
iv. Half open interval : (𝑎, 𝑏 𝑜𝑟 𝑎 < 𝑥 ≤ 𝑏
e.g. (1,5 = 2,3,4,5
𝑏 contains, 𝑎 does not contain

47. Applied Calculus
Euler Identity
Euler's formula, named after Leonhard Euler, is a
mathematical formula in complex analysis that establishes the
fundamental relationship between the trigonometric functions
and the complex exponential function. The Euler Identity is
defined as:

48. Applied Calculus
Proof of Euler Identity
There are certain functions, which can be expressed as infinite
series of non – negative increasing powers of x. Such series
are called “power series.” Some of them are
Put 𝑥 = 𝑥𝑖 in equation (3), we get by calculating 𝑖2
, 𝑖3
, 𝑖4
,…

49. Applied Calculus
Using the first two series, we get This identity was
established by the
mathematician Euler
and therefore is
known as “Euler’s
Identity” or “Euler’s
Formula

50. Applied Calculus
Circular and hyperbolic functions
The Euler’s Identity” or “Euler’s Formula is
𝑒𝑖𝑥 = 𝑐𝑜𝑠𝑥 + 𝑖 𝑠𝑖𝑛𝑥, (1)
By using the concept of even and odd functions:
𝑐𝑜𝑠(−𝑥) = 𝑐𝑜𝑠𝑥 and 𝑠𝑖𝑛 −𝑥 = −𝑠𝑖𝑛𝑥. Now put 𝑥 = −𝑥 in
equation (1), we get
𝑒−𝑖𝑥 = 𝑐𝑜𝑠𝑥 − 𝑖 𝑠𝑖𝑛𝑥, (2)
Adding equations (1) and (2) as:
𝑒𝑖𝑥 = 𝑐𝑜𝑠𝑥 + 𝑖 𝑠𝑖𝑛𝑥
𝑒−𝑖𝑥 = 𝑐𝑜𝑠𝑥 − 𝑖 𝑠𝑖𝑛𝑥

51. Applied Calculus
𝑐𝑜𝑠𝑥 =
𝑒𝑖𝑥 + 𝑒−𝑖𝑥
2
.
Now subtracting equations (1) and (2) as:
𝑒𝑖𝑥
= 𝑐𝑜𝑠𝑥 + 𝑖 𝑠𝑖𝑛𝑥
𝑒−𝑖𝑥 = 𝑐𝑜𝑠𝑥 − 𝑖 𝑠𝑖𝑛𝑥
𝑠𝑖𝑛𝑥 =
𝑒𝑖𝑥 − 𝑒−𝑖𝑥
2𝑖
.
Since (𝑠𝑖𝑛𝑥, 𝑐𝑜𝑠𝑥) satisfies the equation of a circle 𝑥2
+ 𝑦2
=
1, therefore these functions are called circular functions. Now

52. Applied Calculus
Hence, we have
six circular
functions, defined
on the next slide.

53. Applied Calculus
By eliminating 𝑖 from circular functions, we get hyperbolic
functions as

54. Applied Calculus

55. Applied Calculus
PHYSICAL APPLICATIONS OF FUNCTIONS
There is perhaps no field or area where functions are not used.
In real life, where there is a relation between two variables, the
application of function is must.
These applications are found in social and natural sciences,
engineering and technology. Applications of functions are also
known as mathematical modeling of functions.
The mathematical modeling of functions provides the
information about cost, profit, revenue.

56. Applied Calculus
Functions that provide information about cost, revenue, and
profit can be of great value to management. This slide offers
an introduction to the cost function revenue function and profit
function We begin by establishing the notation for three
important types of functions. Using x for the number of units
produced or sold, we have
Cost function is denoted by 𝐶 𝑥
Profit function is denoted by 𝑃 𝑥
Revenue function is denoted by 𝑅(𝑥).

57. Applied Calculus
The mathematical notations for cost function is denoted by
𝐶 𝑥 , profit function is denoted by 𝑃 𝑥 and revenue function
is denoted by 𝑅(𝑥). It can be described by
The formula for profit function, cost function and revenue
function is
𝑃 𝑥 = 𝑅 𝑥 − 𝐶 𝑥 →→ Profit = Revenue − Cost
𝑅 𝑥 = 𝑃 𝑥 − 𝐶 𝑥 →→ Revenue = Profit − Cost

58. Applied Calculus

59. Applied Calculus

60. Applied Calculus
Break-Even Point: It represents no loss and no gain on the
manufacturing of commodities.

61. Applied Calculus

62. Applied Calculus
Work sheet
1.
2.

63. END