Mathematics model

1. Er. Rahul Jarariya

2. Introduction:
• Rate of Change is called Calculus. It was developed by Newton and Leibniz.
• Understand the changes between the values which are related by a function.
• It covers by some important topics like differentiation, Limits, functions, integration.
• Classification based on Differential and Integral Calculus.
• Calculus mathematics vital role in Science and Technologies and Physics.
• Calculus Formulas given below:
• Calculus-formulas.pdf (slideshare.net)
𝛿y = 2𝛿x
y
x
rate of change = 𝛿y / 𝛿x = rise / run = slope
slope = 𝛿y / 𝛿x = (y2 – y1) / (x2 – x1) = (16 – 4) / (8 – 2) = 2
𝑑𝑦
𝑑𝑥
lim
𝑛→∞
1 +
1
𝑛
𝑛
𝑓(𝑥)

4. Why – How, Calculus in Mathematics?
• Calculus means the part of mathematics that deals with the properties of derivatives and integrals of quantities such as area,
volume, velocity, acceleration, etc., by processes initially dependent on the summation of infinitesimal differences.
• It helps in determining the changes between the values that are related to the functions.
• Basic Calculus is the study of differentiation and integration.
• Both concepts are based on the idea of limits and functions. Some concepts, like continuity, exponents, are
the foundation of advanced calculus.
• Basic calculus explains about the two different types of calculus called “Differential Calculus” and “Integral Calculus”.
Differential Calculus helps to find the rate of change of a quantity, whereas integral calculus helps to find the quantity when
the rate of change is known.

5. Differences
LIMITS
• The degree of closeness to any value or the
approaching term. A limit is normally expressed
using the limit formula as-
• It is read as “the limit of f of x as x approaches c
equals A”.
DERIVATIVES
• Instantaneous rate of change of a quantity with
respect to the other. The derivative of a
function is represented as:
• Go through the links given below for Calculus
Class 11 and class 12 concepts and get a
detailed knowledge of basic calculus.

6. Continuity and Differentiability
A Function is always continuous if it is differentiable at any point, whereas the vice-versa condition is
not always true.
Definition of Continuity:
• (i) The continuity of a real function (f) on a subset of the real numbers is
defined when the function exists at point c and is given as-
lim
𝑛→𝑐
𝑓 𝑥 = 𝑓(𝑐)
• (ii) A real function (f) is said to be continuous if it is continuous at every point in
the domain of f.
• Consider a function f(x), and the function is said to be continuous at every point
in [a, b] including the endpoints a and b.
• Continuity of “f” at a means,
• lim
𝑛→𝑎
𝑓 𝑥 = 𝑓(𝑎)
• Continuity of “f” at b means,
• lim
𝑛→𝑏
𝑓 𝑥 = 𝑓(𝑏)
Differentiability formula
• Assume that if f is a real function and c is a point in its domain. The derivative
of f at c is defined by 0
• The derivative of a function f at c is defined by-
• lim
ℎ→0
𝑓 𝑥+ℎ −𝑓(𝑐)
ℎ

7. Theorem’s
Theorem 1: Algebra of continuous functions:
If the two real functions, say f and g, are continuous at a real number c, then
• (i) f + g is continuous at x=c.
• (ii) f – g is continuous at x=c.
• (iii) f. g is continuous at x=c.
• (iv)f/g is continuous at x=c, (provided g(c) ≠ 0).
Theorem 2: Suppose f and g are real-valued functions such that (f o g) is defined at c. If g is continuous at c and if f is continuous at g (c), then (f o
g) is continuous at c.
Theorem 3: If a function f is differentiable at a point c, then it is also continuous at that point.
Theorem 4 (Chain Rule): Let f be a real-valued function which is a composite of two functions u and v; i.e., f = v o u.
• Suppose t = u(x) and if both dt/dx and dv/dt exist, we have df/dx = (dv/dt). (dt/dx)
Theorem 5:
(1) The derivative of ex with respect to x is ex; i.e., d/dx(ex) = ex.
(2) The derivative of log x with respect to x is 1/x.
i.e., d/dx(log x) =1/x.
Theorem 6 (Rolle’s Theorem): Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b), such that f(a) = f(b), where a and b are some
real numbers. Then there exists some c in (a, b) such that f'(c) = 0.

8. Advanced Calculus
• Advanced calculus includes some topics such as infinite
series, power series, and so on which are all just the
application of the principles of some basic calculus topics
such as differentiation, derivatives, rate of change and o on.
the important areas which are necessary for advanced
calculus are vector spaces, matrices, linear transformation.
advanced calculus helps us to gain knowledge on a few
important concepts such as:
• Quadratic forms
• Generalized stokes theorem
• Vector fields as derivatives
• Integration of forms
• Multilinear algebra
• Continuous differentiability
• Tangent space and normal space via gradients
• Dual space and dual basis
• Critical point analysis for multivariate functions,
etc.

9. Applications of Calculus
• Calculus is a mathematical model that aids in system analysis and the
discovery of the best possible solution for future prediction.
• Calculus principles are often used in everyday life to address problems
including sophisticated shapes, automobile safety, survey data analysis for
company planning, credit card payment records, determining how
changing conditions in a system effect humans, etc.
• Doctors, economists, biologists, architects, medical professionals, and
statisticians frequently employ calculus as their language of choice.
• To construct bridges, highways, and tunnels, for instance, architects and
engineers employ calculus concepts to determine the magnitude and shape
of curves.
• Some of the ideas, such as birth and death rates, radioactive decay,
reaction rates, heat and light, motion, electricity, etc., are nicely depicted
using calculus.