Applied Mathematics - Continuity and Differentiability

Name: Applied Mathematics - Continuity and Differentiability
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Last updated 4/2022

English

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What you'll learn

Introduction
Continuity
Differentiability
Exponential and Logarithmic Functions
Logarithmic Differentiation
Derivatives of Functions in Parametric Forms
Second Order Derivative
Mean Value Theorem

Course content

2 sections • 20 lectures • 3h 0m total length

Introduction9:31
Explore the concept of continuity and its link to limits, examining left-hand and right-hand limits and why the function value must match the limit for continuity at a point.
Types of Sums In Continuity5:06
Explore continuity by analyzing type one and type two sum problems, using limits and function values at points like 1 and 4 to determine left-hand, right-hand, and overall continuity.
Removable and Non - Removable Discontinuity6:33
Explore removable and non-removable discontinuities by comparing limit values with function values in type one and type two questions. Redefine the function to remove the discontinuity.
Properties of Continuous Functions6:28
Explore the properties of continuous functions: closure under addition, subtraction, multiplication, and division (denominator nonzero); the intermediate value property; and continuity of composition.

Numericals on Continuity (Level - 1) Question - 14:02
Examine level one continuity questions by evaluating the limit of (x^2+18x-19)/(x-1) as x approaches 1 and confirm it equals f(1)=20, proving continuity.
Numericals on Continuity (Level - 1) Question - 23:52
Analyze the continuity at x=2 for a piecewise function: x^3-2x+1 for x<=2 and 3x-2 for x>2; left-hand limit is 5 and right-hand limit is 4, proving discontinuity at 2.
Numericals on Continuity (Level - 1) Question - 35:55
calculate the limit as x approaches zero to identify a removable discontinuity, then redefine the function at zero with the limit value to obtain a continuous extension.
Numericals on Continuity (Level - 1) Question - 47:13
Analyze the continuity of the piecewise function at x = 5/3 by evaluating the limit with substitution and conclude a discontinuity since the limit does not equal the function value.
Numericals on Continuity (Level - 1) Question - 56:11
Evaluate the limit of the given function as x approaches zero and compare it with f(0) to determine continuity. The caption shows the limit equals the function value, proving continuity.
Numericals on Continuity (Level - 1) Question - 63:44
Identify the discontinuity at x = 4 as a jump. The left-hand limit is 26 and the right-hand limit is 23, so the function is not continuous and cannot be removed.
Numericals on Continuity (Level - 1) Question - 78:00
Identify a removable discontinuity at x = 0 in the given function and compute the limit as x approaches 0, which equals 3/2. Redefine f(0) to 3/2 to achieve continuity.
Numericals on Continuity (Level - 1) Question - 84:40
Show how to evaluate the continuity limit at x=0 by rationalizing the denominator and using a conjugate, concluding the limit equals -4/3.
Numericals on Continuity (Level - 1) Question - 94:23
Apply continuity at x = b by evaluating the limit of f(x) as x approaches b. Conclude that f(b) equals (log 4)^2.
Numericals on Continuity (Level - 1) Question - 103:44
examine a numericals on continuity (level 1), solving for k by equating the function value to the limit at zero and applying factoring to simplify.
Numericals on Continuity (Level - 1) Question - 117:16
Analyze a piecewise function and enforce continuity at x=2 and x=3 by equating left-hand and right-hand limits. Solve for A and B, obtaining A = 2 and B = 1/2.
Numericals on Continuity (Level - 1) Question - 126:32
tackle numericals on continuity by analyzing a removable discontinuity at x=0 in a logarithmic function, and show how left- and right-hand limits guide redefining f(0) to restore continuity.
Numericals on Continuity (Level - 1) Question - 133:06
Use continuity at zero to equate left and right limits with f(0) and solve for a and b in the piecewise function.
Numericals on Continuity Level - 2 Solved Questions - 615:55
Solve level two continuity questions by evaluating limits and defining f(0) to ensure continuity, and identify discontinuities in rational and modulus functions.
Numericals on Continuity Level - 3 Solved Questions - 715:44
Explore level-3 numericals on continuity in applied mathematics, solving problems with limits and L'Hôpital's rule. Analyze piecewise, greatest-integer, and other continuity cases to determine when functions are continuous.
Numericals on Differentiability Level - 3 Solved Questions - 1252:49
Explore level-3 differentiability problems with solved examples on continuity, derivatives, and limits. Apply piecewise function analysis, left and right derivatives, and functional equations to determine differentiability and constant behavior.

Requirements

Basic knowledge of mathematics of 9th and 10th std Mathematics

Description

Continuity and Differentiability

Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions
Concept of exponential and logarithmic functions.
Derivatives of logarithmic and exponential functions
Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives
Rolle's and Lagrange's Mean Value Theorems (without proof) and their geometric interpretation

SUMMARY

1. A real valued function is continuous at a point in its domain if the limit of the function at that point equals the value of the function at that point. A function is continuous if it is continuous on the whole of its domain.

2. Sum, difference, product and quotient of continuous functions are continuous. i.e., if f and g are continuous functions, then (f ± g) (x) = f (x) ± g(x) is continuous. (f . g) (x) = f (x) . g(x) is continuous.

3. Every differentiable function is continuous, but the converse is not true.

4. Chain rule is rule to differentiate composites of functions. If f = v o u, t = u (x) and if both dt/dx and dv/dt exist then df/dv = dt/dx ⋅ dt/dx

5. Logarithmic differentiation is a powerful technique to differentiate functions of the form f (x) = [u (x)] raise to v (x) . Here both f(x) and u (x) need to be positive for this technique to make sense.

6. Rolle’s Theorem: If f : [a, b] → R is continuous on [a, b] and differentiable on (a, b) such that f (a) = f (b), then there exists some c in (a, b) such that f ′(c) = 0.

7. Mean Value Theorem: If f : [a, b] → R is continuous on [a, b] and differentiable on (a, b). Then there exists some c in (a, b) such that f'c = [f(b) - f(a)] / (b - a)

Who this course is for:

Complete Mathematics for Engineering Entrance Exam Preparation. ( IIT-JEE Main | Advanced | BITSAT | SAT | etc.)
Those preparing for board and competitive exams State Board, CBSE, ICSE , IGCSE, MHT-CET & NEET
Courses are suitable for 160 countries from Europe, America, Middle East, Asia, Africa and APAC. Notably England, Germany, France, Sweden, Ireland, Scotland, USA, Canada, UAE, Saudi, Qatar, Kuwait, Malaysia, Indonesia, Myanmar, Newzealand, Australia, South Africa, South Korea, Nigeria, Nepal, Sri Lanka, etc

Applied Mathematics - Continuity and Differentiability

What you'll learn

Explore related topics

Course content

Continuity4 lectures • 28min

Numericals (level 1 | level 2 | level 2)16 lectures • 2hr 33min

Requirements

Description

Who this course is for: