Applied Mathematics - Continuity and Differentiability

IIT-JEE Main & Advanced | BITSAT | SAT | MSAT | MCAT | State Board | CBSE | ICSE | IGCSE
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
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) = (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.)
State Board | CBSE | ICSE | IGCSE | Course for High School & College
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