Applied Mathematics - Continuity and Differentiability

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)