Applied Mathematics - Continuity and Differentiability

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)

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