Rudin Analysis Chapter 5: Differentiation

Questions and Answers

What is the definition of f'(x) for a function f defined and real-valued on [a,b]?

lim t-> x (f(t)-f(x))/(t-x) or lim h-> 0 (f(x+h)-f(x))/h

If f is differentiable at x∈[a,b], what can we say about f?

f is continuous at x

What are the derivative formulas for (f+g)'(x), (fg)'(x), and (f/g)'(x) if f and g are differentiable at x?

(f+g)'(x)=f'(x)+g'(x), (fg)'(x)=f'(x)g(x)+f(x)g'(x), (f/g)'(x)=(g(x)f'(x)-g'(x)f(x))/g(x)^2

If f is continuous and f'(x) exists and g is differentiable at f(x), what can we say about g(f(x))?

g(f(x)) is differentiable and its derivative equals g'(f(x))*f'(x) Signup and view all the answers

If x is a local max of f and f'(x) exists, what can we conclude about f'(x)?

f'(x)=0 Signup and view all the answers

What is the condition for functions f and g that are continuous and differentiable on [a,b] and (a,b) respectively?

There exists x∈(a,b) such that [f(b)-f(a)]g'(x)=[g(b)-g(a)]f'(x) Signup and view all the answers

If f is real, continuous on [a,b] and differentiable on (a,b), what condition exists for x∈(a,b)?

There exists x∈(a,b) such that f(b)-f(a)=(b-a)f'(x) Signup and view all the answers

If f is differentiable on (a,b) and f'(x)>= 0 for all x∈(a,b), then _____

f is non-decreasing Signup and view all the answers

If f is differentiable on (a,b) and f'(x)=0 for all x, then _____

f is constant Signup and view all the answers

Study Notes

Differentiation in Analysis

The derivative of a function f at point x, denoted f'(x), is defined as the limit of the difference quotient as t approaches x, or alternatively the limit as h approaches 0 of the difference ( \frac{f(x+h)-f(x)}{h} ).

Continuity and Differentiability

If a function f is differentiable at a point x within the interval [a, b], it must also be continuous at that point x.

Derivative Rules

The sum of derivatives: If f and g are differentiable at x, then the derivative of their sum is given by ((f+g)'(x) = f'(x) + g'(x)).
The product rule: For differentiable functions f and g, ((fg)'(x) = f'(x)g(x) + f(x)g'(x)).
The quotient rule: The derivative of the quotient of f and g is ((f/g)'(x) = \frac{g(x)f'(x) - g'(x)f(x)}{(g(x))^2}).

Composite Functions

If f is continuous and f' exists, and g is differentiable at f(x), then the composition g(f(x)) is differentiable, and its derivative is given by (g'(f(x)) \cdot f'(x)).

Local Extrema

At a local maximum x of a function f, if f'(x) exists, then the derivative f'(x) equals zero.

Mean Value Theorem

For continuous functions f and g on [a, b] that are differentiable on (a, b), there exists at least one point (x \in (a, b)) that satisfies the equation ([f(b) - f(a)] g'(x) = [g(b) - g(a)] f'(x)).

Fundamental Theorem of Calculus

For a real and continuous function f on [a, b] that is differentiable on (a, b), there exists some (x \in (a, b)) such that the integral of f over [a, b] equals ((b-a)f'(x)).

Properties of the Derivative

If f is differentiable on (a, b) and its derivative f'(x) is non-negative for all x in (a, b), then f is non-decreasing on that interval.
If f'(x) equals zero for all x, then f is constant.

Description

Test your knowledge of differentiation from Chapter 5 of Rudin's Analysis with these flashcards. Each card provides a critical definition or theorems related to differentiable functions. Perfect for mastering calculus concepts!