Calculus III Part 1

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Differentiability implies continuity

Calculus III (Part 1)

Continuity

When function is continuous, we can use

Intermediate Value Theorem

If y=f(x) is continous on interval [a,b], and u is a number between f(a) and f(b), then there is a c between [a,b] st. f(c)=u

If a function f is continous on a finate closed interval [a,b], then f has both an absolute maximum and an abolute minimum

Epsilon-delta defition of Continuity

Differentiation

Differentiating

Techniques

inverse functions

Implicit differentiation

link to e.g. Folium of descartes

derivatives of inverse trigonometric functions

Derivatives of exponential and logarithmic functions

Chain Rule

Derivatives of Trigonometric functions

Quotient Rule

Product Rule

Basic differentiation

Derivatives and Applications

Interpretations of f(x) and f'(x) on the graph of f(x)

Critical points

f'(x)=0 or does not exist

Second Derivative Test

First Derivative Test

Locating and Identifying critical points

Concavity

Inflection point is when the graph of f(x) changes concavity

occurs when f''(x) changes sign

Theorems

Extreme Value Theorem

Locating Absolute extrema

To locate absolute extrema on a finite, closed interval [a,b], evaluate f at all critical points and both endpoints. The largest of these values is the absolute maximum value of f on [a,b] while the smallest value is the absolute minimum.

If f(x) has only 1 relative maximum(minimum) at x=c in any given interval, then x=c is also the absolute maximum(minimum) of f(x) in that given interval.

Mean Value Theorem

Let f be continous on the closed interval [a,b] and differentiable on the open interval (a,b). Then there is at least one point c in (a,b) such that f'(c) = (f(b)-f(a))/(b-a)

Rolle's Theorem

Let f be continous on the closed interval [a,b] and differentiable on the opne interval (a,b). If f(a)=0 and f(b)=0, then there is at least one point c in (a,b) such that f'(c)=0

Vector Functions

Useful link

http://ltcconline.net/greenl/courses/202/vectorFunctions/vectorDerivativeIntegral.htm

Definition

Suppose the curve C is the graph of a vector- valued function r(t)

Normal vector

Tangent Vectors

Derivatives

suppose r(t)=, then r'(t)=

Polar Functions

Arc Length of the Polar Curve

Tangent Lines to Polar Curve

Parametric Functions

Tangent lines in Parametric curves

Arc length of the Curve

if y=f(x) is a somooth curve on (a,b)

sometimes, a curve would be in parametric forms, X=f(t),Y=g(t)

Converting to Cartesian form

tan@=y/x,@ represents the angle

x=rcos@, y=rsin@,@ reprsents the angle

Finding the Derivative

Related Rates

useful link:http://en.wikipedia.org/wiki/Related_rates

Maximizing and Minimizing variable quantities

Formulate appropriate equation

Set required domain.

Make use of Extreme Value Theorem (see "Theorems")

Equations of Tangent and Normal lines

Slope of normal = -1/m

At a point of xy-coordinate of (a,b) Equation of tangent line is: y-b = -1/m (x-a)

Slope of tangent = m = dy/dx

At a point of xy-coordinate of (a,b) Equation of tangent line is: y-b = m (x-a)

Motion along a Line

Local Linear Approximation

When f(x) is differentiable,

If the above limit exists, then f(x) is differentiable (limit to the difference quotient).

Limits

Defintion

Left and Right limits

Right and Left hand limits

Formal Definition

Normal definition

Finding Limits

Common Techniques

Common useful identities

Using a graph or table of values of the given function

For eg, finding the above limit:

A table of values of f(x) as x approaches 2

Multiplying by a conjugate surd

Divide by highest power

Factorising and simplifying

Direct substitution

When limit exists

Infinite limits

When both numerator and denominator = 0 or ∞

L'Hopital's Rule

Convert to either of the other forms

Squeeze Theorem

e.g.

Suppose g(x)≤f(x)≤h(x) and if the limit of g(x) at x=c is Y and the limit of h(x) at x=c is Y, then the limit of f(x) at x=c is also equal to Y.