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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
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
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)=
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,
Right and Left hand limits
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
