Critical points of f

Endpoints of domain Where f’(x) = 0 Where f’(x) is undefined

Relative/Local Maximum

f’(x) = 0 f’(x) goes from positive to

negative

Relative/Local Minimum

f’(x) = 0

f’(x) goes from negative to positive

Global/Absolute Maximum

Largest local maximum value of f(x) including endpoints

Global/Absolute Minimum

Smallest local minimum value of f(x) including endpoints

Point of Inflection

f”(x) = 0 f”(x) is undefined

indicates a change in concavity (when f”(x) changes sign)

Concavity

When f”(x) > 0, concave up When f”(x) < 0, concave

down

Plateau

When f’(x) = 0 but does not change signs

Cusp

A point at which f(x) is continuous but f’(x) is discontinuous

Area between two functions

A=∫a

b

( y1− y2 )dx

where y1 is the curve on the right or top, and y2 is the curve on the left or the bottom

Volume by disks

V=π∫a

b

r2h

where r is the radius and h is dy or dx

Volume by Washers

V=π∫a

b

(R ¿¿2−r2)h¿

Where R is the top or right-most function, and r is the bottom or left-most function; h is dy or dx

Volume by Cross Sections

∫a

b

( Area of the cross section )(dy∨dx)

dy or dx determined by axis perpendicular to cross sections

Note: π should only be present if the cross sections involve circles

Volume by Cylindrical Shells

2π∫a

b

rh( thickness)

where the thickness is dy or dx determined by the axis parallel to the axis of rotation

General Rules of Area and Volume by Definite Integral

When subtracting functions: (top or rightmost function) - (bottom or leftmost function)

Cannot slice from curve to curve (use two distinct functions)

General Rules of Area and Volume by Definite Integral

(Continued)

Limits of integration: points of intersection of the two functions.

When creating shells, slice parallel to axis of rotation

When creating disks and washers, slice perpendicular to the axis of rotation

Increasing Function

If f’(x) > 0, then the function is increasing

Decreasing Function

If f’(x) < 0, then the function is decreasing.

Optimization: Maximizing and Minimizing

1) Rewrite the equation in terms of one variablea. Solve another equation relating the two

variables in terms of one variableb. Substitute this expression into the original

equation to minimize/maximize2) Take the derivative of the equation to

minimize/maximize3) Set the derivative equal to zero.4) Solve the equation for the variable.