Review I

Math 1231: Single-Variable Calculus

Theorems

The Fundamental Theorem of Calculus If f is continuous on [a, b], then

where F is any anti-derivative of f .

LimitsDirect ‘plug-in’

Kill ‘singularity’

Squeeze Thm

Trig identities

Limits from two sides

Def of Derivative

Example:

Derivative LawsList of Derivatives

Sum RuleProduct RuleQuotient RuleChain Rule

Application of Derivative Laws: Implicit Differentiation

Example:

Global Max/Min

Local Max/Min

a bx1 x2

f ’(x)

a bx1 x2

f(x)

a bx1 x2

f ’’(x)

a bx1 x2

f(x)

1st deri t

est2 nd deri test

Word ProblemsRelated Rates

• Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing when the area of the square is 16 cm2?

• A cylindrical tank with radius 5 m is being filled with water at a rate of 3 m3/min. How fast is the area of the rectangle increasing?

Optimization Problems

• Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible.

• If 1200 cm2 of material is

available to make a box with a square base and an open top, find the largest possible volume of the box.

IntegralsList of Integrals

Substitution Rule (U-sub)

Application of Integrals

• Area• Volume