1. Math 1231: Single-Variable Calculus
Review I
Math 1231: Single-Variable Calculus

2. Theorems
The Fundamental Theorem of Calculus If f is continuous on [a, b], then
where F is any anti-derivative of f .

3. Limits Direct ‘plug-in’ Squeeze Thm Def of Derivative Example:
Kill ‘singularity’
Trig identities
Limits from two sides

4. Derivative Laws List of Derivatives Sum Rule Product Rule
Quotient Rule
Chain Rule
Application of Derivative Laws: Implicit Differentiation
Example:

5. Global Max/Min

6. Local Max/Min 2nd deri test 1st deri test f ’(x) f ’’(x) a x1 x2 b a

7. Word Problems Related Rates Optimization Problems
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.

8. Integrals List of Integrals Application of Integrals Area Volume
Substitution Rule (U-sub)
Application of Integrals
Area
Volume