Stuff you MUST know for the AP Calculus Exam

1. Stuff you MUST know for the AP Calculus Exam on the morning of Tuesday, May 9, 2007 By Sean Bird

2. Curve sketching and analysis y = f(x) must be continuous at each: • critical point: = 0 or undefined. And don’t forget endpoints • local minimum: goes (–,0,+) or (–,und,+) or > 0 • local maximum: goes (+,0,–) or (+,und,–) or < 0 • point of inflection: concavity changes goes from (+,0,–), (–,0,+), (+,und,–), or (–,und,+)

3. Basic Derivatives

4. Basic Integrals Plus a CONSTANT

5. More Derivatives

6. Differentiation Rules Chain Rule Product Rule Quotient Rule

7. The Fundamental Theorem of Calculus Corollary to FTC

8. Intermediate Value Theorem • If the function f(x) is continuous on [a, b], and y is a number between f(a) and f(b), then there exists at least one number x = c in the open interval (a, b) such that f(c) = y. Mean Value Theorem . . • If the function f(x) is continuous on [a, b], AND the first derivative exists on the interval (a, b), then there is at least one number x = c in (a, b) such that

9. Mean Value Theorem & Rolle’s Theorem If the function f(x) is continuous on [a, b], AND the first derivative exists on the interval (a, b), then there is at least one number x = c in (a, b) such that If the function f(x) is continuous on [a, b], AND the first derivative exists on the interval (a, b), AND f(a) = f(b), then there is at least one number x = c in (a, b) such that f '(c) = 0.

10. Approximation Methods for Integration Trapezoidal Rule Simpson’s Rule Simpson only works for Even sub intervals (odd data points) 1/3 (1 + 4 + 2 + 4 + 1 )

11. Theorem of the Mean Valuei.e. AVERAGE VALUE • If the function f(x) is continuous on [a, b] and the first derivative exists on the interval (a, b), then there exists a number x = c on (a, b) such that • This value f(c) is the “average value” of the function on the interval [a, b].

12. Solids of Revolution and friends • Disk Method • Arc Length *bc topic • WasherMethod • General volume equation (not rotated)

13. Distance, Velocity, and Acceleration velocity = (position) average velocity = (velocity) acceleration = speed = *velocity vector = displacement = *bc topic

14. Values of Trigonometric Functions for Common Angles π/3 = 60° π/6 = 30° θ sin θ cos θ tan θ 0° 0 1 0 sine ,30° cosine 37° 3/5 4/5 3/4 ,45° 1 53° 4/5 3/5 4/3 ,60° ,90° 1 0 ∞ π,180° 0 –1 0

15. Trig Identities Double Argument

16. Trig Identities Double Argument Pythagorean sine cosine