Must-Know Concepts for AP Calculus Exam Preparation

1. Stuff you MUST know Cold for the AP Calculus Exam In preparation for Wednesday May 9, 2012. Sean Bird Updated by Mrs. Shak May 2012 AP Physics & Calculus Covenant Christian High School 7525 West 21st Street Indianapolis, IN 46214 Phone: 317/390.0202 x104 Email:seanbird@covenantchristian.org Website:http://cs3.covenantchristian.org/bird Psalm 111:2

2. Curve sketching and analysis y = f(x) must be continuous at each: • critical point: = 0 or undefined. And don’t forget endpoints for absolute min/max • 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,+) or (+,und,–), or (–,und,+) goes from incr to decr or decr to incr

3. Basic Derivatives

4. Basic Integrals Plus a CONSTANT

5. Some more handy integrals

6. More Derivatives Recall “change of base”

7. Differentiation Rules Chain Rule Product Rule Quotient Rule

8. The Fundamental Theorem of Calculus Corollary to FTC

9. 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

10. 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.

11. Approximation Methods for Integration Trapezoidal Rule Non-Equi-Width Trapezoids

12. 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].

13. AVERAGE RATE OF CHANGEof f(x) on [a, b] • This value is the “average rate of change” of the function on the interval [a, b]. We use the difference quotient to approximate the derivative in the absence of a function

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

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

16. 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

17. Trig Identities Double Argument

18. Trig Identities Double Argument Pythagorean sine cosine

19. Slope – Parametric & Polar Parametric equation • Given a x(t) and a y(t) the slope is Polar • Slope of r(θ) at a given θ is What is y equal to in terms of r and θ ? x?

20. Polar Curve For a polar curve r(θ), the AREA inside a “leaf” is (Because instead of infinitesimally small rectangles, use triangles) where θ1 and θ2 are the “first” two times that r = 0. We know arc length l = rθ and

21. l’Hôpital’s Rule If then

22. Integration by Parts We know the product rule Antiderivative product rule (Use u = LIPET) e.g. L I P E T Logarithm Inverse Polynomial Exponential Trig Let u = ln x dv = dx du = dx v = x

23. Maclaurin SeriesA Taylor Series about x = 0 is called Maclaurin. Taylor Series If the function f is “smooth” at x = a, then it can be approximated by the nth degree polynomial