2. Stuff you MUST know Cold for the AP Calculus Exam in the morning of Wednesday, May 7, 2008. AP Physics & Calculus Covenant Christian High School 7525 West 21st Street Indianapolis, IN 46214 Phone: 317/390.0202 x104 Email: seanbird@covenantchristian.orgseanbird@covenantchristian.org Website: http://cs3.covenantchristian.org/birdhttp://cs3.covenantchristian.org/bird Psalm 111:2 Sean Bird Updated April 24, 2009

3. 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,+)

4. Basic Derivatives

5. Basic Integrals

6. Some more handy integrals

7. More Derivatives Recall “change of base”

8. Differentiation Rules Chain Rule Product Rule Quotient Rule

9. The Fundamental Theorem of Calculus Corollary to FTC

10. Intermediate Value Theorem. 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 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.

11. 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. Mean Value Theorem & Rolle’s Theorem

12. 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 )

13. Theorem of the Mean Value i.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].

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

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

16. Values of Trigonometric Functions for Common Angles 0–10π,180° ∞ 01,90°,60° 4/33/54/553° 1,45° 3/44/53/537°,30° 0100° tan θcos θsin θθ π/3 = 60° π/6 = 30° sine cosine

17. Trig Identities Double Argument

18. 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 θ 1 and θ 2 are the “first” two times that r = 0. and We know arc length l = r θ

21. l’Hôpital’s Rule If then

22. Integration by Parts Antiderivative product rule (Use u = LIPET) e.g. We know the product rule Let u = ln xdv = dx du = dx v = x LIPETLIPET Logarithm Inverse Polynomial Exponential Trig

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