2. Stuff you MUST know Cold for the AP Calculus Exam

3. Curve sketching and analysis y = f(x) must be continuous at each: critical point: = 0 or undefined. local minimum: goes (–,0,+) or (–,und,+) or > 0 at stationary pt local maximum: goes (+,0,–) or (+,und,–) or < 0 at stationary pt 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 Other part of the 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. If the function f(x) is continuous on [a, b], then f has both an absolute maximum and an absolute minimum on [a,b] The absolute extremes occur either at the critical points or at the endpoints. Extreme Value Theorem

13. Approximation Methods for Integration Trapezoidal Rule Riemann Sum LRAM when c k is a LEFT endpoint RRAM when c k is a RIGHT endpoint MRAM when c k is a MIDPOINT

14. 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].

15. Solids of Revolution and friends Disk Method Washer Method General volume equation (not rotated) Arc Length

16. Distance, Velocity, and Acceleration velocity =(position) (velocity) speed = displacement = average velocity = acceleration = velocity vector =

17. Values of Trigonometric Functions for Common Angles 0–10π,180° ∞ 01,90°,60° 1,45°,30° 0100° tan θcos θsin θθ π/3 = 60° π/6 = 30° sine cosine

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 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. Other Indeterminate forms: Write as a ratio Use Logs

23. 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

24. 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