1. Stuff you MUST know Cold for the AP Calculus Exam
In preparation for Wednesday May 8, 2013.
Sean Bird
Updated by Mrs. Reynolds
April 2014
AP Physics & Calculus
Covenant Christian High School
7525 West 21st Street
Indianapolis, IN 46214
Phone: 317/ x104
Website:
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
Make the box slid and stay over the C. The reveal the rest.

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 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].

13. AVERAGE RATE OF CHANGE of 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
Washer Method
General volume equation (not rotated)

15. Distance, Velocity, and Acceleration
(position)
average velocity =
acceleration =
(velocity)
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° 1
sine
,30°
cosine
37°
3/5
4/5
3/4
Pi/3 is 60 degrees
Pi/6 is 30 degrees
,45° 1
53°
4/5
3/5
4/3
,60°
,90° 1 ∞
π,180° –1

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 = r θ
and

21. l’Hôpital’s Rule If
then

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

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 nth degree polynomial