Presentation on theme: "Stuff you MUST know Cold for the AP Calculus Exam"— Presentation transcript:

1 Stuff you MUST know Cold for the AP Calculus Exam

In the morning of Tuesday, May 3, 2005. Sean Bird 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 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 ( )



11 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].



12 Solids of Revolution and friends

Disk Method Arc Length *bc topic Washer Method General volume equation (not rotated)



13 Distance, Velocity, and Acceleration

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



15 Trig Identities Double Argument



16 Trig Identities Double Argument Pythagorean sine cosine



17