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: 317/390.0202 x104 Email: seanbird@covenantchristian.org Website: http://cs3.covenantchristian.org/bird Psalm 111:2

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

Basic Derivatives

Basic Integrals Plus a CONSTANT

Some more handy integrals Make the box slid and stay over the C. The reveal the rest.

More Derivatives Recall “change of base”

Differentiation Rules Chain Rule Product Rule Quotient Rule

The Fundamental Theorem of Calculus Corollary to FTC

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

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.

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

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].

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

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

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

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

Trig Identities Double Argument

Trig Identities Double Argument Pythagorean sine cosine

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?

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

l’Hôpital’s Rule If then

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

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