Stuff you MUST know Cold for the AP Calculus Exam

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Stuff you MUST know Cold for the AP Calculus Exam in the morning of Wednesday, May 7, 2008. Sean Bird AP Physics & Calculus Covenant Christian High School
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Title: Stuff you MUST know Cold for the AP Calculus Exam 1 Stuff you MUST know Cold for the AP Calculus Exam
• in the morning of Wednesday, May 7, 2008.
• Sean Bird AP Physics Calculus Covenant Christian High School 7525 West 21st Street Indianapolis, IN 46214 Phone 317/390.0202 x104 Email seanbird_at_covenantchristian.org Website http//cs3.covenantchristian.org/bird Psalm 1112 Updated April 24, 2009 2 Curve sketching and analysis
• y f(x) must be continuous at each
• critical point 0 or undefined. And dont forget endpoints
• local minimum goes (,0,) or (,und,) or gt 0
• local maximum goes (,0,) or (,und,) or lt 0
• point of inflection concavity changes
• goes from (,0,), (,0,), (,und,), or (,und,)
• 3 Basic Derivatives 4 Basic Integrals Plus a CONSTANT 5 Some more handy integrals 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 Rolles 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 Simpsons Rule Simpson only works for Even sub intervals (odd data points) 1/3 (1 4 2 4 1 ) 12 Theorem of the Mean Valuei.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 Solids of Revolution and friends
• Disk Method
• Arc Length bc topic
• Washer Method
• General volume equation (not rotated)
• 14 Distance, Velocity, and Acceleration velocity (position) average velocity (velocity) acceleration speed velocity vector displacement bc topic 15 Values of Trigonometric Functions for Common Angles p/3 60 p/6 30 tan ? cos ? sin ? ? 0 1 0 0 sine ,30 cosine 3/4 4/5 3/5 37 1 ,45 4/3 3/5 4/5 53 ,60 0 1 ,90 8 0 1 0 p,180 16 Trig Identities Double Argument 17 Trig Identities Double Argument Pythagorean sine cosine 18 Slope Parametric Polar
• Parametric equation
• Given a x(t) and a y(t) the slope is
• Polar
• Slope of r(?) at a given ? is
• What is y equal to in terms of r and ? ? x? 19 Polar Curve
• For a polar curve r(?), the AREA inside a leaf is
• (Because instead of infinitesimally small rectangles, use triangles)
• where ?1 and ?2 are the first two times that r 0.
• We know arc length l r ? and 20 lHôpitals Rule
• If
• then
• 21 Integration by Parts We know the product rule L I P E T Logarithm Inverse Polynomial Exponential Trig
• Antiderivative product rule
• (Use u LIPET)
• e.g.
• Let u ln x dv dx du dx v x 22 Maclaurin SeriesA Taylor Series about x 0 is called Maclaurin. Taylor Series
• If the function f is smooth at x a, then it can be approximated by the nth degree polynomial
• 23 (No Transcript)
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