(2020年整理)AP Calculus AB review AP微积分复习提纲.pptx

, f’(g(x))≠0.
*The Derivative of the Natural Exponential Function Let u be a differentiable function of x.
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*Integration Rules for Exponential Functions Let u be a differentiable function of x.
is
Moreover, as n→∞, the right-hand side approaches
*Inverse functions(y=f(x), switch y and x, solve for x)
*The Derivative of an Inverse Function Let f be a function that is differentiable on an interval I. If f has an inverse function g, then g is differentiable at any x for which f’(g(x))≠0. Moreover,
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AP CALCULUS AB REVIEW
Chapter 2 Differentiation Definition of Tangent Line with Slop m If f is defined on an open interval containing c, and if the limit
*The Mean Value Theorem If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that f ’(c) =
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*Definition of a Critical Number Let f be defined at c. If f ’(c) = 0 OR IF F IS NOT DIFFERENTIABLE AT C, then c is a critical number of f.
*Rolle’s Theorem If f is differentiable on the open interval (a, b) and f (a) = f (b), then there is at least one number c in (a, b) such that f ’(c) = 0.
*Integration of Even and Odd Functions
1) If f is an even function, then
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2) If f is an odd function, then
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*The Trapezoidal Rule Let f be continuous on [a, b]. The trapezoidal Rule for
*Definition of the Hyperbolic Functions
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*Limits at Infinity
*Curve Sketching (take first and second derivative, make sure all the characteristics of a function are clear)
♫ Optimization Problems
*The second fundamental theorem of calculus If f is continuous on an open internal I containing a, then, for every x in the interval, .
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*Integration by Substitution
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*Definition of the Average Value of a Function on an Interval If f is integrable on the closed interval [a, b], then the average value of f on the interval is .
provided the limit exists. For all x for which this limit exists, f ’ is a function of x. *The Power Rule *The Product Rule
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*The Chain Rule ☺Implicit Differentiation (take the derivative on both sides; derivative of y is y*y’) Chapter 3 Applications of Differentiation *Extrema and the first derivative test (minimum: − → + , maximum: + → − , + & − are the sign of f’(x) )
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♠Derivatives for Bases other than e Let a be a positive real number (a ≠1) and let u be a differentiable function of x.
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*Derivatives of Inverse Trigonometric Functions Let u be a differentiable function of x.
exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)). Definition of the Derivative of a Function The Derivative of f at x is given by
approximating
is given by
Moreover, a n → ∞, the right-hand side approaches
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*Simpson’s Rule (n is even) Let f be continuous on [a, b]. Simpson’s Rule for approximating
*Newton’s MethodLeabharlann Baidu(used to approximate the zeros of a function, which is tedious and stupid, DO NOT HAVE TO KNOW IF U DO NOT WANT TO SCORE 5)
Chapter 4 & 5 Integration
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*Be able to solve a differential equation *Basic Integration Rules
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4) *Integral of a function is the area under the curve *Riemann Sum (divide interval into a lot of sub-intervals, calculate the
*Points of Inflection (take second derivative and set it equal to 0, solve the equation to get x and plug x value in original function)
*Asymptotes (horizontal and vertical)
area for each sub-interval and summation is the integral). *Definite integral *The Fundamental Theorem of Calculus
If a function f is continuous on the closed interval [a, b] and F is an anti-derivative of f on the interval [a, b], then
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*Increasing and decreasing interval of functions (take the first derivative)
*Concavity (on the interval which f’’ > 0, concave up)
*Second Derivative Test Let f be a function such that f ’(c) = 0 and the second derivative of f exists on an open interval containing c. 1. If f’’(c) > 0, then f(c) is a minimum 2. If f’’(c) < 0, then f(c) is a maximum