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AP Calculus AB Formulas
Theorems, formulas, etc.
28
Mathematics
05/02/2010

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 Intermediate Value Theorem
Definition
 If ƒ is continuous on the closed interval [a,b] and k is any number between ƒ(a) and ƒ(b), then there is at least one number c in [a,b] such that ƒ(c)=k.
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 Definition of the Derivative of a Function
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 Alternative form of the derivative
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 Average velocity or Average rate of change
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 Guidelines for Implicit Differentiation
Definition
 Differentiate both sides of the equation with respect to x. Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation. Factor dy/dx out of the left side of the equation. Solve for dy/dx by dividing both sides of the equation by the left-hand factor that does not contain dy/dx.
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 Steps for solving related rate problems
Definition
 Make a sketch and label with appropriate variables Write Find, When, Given Write equation-involving variables whose rate of change either are given or are to be determined. Implicitly differentiate both sides with respect to time. After differentiating, substitute known values and solve for required rate of change.
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 Mean Value Theorem
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 Second Derivative Test
Definition
 Substitute critical numbers in the 2nd derivative. If the answer is positive, there is a minimum at that x value. If the answer is negative, there is a maximum at that x value.
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 Steps for optimization word problems:
Definition
 Draw a sketch and label with appropriate variable. Write a primary equation for the quantity that is to be maximized or minimized. Reduce the primary equation to one having a single variable. This may involve the sum of a secondary equation. Find the derivative of the equation and determine the correct value.
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 Linear Approximation
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 Fundamental Theorem of Calculus
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 Definition of the Average Value of a Function on an Interval
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 Second Fundamental Theorem of Calculus
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 The Trapezoidal Rule
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 Properties of the Natural Logarithmic Function
Definition
 Domain:   (0,∞)   Range:     (-∞,∞)   ln(1) = 0   ln(ab) = ln(a) + ln(b)   ln(an) = n ln(a)   ln(a/b) = ln(a) - ln(b)
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 Steps for finding the derivative of an inverse function at a given value
Definition
 Set function equal to given value and solve Find derivative of given function Substitute the answer from Step 1 into the reciprocal of the derivative
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 Properties of the Natural Exponential Function
Definition
 Domain: (-∞, ∞)   Range: (0, ∞)   [image]   [image]   eaeb = ea+b   [image]
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 Exponential Growth and Decay Model
Definition
 y = Cekt
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 Area of region between two curves
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 Volume: Disk method
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 Volume: Washer method
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 Volume: Shell method
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 Volume: Cross sections
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