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Intermediate Value Theorem 

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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 lefthand 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 equationinvolving 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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Substitute critical numbers in the 2^{nd} 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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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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Properties of the Natural Logarithmic Function 

Definition
Domain: (0,∞)
Range: (∞,∞)
ln(1) = 0
ln(ab) = ln(a) + ln(b)
ln(a^{n}) = 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]
e^{a}e^{b} = e^{a+b}
[image]



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Exponential Growth and Decay Model 

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Area of region between two curves 

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