Calculus Cheat Sheet.pdf

2 if f x for all x in an interval i then f x is

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2. If ( ) 0 f x ′′ < for all x in an interval I then ( ) f x is concave down on the interval I . Inflection Points x c = is a inflection point of ( ) f x if the concavity changes at x c = .
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Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Extrema Absolute Extrema 1. x c = is an absolute maximum of ( ) f x if ( ) ( ) f c f x for all x in the domain. 2. x c = is an absolute minimum of ( ) f x if ( ) ( ) f c f x for all x in the domain. Fermat’s Theorem If ( ) f x has a relative (or local) extrema at x c = , then x c = is a critical point of ( ) f x . Extreme Value Theorem If ( ) f x is continuous on the closed interval [ ] , a b then there exist numbers c and d so that, 1. , a c d b , 2. ( ) f c is the abs. max. in [ ] , a b , 3. ( ) f d is the abs. min. in [ ] , a b . Finding Absolute Extrema To find the absolute extrema of the continuous function ( ) f x on the interval [ ] , a b use the following process. 1. Find all critical points of ( ) f x in [ ] , a b . 2. Evaluate ( ) f x at all points found in Step 1. 3. Evaluate ( ) f a and ( ) f b . 4. Identify the abs. max. (largest function value) and the abs. min.(smallest function value) from the evaluations in Steps 2 & 3. Relative (local) Extrema 1. x c = is a relative (or local) maximum of ( ) f x if ( ) ( ) f c f x for all x near c . 2. x c = is a relative (or local) minimum of ( ) f x if ( ) ( ) f c f x for all x near c . 1 st Derivative Test If x c = is a critical point of ( ) f x then x c = is 1. a rel. max. of ( ) f x if ( ) 0 f x > to the left of x c = and ( ) 0 f x < to the right of x c = . 2. a rel. min. of ( ) f x if ( ) 0 f x < to the left of x c = and ( ) 0 f x > to the right of x c = . 3. not a relative extrema of ( ) f x if ( ) f x is the same sign on both sides of x c = . 2 nd Derivative Test If x c = is a critical point of ( ) f x such that ( ) 0 f c = then x c = 1. is a relative maximum of ( ) f x if ( ) 0 f c ′′ < . 2. is a relative minimum of ( ) f x if ( ) 0 f c ′′ > . 3. may be a relative maximum, relative minimum, or neither if ( ) 0 f c ′′ = . Finding Relative Extrema and/or Classify Critical Points 1. Find all critical points of ( ) f x . 2. Use the 1 st derivative test or the 2 nd derivative test on each critical point. Mean Value Theorem If ( ) f x is continuous on the closed interval [ ] , a b and differentiable on the open interval ( ) , a b then there is a number a c b < < such that ( ) ( ) ( ) f b f a f c b a = . Newton’s Method If n x is the n th guess for the root/solution of ( ) 0 f x = then ( n +1) st guess is ( ) ( ) 1 n n n n f x x x f x + = provided ( ) n f x exists.
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Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Related Rates Sketch picture and identify known/unknown quantities. Write down equation relating quantities and differentiate with respect to t using implicit differentiation ( i.e. add on a derivative every time you differentiate a function of t ). Plug in known quantities and solve for the unknown quantity.
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