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Calculus - Cheat - Sheet - All 5

This cheat sheet covers key concepts in calculus related to extrema, including absolute and relative extrema, Fermat’s Theorem, the Extreme Value Theorem, and methods for finding and classifying critical points. It also outlines the Mean Value Theorem and Newton's Method for approximating roots of functions. Each section provides definitions and tests to identify maxima and minima of functions.

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93 views1 page

Calculus - Cheat - Sheet - All 5

This cheat sheet covers key concepts in calculus related to extrema, including absolute and relative extrema, Fermat’s Theorem, the Extreme Value Theorem, and methods for finding and classifying critical points. It also outlines the Mean Value Theorem and Newton's Method for approximating roots of functions. Each section provides definitions and tests to identify maxima and minima of functions.

Uploaded by

Soderberg
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Calculus Cheat Sheet

Extrema
Absolute Extrema Relative (local) Extrema
1. x = c is an absolute maximum of f (x) if 1. x = c is a relative (or local) maximum of f (x)
f (c) ≥ f (x) for all x in the domain. if f (c) ≥ f (x) for all x near c.
2. x = c is an absolute minimum of f (x) if 2. x = c is a relative (or local) minimum of f (x)
f (c) ≤ f (x) for all x in the domain. if f (c) ≤ f (x) for all x near c.

Fermat’s Theorem 1st Derivative Test


If f (x) has a relative (or local) extrema at x = c, If x = c is a critical point of f (x) then x = c is
then x = c is a critical point of f (x). 1. a relative maximum of f (x) if f 0 (x) > 0 to the
left of x = c and f 0 (x) < 0 to the right of x = c.
Extreme Value Theorem
If f (x) is continuous on the closed interval [a, b] then 2. a relative minimum of f (x) if f 0 (x) < 0 to the
there exist numbers c and d so that, left of x = c and f 0 (x) > 0 to the right of x = c.
1. a ≤ c, d ≤ b, 3. not a relative extrema of f (x) if f 0 (x is the
2. f (c) is the absolute maximum in [a, b], same sign on both sides of x = c.
3. f (d) is the absolute minimum in [a, b].
2nd Derivative Test
Finding Absolute Extrema If x = c is a critical point of f (x) such that f 0 (c) = 0
then x = c
To find the absolute extrema of the continuous
function f (x) on the interval [a, b] use the following 1. is a relative maximum of f (x) if f 00 (c) < 0.
process. 2. is a relative minimum of f (x) if f 00 (c) > 0.
1. Find all critical points of f (x) in [a, b].
3. may be a relative maximum, relative
2. Evaluate f (x) at all points found in Step 1. minimum, or neither if f 00 (c) = 0.
3. Evaluate f (a) and f (b).
Finding Relative Extrema and/or
4. Identify the absolute maximum (largest
Classify Critical Points
function value) and the absolute minimum
1. Find all critical points of f (x).
(smallest function value) from the
2. Use the 1st derivative test or the
evaluations in Steps 2 & 3.
2nd 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
f (b) − f (a)
number a < c < b such that f 0 (c) = .
b−a

Newton’s Method
f (xn )
If xn is the nth guess for the root/solution of f (x) = 0 then (n + 1)st guess is xn+1 = xn − provided
f 0 (xn )
f 0 (xn ) exists.

© Paul Dawkins - https://tutorial.math.lamar.edu

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