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# MATH 1225 - Test 3 Study Guide MATH 1225

Virginia Tech

GPA 4.0

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## Popular in Calculus of a Single Variable

## Popular in Mathematics (M)

This 3 page Study Guide was uploaded by Gavin B on Tuesday March 1, 2016. The Study Guide belongs to MATH 1225 at Virginia Polytechnic Institute and State University taught by Eun Chang in Fall 2015. Since its upload, it has received 129 views. For similar materials see Calculus of a Single Variable in Mathematics (M) at Virginia Polytechnic Institute and State University.

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Date Created: 03/01/16

VT – MATH 1225 - UNIT 3 TEST 3 STUD Y GUIDE Always determine and write the domain, if its continuous, and if its differentiable for any problem. Absolute Minimum and Absolute Maximum. If f is continuous on closed interval [a, b] then f attains both an absolute maximum “M” and an absolute minimum “m” in [a, b]. m≤ f (xfor every x in [a, b]. Fermat’s Theorem: If f has a local max or local min at c, if f ‘(c) exists then f ‘(c)=0. ‘(c)=DNE.Point: Any point c in the domain such that either f ‘(c)=0 or f Closed Interval Method: VT – MATH 1225 - UNIT 3 Only places where a function f can have an extreme value (local or absolute) are: 1. f ‘(c)=0 (C.P.) 2. f ‘(c)=DNE (C.P.) 3. End-point of the domain of f. Rolle’s Theorem: (can use to find C.P. and used with IVT to find number and value or roots). 1. f is continuous on [a, b] (closed). 2. f is differentiable on (a, b) (open). 3. f(a)=f(b) Then there is a number c in (a, b) (open) such that f ‘(c)=0. Mean Value Theorem: (can use to find C.P.). 1. f is continuous on [a, b] (closed). 2. f is differentiable on (a, b) (open). ' f b − f (a) Then there is a number c in (a, b) such that f c)= b−a Answer format: x=d∈(a,b) If f ‘(x)=0, ∀ x∈(a,b) then f is constant on (a, b). Corollary: If f ‘(x)=g‘(x), ∀ x∈(a,b) then f-g=c (is constant) on (a, b). Increasing/ Decreasing Test: If f ‘(x)¿ 0 on an interval, then f is increasing on that interval. If f ‘(x)¿ 0 on an interval, then f is decreasing on that interval. The First Derivative Test: Suppose that c is a critical number of a continuous function f. a) If f ' changes from positive to negative at c, then f has a local max at c. b) If f ' changes from negative to positive at c, then f has a local min at c. c) If f ' does not change signs at c, then no local in, or local max, at c. Concavity Test: When f '' is positive f is concave up. When f '' is negative f is concave down. Second Derivative Test: If x=k is ''critical point on a differentiable function f and f (k>0 , then x=k is a local minimum. f ''(k)<0 , then x=k is a local maximum. VT – MATH 1225 - UNIT 3 f 'k =0 or undefined, then the test is inconclusive. POI (Points of Inflection) at (c, f(c)) if f(x) is continuous and the curve changes from concave up to concave down or from concave down to concave up at the point (c, f(c)). Guidelines for sketching curves: Domain y int., x=0 x int., y=0 Asymptotes: HA: Use x→−∞ f (x) (Tip: Divide top and bottom by largest x in 2 both so for example: x . VA: Set denominator equal to zero. 1 Derivative Test: C.N., max, min. nd 2 Derivative Concavity/ POI. Symmetry about the y-axis. (If f: even function ( f(-x)=f(x) )). Symmetry about the origin (rotate 180 degrees). (If f: odd function ( f(- x)=-f(x) )). Period (P): ( f π+P )= f (x) . ' ' f is increasing if f >0 , f is decreasing if f <0 . Optimization. First Derivative Test for Absolute Extreme Values: If f(x)>0,∀ x<c and f'(x)<0,∀ x>c , then f(c) is the absolute maximum value of f. ' ' If f(x)<0,∀ x<c and f (x)>0,∀ x>c , then f(c) is the absolute minimum value of f.

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