sec41 - Sections 4.1 & 4.2: Using the Derivative to...

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1 Sections 4.1 & 4.2: Using the Derivative to Analyze Functions f ’ ( x ) indicates if the function is: Increasing or Decreasing on certain intervals. Critical Point c is where f ’ ( c) = 0 (tangent line is horizontal), or f ’ ( c) = undefined (tangent line is vertical) f ’’ ( x ) indicates if the function is concave up or down on certain intervals. Inflection Point : where f '' ( x) = 0 or where the function changes concavity, no Min no Max. If the sign of f ‘ ( c ) changes: from + to - , then: f ( c ) is a local Maximum If the sign of f ‘ ( c ) changes: from - to + , then: f ( c ) is a local Minimum If there is no sign change for f ‘ ( c ): then f ( c ) is not a local extreme, it is: An Inflection Point ( concavity changes ) Critical points, f ' (x) = 0 at : x = a , x = b Increasing, f ' (x) > 0 in : x < a and x > b Decreasing, f ' (x) < 0 in : a < x < b Max at: x = a , Max = f ( a ) Min at: x = b , Min = f ( b ) Inflection point, f '' (x) = 0 at : x = i Concave up, f '' (x) > 0 in: x > i Concave Down, f '' (x) < 0 in: x < i Critical points, f ' (x) = 0 at : x = a, x = b, x = c Increasing, f ' (x) > 0 in : a < x < b and x > c Decreasing, f ' (x) < 0 in : x < a and b < x < c Max at: x = b , Max = f ( b ) Min at: x = a , x = c , Min = f ( a ) and f ( c ) Inflection point, f '' (x) = 0 at : x = i, x = j Concave up, f '' (x) > 0 in: x < i and x > j Concave Down, f '' (x) < 0 in: i < x < j f ’(x) < 0 decreasing f ’’(x) > 0 concave up b f(c) j i c a f(a) f(a) f(b) a b i f ’(x) < 0 decreasing f ’’(x) < 0 concave down f ’(x) > 0 increasing f ’’(x) > 0 concave up f ’(x) > 0 increasing f ’’(x) < 0 concave down f(c) f ' (c) =0 + - + + - - - - - f(c) f ' (c) =0 + + + + - - - f(c) + + + + + + + f ' (c) =0 - - - - - - f ' (c) =0 f(c)
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2 I) Applications of The First Derivative: Finding the critical points Determining the intervals where the function is increasing or decreasing Finding the local maxima and local minima Step 1: Locate the critical points where the derivative is = 0; find f '( x ) and make it = 0 f ' ( x ) = 0 x = a , b , c , …. .. Step 2: Divide f ' ( x ) into intervals using the critical points found in the previous step: then choose a test point in each interval. Step 3: Find the derivative for the function in each test point: Sign of f ' (test point) Label the interval of the test point: > 0 or positive increasing , + + + + + , < 0 or negative decreasing , - - - - - - - , Step 4: Look at both sides of each critical point, take point a for example: or then it is a local Min. Min = f ( a ) or then it is a local Max. Max = f ( a ) or No local Min. or Max .
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This note was uploaded on 02/29/2012 for the course MATH 119 taught by Professor Maanomran during the Fall '09 term at Indiana University-Purdue University Fort Wayne.

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sec41 - Sections 4.1 &amp; 4.2: Using the Derivative to...

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