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1001_4.2_Lab_17

# 1001_4.2_Lab_17 - "" f c>" f(c is a Relative...

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CALCULUS 1 LAB 17 NAME: ___________________PARTNER: _________________ 4.2 Analysis of Functions II GSA:______________________Lab Time: ____________ Relative Extrema, First and Second Derivative Tests Basic Guidelines: Finding Relative Extrema with the First Derivative Test (FDT) : Let c be a critical point of f. * For x < c , " f ( x ) < 0 , f(x) is decreasing, and for x > c , " f ( x ) > 0 , f(x) is increasing " f(c) is a Relative Minimum. " f ( x ) changes sign about x=c. * For x < c , " f ( x ) > 0 , f(x) is increasing, and for x > c , " f ( x ) < 0 , f(x) is decreasing " f(c) is a Relative Maximum. " f ( x ) changes sign about x=c. Finding Relative Extrema with the Second Derivative Test (SDT) : Let " f ( c ) = 0 , c is a stationary point. * Substitute the critical point c into the second derivative to test the concavity. * " " f ( c ) < 0 " f(c) is a Relative Maximum since it is concave down about x=c.
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Unformatted text preview: * " " f ( c ) > " f(c) is a Relative Minimum since it is concave up about x=c. 1. f ( x ) = ln( x ) x 2. 3 2 2 3 3 ) ( x x x f ! + = I. The First Derivative Test conclusions depend on the signs of the first derivative. For each function above: a) find the critical points, b) make a sign analysis of the first derivative and c) apply the FDT to find the relative extrema. a) b) c) The Second Derivative Test is used only on stationary points, c. For each function above d) find the second derivative and make a sign analysis. Apply the SDT, e) determine if " " f ( c ) < or " " f ( c ) > , then f) state the relative extrema. d) e) f) II. On the back make a s ketch with critical points and inflection points labeled as time permits....
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