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Extremas – First Derivative Test & Second Derivative Test

Extremas – First Derivative Test & Second Derivative Test. Fast Five. 1. Determine the first and second derivatives of the function f(x) = x 4 – 4x 3 + 4x 2 2. Determine the first and second derivatives of the function f(x) = x 4 – 4x 3

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Extremas – First Derivative Test & Second Derivative Test

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  1. Extremas – First Derivative Test & Second Derivative Test Calculus - Santowski

  2. Fast Five • 1. Determine the first and second derivatives of the function f(x) = x4 – 4x3 + 4x2 • 2. Determine the first and second derivatives of the function f(x) = x4 – 4x3 • 3. Determine the first and second derivatives of the function f(x) = x2e-x • 4. Determine the first and second derivatives of the function f(x) = 2sin(x) + sin2(x) Calculus - Santowski

  3. Lesson Objectives • 1. Explain what the First Derivative Test is and why it “works” • 2. Explain what the Second Derivative Test is and why it “works” • 3. Work with the FDT & SDT to classify extrema Calculus - Santowski

  4. (A) First Derivative Test • Let f be a differentiable function with f '(c) = 0 then • 1. If f '(x) changes from positive to negative, then f has a relative maximum at c. • 2. If f '(x) changes from negative to positive, then f has a relative minimum at c. • 3. If f '(x) does not change sign at c, then f has neither an maximum or minimum at c. (Stationary point) Calculus - Santowski

  5. (A) First Derivative Test • Use the FDT to classify all extrema of the function f(x) = x4 – 4x3 + 4x2 Calculus - Santowski

  6. (B) Second Derivative Test • Let f be a twice differentiable function near x = c such that f '(c) = 0. Then • 1. If f ''(x) > 0 then f(c) is a relative minimum. • 2. If f ''(x) < 0 then f(c) is a relative maximum. • 3. If f ''(x) = 0 then use the first derivative test to classify the extrema. Calculus - Santowski

  7. (B) Second Derivative Test • Your task: Write an explanation/clarification/rationalization of the FDT. Include diagrams in your explanation Calculus - Santowski

  8. (B) Second Derivative Test • Use the function f(x) = x4 – 4x3 to show algebraically HOW the SDT can be used to classify the extremas as either (i) maximums, (ii) minimums, or (iii) stationary points Calculus - Santowski

  9. (C) Examples • ex 1. Find and classify all extrema using FDT of f(x) = 3x5 - 25x3 + 60x. • ex 2. Find and classify all extrema using SDT of f(x) = 3x4 - 16x3 + 18x2 + 2. Calculus - Santowski

  10. (C) Examples - FDT • Ex 3. Find the intervals of increase and decrease and max/min values of f(x) = cos(x) – sin(x) on (-,) • Ex 4. Find the intervals of increase/decrease and max/min points of f(x) = x2e-x • Ex 5. Find the local and absolute maximum & minimum points for f(x) = x(ln(x))2 Calculus - Santowski

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