Extremas – First Derivative Test & Second Derivative Test Calculus - Santowski Calculus - Santowski 9/11/2018

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 9/11/2018

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 9/11/2018

(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 9/11/2018

(A) First Derivative Test Your task: Write an explanation/clarification/rationalization of the FDT. Include diagrams in your explanation Calculus - Santowski 9/11/2018

(A) First Derivative Test Use the FDT to classify all extrema of the function f(x) = x4 – 4x3 + 4x2 (NOTE: Sign charts are NOT allowed!) Calculus - Santowski 9/11/2018

(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 9/11/2018

(B) Second Derivative Test Your task: Write an explanation/clarification/rationalization of the FDT. Include diagrams in your explanation Calculus - Santowski 9/11/2018

(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 9/11/2018

(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 9/11/2018

(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 9/11/2018

(C) Examples - SDT Ex 6. For the function f(x) = xex determine the co-ordinates of and classify the extrema using the SDT Ex 7. For the function f(x) = 2sin(x) + sin2(x) determine the co-ordinates of and classify the extrema using the SDT Ex 8. Determine the co-ordinates of and classify the extrema using the SDT for Calculus - Santowski 9/11/2018

(D) Homework Textbook, S5.1 (FDT), p278 - 282 (i) Q7,8 (graphs) (ii) Q9-26 (algebra, ANV) Textbook, S5.2 (SDT), p307-310, (i) Graphs: Q27-32 (ii) Algebra: max/min; Q33-44, ANV (iii) Algebra: SDT; Q48-53, ANV Calculus - Santowski 9/11/2018

(E) Internet Links - FDT Visual Calculus - Maxima and Minima from UTK Visual Calculus - Mean Value Theorem and the First Derivative Test from UTK First Derivative Test -- From MathWorld Tutorial: Maxima and Minima from Stefan Waner at Hofstra U Calculus - Santowski 9/11/2018

(E) Internet Links - SDT Graphing Using First and Second Derivatives from UC Davis Visual Calculus - Graphs and Derivatives from UTK Calculus I (Math 2413) - Applications of Derivatives - The Shape of a Graph, Part II Using the Second Derivative - from Paul Dawkins http://www.geocities.com/CapeCanaveral/Launchpad/2426/ page203.html Calculus - Santowski 9/11/2018