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

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
9/11/2018

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
9/11/2018

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)
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9/11/2018

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

6. (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

7. (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

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

9. (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

10. (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

11. (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

12. (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

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

14. (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

15. (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
page203.html
Calculus - Santowski
9/11/2018