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Year 6 Relative Min, Max, Points of Inflection, First and Second Test.notebookNovember 13, 2018

New Divider! Uses of Derivative 2: Sketching

1 Year 6 Relative Min, Max, Points of Inflection, First and Second .notebookNovember 13, 2018

Your Mission:

Using your markers and BIG piece of paper, draw a continuous . Be crazy!!! Make it super interesting that includes a lot of the funky behaviors we have discussed that a may have.

2 Year 6 Relative Min, Max, Points of Inflection, First and Test.notebookNovember 13, 2018

Next, find as many lines and draw them in as you and your group see fit.

Make observations about the tangent lines. Be prepared to share your findings!

3 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

4 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Based on your findings we should be able to answer these essential questions.

If not we will revisit your findings!

­function is increasing the of the tangent is positive and if it is decreasing the slope of the tangent is negative

­as you approach maxima/minima that are differentiable the of the tangent lines go to zero ­slopes of that are undefined as possible max/min

(1) ­when a graph is concave down the slope of the tangent line going from to left to right decreases

(2) ­when a graph is concave up the slope of the tangent line going from left to right increases

(3) ­differentiating the function

5 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Goals of this lesson:

1) At the local , if the function is differentiable, the slope of the tangent line will equal______.

2) If the slope of the tangent lines switch from: a) positive to negative then the graph of the function has a ______.

b) negative to positive then the graph of the function has a ______.

3) If the slope of the tangent line is positve, then the function is ______.

If the slope of the tangent line is negative, then the function is ______.

4) If there is a point of non differentiability then

6 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Packet

How are you determining these intervals?

How can help to determine these intervals?

7 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Sign Graphical Representation of f '(x)

8 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

(a) Find the derivative with respect to x of f(x) = x3 ­ x. (b) Graph f and f ' together, and discuss the relationship between the two graphs.

9 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Answer:

10 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

11 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Use this language for AP exam

State and justify any relative maxima/minima.

12 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

13 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

14 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

15 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

16 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Match the capital letter graph of f '(x) with the graph of f(x).

17 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Let's look at certain functions that have interesting behaviors and then the graphs of their .

1) f(x) = 1/x

2) f(x) = x

3) f(x) = ex

18 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

4) f(x) = ln x

5) f(x) = x1/3

6) f(x) = (x2 ­ 25)/(x ­ 5)

7) f(x) = (x ­ 5)/(x2 ­ 25)

19 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

20 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

21 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

22 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

How does the first derivative help in sketching the graph of a function?

How does the second derivative help in sketching the graph of a function?

23 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

24 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

NOTE: MAKE A STATEMENT ABOUT TANGENT APPROXIMATIONS AND CONCAVITY!!!

25 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Without a display calculator, draw a sketch of the function using all of the math that you have learned over the course of your mathematical career.

y = x3 ­ 3x2 + 1

26 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Deeper understanding of calculus and the graph of y = x3.

27 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Sketch the graph of y = x3 ­ 3x + 2

f '(x)

f(x)

f ''(x)

f(x)

28 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Sketch the graph of y = 3x5 ­ 5x3

f '(x)

f(x)

f ''(x)

f(x)

29 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

30 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Card activity!!!

Arrange the cards in order from f(x), f'(x), f''(x)

31 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

32 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

33 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

(d) Using all of the above, include a sketch of f

34 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

35 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

or maximum?

(f) skip (e) and instead include a sketch of f

36 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Question The graph of y=f(x) is shown below, where A is a local maximum point and D is a local minimum point.

On the axes below, sketch the graph of the derivative y=f ′ (x) , clearly showing the coordinates of the images of the points A and D, labelling them A ′′ and D ′′ respectively.

37 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Question The diagram below shows a sketch of the function f ′ (x) of the curve f(x) .

On the graph above, sketch the curve y=f(x) given that f(0)=0 . Clearly indicate on the graph any maximum, minimum or inflexion points.

38 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Question Consider the function f(x)=x 3 −3x 2 −9x+10 , x∈R . Find the of the straight line passing through the maximum and minimum points of the graph y=f(x) .

39 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Question

(i) Solve the equation f ′ (x)=0 . (ii) Hence show the graph of f has a local maximum. (iii) Write down the range of the function f .

40 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Question

Show that the curve y=f(x) has one point of inflexion, and find its coordinates.

41 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

Question

Given that the graph of the function has exactly one point of inflexion, find its coordinates.

42 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

43 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

44 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

45 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

AP 2001 Answers

46 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

47 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

48 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

49 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

50 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

51 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

52 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

53 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

54 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

55 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

AP 2004

56 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

57 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018

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