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Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
New Divider!
Uses of Derivative 2: Curve Sketching
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
Your Mission:
Using your markers and BIG piece of paper, draw a continuous function. Be crazy!!! Make it super interesting that includes a lot of the funky behaviors we have discussed that a continuous function may have.
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
Next, find as many tangent lines and draw them in as you and your group see fit.
Make observations about the tangent lines. Be prepared to share your findings!
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Based on your findings we should be able to answer these essential questions.
If not we will revisit your findings!
as you approach maxima/minima that are differentiable the slopes of the tangent lines go to zeroslopes of tangents that are undefined as possible max/min
function is increasing the slope of the tangent is positive and if it is decreasing the slope of the tangent is negative
(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
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Goals of this lesson:
1) At the local maxima and minima, 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
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
How are you determining these intervals?
How can calculus help to determine these intervals?
Packet
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Sign Diagram Graphical Representation of f '(x)
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
(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.
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Use this language for AP exam
State and justify any relative maxima/minima.
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
Match the capital letter graph of f '(x) with the graph of f(x).
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
Let's look at certain functions that have interesting behaviors and then the graphs of their derivatives.
1) f(x) = 1/x
2) f(x) = x
3) f(x) = ex
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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4) f(x) = ln x
5) f(x) = x1/3
6) f(x) = (x2 25)/(x 5)
7) f(x) = (x 5)/(x2 25)
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
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?
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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NOTE: MAKE A STATEMENT ABOUT TANGENT APPROXIMATIONS AND CONCAVITY!!!
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
Without a graphics 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
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Deeper understanding of calculus and the graph of y = x3.
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Sketch the graph of y = x3 3x + 2
f '(x)
f(x)
f ''(x)
f(x)
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Sketch the graph of y = 3x5 5x3
f '(x)
f(x)
f ''(x)
f(x)
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Textbook Page 220221 #3, 7, 13, 21, 23, 39
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Card activity!!!
Arrange the cards in order from f(x), f'(x), f''(x)
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Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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(d) Using all of the above, include a sketch of f
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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or maximum?
(f) skip (e) and instead include a sketch of f
a) y 1 = 2(x 3)
b) Local minimum at x = 1 because f '(1) = 0 and f '(x) < 0 from (3, 1) and f '(x)>0 from (1, 4).
c) slope anywhere between 0.5 and 1
d) x = 3, and x = 1 and x = 3 because f ''(3) = f ''(1) = f ''(3) = 0 and ......discuss the change in slope around each x value in the appropriate interval.
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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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.
Question
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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QuestionThe diagram below shows a sketch of the gradient 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.
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
QuestionConsider the function f(x)=x 3 −3x 2 −9x+10 , x∈R .
Find the equation of the straight line passing through the maximum and minimum points of the graph y=f(x) .
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
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 .
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Question
Show that the curve y=f(x) has one point of inflexion, and find its coordinates.
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
Question
Given that the graph of the function has exactly one point of inflexion, find its coordinates.
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Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
AP 2001 Answers
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
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Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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AP 2004
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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1997 AP Multiple Choice
#2, 4, 5, 7, 8, 11, 22, 79, 85
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Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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November 18, 2019
1997 AP Multiple Choice
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