New Divider! Uses of Derivative 2: Curve Sketching · The diagram below shows a sketch of the...
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Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook 1 November 18, 2019 New Divider! Uses of Derivative 2: Curve Sketching
Transcript of New Divider! Uses of Derivative 2: Curve Sketching · The diagram below shows a sketch of the...
Year 7 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebook
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New Divider!
Uses of Derivative 2: Curve Sketching
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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.
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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!
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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
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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
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How are you determining these intervals?
How can calculus help to determine these intervals?
Packet
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Sign Diagram Graphical Representation of f '(x)
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(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.
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Use this language for AP exam
State and justify any relative maxima/minima.
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Match the capital letter graph of f '(x) with the graph of f(x).
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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
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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)
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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?
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NOTE: MAKE A STATEMENT ABOUT TANGENT APPROXIMATIONS AND CONCAVITY!!!
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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
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Deeper understanding of calculus and the graph of y = x3.
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Sketch the graph of y = x3 3x + 2
f '(x)
f(x)
f ''(x)
f(x)
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Sketch the graph of y = 3x5 5x3
f '(x)
f(x)
f ''(x)
f(x)
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Textbook Page 220221 #3, 7, 13, 21, 23, 39
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Card activity!!!
Arrange the cards in order from f(x), f'(x), f''(x)
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(d) Using all of the above, include a sketch of f
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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.
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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
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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.
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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) .
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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 .
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Question
Show that the curve y=f(x) has one point of inflexion, and find its coordinates.
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Question
Given that the graph of the function has exactly one point of inflexion, find its coordinates.
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AP 2001 Answers
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AP 2004
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1997 AP Multiple Choice
#2, 4, 5, 7, 8, 11, 22, 79, 85
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1997 AP Multiple Choice
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