Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018 New Divider! Uses of Derivative 2: Curve Sketching 1 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018 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. 2 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018 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! 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 slope 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 slopes of the tangent lines go to zero ­slopes of tangents 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 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 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 calculus help to determine these intervals? 7 Year 6 Relative Min, Max, Points of Inflection, First and Second Derivative Test.notebookNovember 13, 2018 Sign Diagram 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 derivatives. 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 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 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 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. 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 equation 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