Slide 1 / 163 Slide 2 / 163 AP Calculus Analyzing Functions Using Derivatives 2015-11-04 www.njctl.org Slide 3 / 163 Table of Contents click on the topic to go to that section Extreme Values - Graphically 1 st Derivative Test Concavity & 2 nd Derivative Test Connecting Graphs of f, f', and f'' Curve Sketching Rolle's Theorem Mean Value Theorem Newton's Method Optimization
Slide 4 / 163 Extreme Values: Graphically Return to Table of Contents Slide 5 / 163 Horizontal Tangents Recall from the previous unit... we analyzed graphs and discovered the locations of horizontal tangent lines. a Looking at locations a, b, and c, while they all share the trait that b they have horizontal tangents, what is different about each point? c Slide 6 / 163 Slopes Surrounding Point a Looking specifically at point a, we know at the peak, the slope is zero. What do you notice about the slope on either side of a? a
Slide 7 / 163 Slopes Surrounding Point c Similarly, we have a change in slopes at point c, however the slope is changing from negative to positive at this point. c Slide 8 / 163 Slopes Surrounding Point b Now, consider point b. We know the slope is zero at b; however, the function's slope does not change signs at this point. b Slide 9 / 163 Local (Relative) Extrema Local Maximum: a high point on any interval relative to points around it. At this point, the slope changes from positive to negative, and the function changes from increasing to decreasing. Local Minimum: a low point on any interval relative to points around it. At this point the slope changes from negative to positive, and the function changes from decreasing to increasing. NOTE: Local max/mins CANNOT occur at endpoints!
Slide 10 / 163 Absolute (Global) Extrema Absolute Maximum: occurs at c if f(c)>f(x) for all x in domain Absolute Minimum: occurs at c if f(c)<f(x) for all x in domain NOTE: Absolute max/mins can occur at endpoints! Slide 11 / 163 Extrema Identify/Label each of the following with Local or Absolute Maximum or Minimum. What do you notice about what is occuring at the star? Slide 12 / 163
Slide 13 / 163 Critical Value A critical value (or critical point) is a point on the interior of the domain of a function at which the slope is zero or undefined. When asked to find local extrema, only critical values must be considered. If asked to find absolute extrema, critical values as well as endpoints are considered. Slide 14 / 163 Extrema & Endpoints An extrema can only occur at critical values or endpoints (absolute); however, the presence of a critical value does not guarantee an extrema at that value. What does this mean? Slide 15 / 163 1 Using the given graph, which of the following are occurring at point b? A local maximum B local minimum C absolute maximum D absolute minimum E slope is zero slope is undefined F
Slide 16 / 163 2 Using the given graph, which of the following are occurring at point d? A local maximum B local minimum C absolute maximum D absolute minimum E slope is zero slope is undefined F Slide 17 / 163 3 Using the given graph, which of the following are occurring at point c? A local maximum B local minimum C absolute maximum D absolute minimum E slope is zero slope is undefined F Slide 18 / 163 4 Using the given graph, which of the following are occurring at point a? A local maximum B local minimum C absolute maximum D absolute minimum E slope is zero slope is undefined F
Slide 19 / 163 5 On which interval(s) is the function increasing? A (a,b) B (b,c) C (c,d) Slide 20 / 163 6 Using the given graph, which of the following are critical values? A E B F G C D H Slide 21 / 163 7 Using the given graph, which of the following are occurring at ? A local maximum B local minimum C absolute maximum D absolute minimum E slope is zero slope is undefined F
Slide 22 / 163 8 On which interval(s) is the function decreasing? A B C D E F G Slide 23 / 163 Slide 24 / 163
Slide 25 / 163 Slide 26 / 163 12 On which interval(s) is the function increasing? Slide 27 / 163 13 If a function has a critical value at x=3, then there must be a local or absolute extrema at that value. True False
Slide 28 / 163 Slide 29 / 163 1 st Derivative Test Return to Table of Contents Slide 30 / 163 Calculating Extrema Algebraically We have discovered how to graphically interpret local and absolute extrema, and now we will extend our understanding to calculate extrema algebraically.
Slide 31 / 163 Recall: A local maxima/minima occurs when the slope of the function changes from positive to negative, or negative to positive at a critical value. Another way of thinking is that the original function changes from increasing to decreasing, or decreasing to increasing. For a function to contain an absolute extrema, it must be the highest or lowest extrema on the interval, including endpoints. Slide 32 / 163 The 1 st Derivative Test Algebraically, we can find critical values and test points on either side to determine the change in slope, if there is one. This is known as the 1 st Derivative Test. Helpful Steps: 1. Find the derivative of the given function. 2. Find x-values where the derivative equals zero or is undefined. These are the critical values (locations of possible extrema). 3. Test x-values on either side of the critical values, substitute into the derivative and observe sign change for maximum or minimum. 4. Substitute x-values into original function to get corresponding y-values for the extrema. Slide 33 / 163 Critical Values First let's practice finding critical values. Remember, a critical value is the potential location for an extreme value. Find the location of any critical value(s):
Slide 34 / 163 Slide 35 / 163 Example For the function, identify the intervals of increasing/decreasing: Slide 36 / 163 Example, Continued For the previous function, find any local extrema.
Slide 37 / 163 Example For the following function, identify the intervals of increasing/decreasing. Slide 38 / 163 Example, Continued For the previous function, find any local extrema: Slide 39 / 163 Example Find any local extrema for
Slide 40 / 163 Example Find any local extrema for Slide 41 / 163 Grading on the AP Exam Drawing a sign chart is a great way to help visualize what is happening; however, a sign chart alone is not enough of an explanation on the AP Exam. Be sure to defend your answer, describing the sign change of the derivative. For example: The function has a local maximum at (-3,7) because f '(x) changes from positive to negative at x=-3. Slide 42 / 163 Absolute Extrema Now that we feel comfortable solving for local extrema, we can go further and determine where absolute extrema occur. Remember, for absolute extrema we must compare all extrema within the given interval, as well as the endpoints of that interval.
Slide 43 / 163 Extreme Value Theorem Given that is continuous on the interval then must attain an absolute maximum and minimum on that interval. Slide 44 / 163 Extreme Value Theorem Which of the following graph(s) meet the criteria for the Extreme Value Theorem? Circle your answer, and for those graph(s) that do not, explain why. Slide 45 / 163 Absolute Extrema Let's take a look back at an example we previously worked with: Example: Find the absolute extrema for on We found this function had a local minimum at and a local maximum at Now, to find the absolute extrema we need to compare these with the endpoints.
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Slide 52 / 163 20 If a function has a local minimum at then True False Slide 53 / 163 Slide 54 / 163
Slide 55 / 163 Concavity & 2 nd Derivative Test Return to Table of Contents Slide 56 / 163 Relationship Between Derivatives & Functions As we saw in the previous section, derivatives can tell us a lot of information about the original function. Next, we will discover what information the 2 nd derivative provides about the original function. Slide 57 / 163 Concavity As we know, in mathematics, not all functions are linear. There are infinite curves created by functions and they take on unique shapes. Let's cover some new vocabulary to describe these curves.
Slide 58 / 163 Concavity Recall that the sign of the 1 st derivative told us whether the function was increasing or decreasing. The sign of the second derivative will tell us where the function is concave up or concave down. Why is this helpful? Slide 59 / 163 Concavity Some functions remain concave up or down for all x in their domain, but many others can change concavity at multiple points. For example: Label each section with the correct concavity. Slide 60 / 163 Inflection Point A point x=c is an inflection point (or point of inflection) if a function is continuous at that point and the function changes concavity at that point. Note: Each star represents an inflection point for the function above.
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