Slide 1 / 163 Slide 2 / 163 AP Calculus Analyzing Functions Using Derivatives 2015-11-04 www.njctl.org Slide 3 / 163 Slide 4 / 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 Extreme Values: Connecting Graphs of f, f', and f'' Curve Sketching Graphically Rolle's Theorem Mean Value Theorem Newton's Method Return to Optimization Table of Contents Slide 5 / 163 Slide 5 (Answer) / 163 Horizontal Tangents Horizontal Tangents Allow students to discuss what they Recall from the previous unit... we analyzed graphs and discovered Recall from the previous unit... we analyzed graphs and discovered notice about each point. They may Teacher Notes comment that the function is changing the locations of horizontal tangent lines. the locations of horizontal tangent lines. direction at points a and c, but not at b. Hopefully they will conclude that at point a, the function reaches a "high point" or a a Looking at locations a, b, and c, Looking at locations a, b, and c, maximum, and at point c, the function while they all share the trait that while they all share the trait that hits a "low point" or minimum, while at b b point b it does not. they have horizontal tangents, they have horizontal tangents, what is different about each point? what is different about each point? [This object is a pull tab] c c
Slide 6 / 163 Slide 6 (Answer) / 163 Slopes Surrounding Point a Slopes Surrounding Point a a Looking specifically at point a, we know at the peak, the slope is zero. 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? What do you notice about the slope on either side of a? Teacher Notes a a You may wish to have students draw tangent lines on either side of point a and make observations. The slope, or derivative, changes from positive to negative at point a. [This object is a pull tab] Slide 7 / 163 Slide 8 / 163 Slopes Surrounding Point c Slopes Surrounding Point b Similarly, we have a change in slopes at point c, however the slope is Now, consider point b. We know the slope is zero at b; however, the changing from negative to positive at this point. function's slope does not change signs at this point. b c Slide 9 / 163 Slide 10 / 163 Local (Relative) Extrema Absolute (Global) Extrema Local Maximum: a high point on any interval relative to points Absolute Maximum: occurs at c if f(c)>f(x) for all x in domain 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 Absolute Minimum: occurs at c if f(c)<f(x) for all x in domain it. At this point the slope changes from negative to positive, and the function changes from decreasing to increasing. NOTE: Absolute max/mins can occur at endpoints! NOTE: Local max/mins CANNOT occur at endpoints!
Slide 11 / 163 Slide 11 (Answer) / 163 Extrema Extrema Abs. & Local Max. Identify/Label each of the following with Local or Absolute Maximum or Identify/Label each of the following with Local or Absolute Maximum or Minimum. What do you notice about what is occuring at the star? Minimum. What do you notice about what is occuring at the star? Local Max. Answer nothing Local Min. Abs. Min. At the star, although the slope is zero, there is neither a maximum or minimum. Also, at point e, it cannot be a local max/min and isn't [This object is a pull tab] the lowest point, therefore not an abs. min. Slide 12 / 163 Slide 12 (Answer) / 163 Slide 13 / 163 Slide 14 / 163 Extrema & Endpoints Critical Value An extrema can only occur at critical values or endpoints (absolute); however, the presence of a critical value does not guarantee an A critical value (or critical point) is a point on the interior of the domain extrema at that value. of a function at which the slope is zero or undefined. What does this mean? 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 (Answer) / 163 Slide 15 / 163 1 Using the given graph, which of the following are Extrema & Endpoints It may be helpful to use the graphs of occurring at point b? and to emphasize this point. An extrema can only occur at critical values or endpoints (absolute); Another analogy that may help: If you live in New Teacher Notes however, the presence of a critical value does not guarantee an Jersey, it is guaranteed that you live in the A local maximum extrema at that value. United States. However, the converse is not always true. If you live in the United States, you B local minimum What does this mean? don't necessarily live in New Jersey. C absolute maximum D absolute minimum E slope is zero [This object is a pull tab] slope is undefined F Slide 15 (Answer) / 163 Slide 16 / 163 1 Using the given graph, which of the following are 2 Using the given graph, which of the following are occurring at point b? Answer occurring at point d? A, E A local maximum A local maximum B local minimum B local minimum [This object is a pull tab] C absolute maximum C absolute maximum D absolute minimum D absolute minimum E slope is zero E slope is zero slope is undefined F slope is undefined F Slide 16 (Answer) / 163 Slide 17 / 163 2 Using the given graph, which of the following are 3 Using the given graph, which of the following are Answer occurring at point c? occurring at point d? A, C, F A local maximum A local maximum B local minimum B local minimum [This object is a pull tab] C absolute maximum C absolute maximum D absolute minimum D absolute minimum E slope is zero E slope is zero slope is undefined F slope is undefined F
Slide 17 (Answer) / 163 Slide 18 / 163 4 Using the given graph, which of the following are 3 Using the given graph, which of the following are occurring at point a? occurring at point c? Answer B, D, F A local maximum A local maximum B local minimum B local minimum [This object is a pull tab] C absolute maximum C absolute maximum D absolute minimum D absolute minimum E slope is zero E slope is zero slope is undefined F slope is undefined F Slide 18 (Answer) / 163 Slide 19 / 163 5 On which interval(s) is the function increasing? 4 Using the given graph, which of the following are occurring at point a? Answer B, E A (a,b) A local maximum B (b,c) B local minimum [This object is a pull tab] C (c,d) C absolute maximum D absolute minimum E slope is zero slope is undefined F Slide 19 (Answer) / 163 Slide 20 / 163 5 On which interval(s) is the function increasing? 6 Using the given graph, which of the following are critical A, C values? Answer emphasize that the A (a,b) function is increasing on A E these intervals because B (b,c) B F the slope is positive [This object is a pull tab] C (c,d) G C H D
Slide 20 (Answer) / 163 Slide 21 / 163 7 Using the given graph, which of the following are 6 Using the given graph, which of the following are critical occurring at ? values? Answer B, C, D, A local maximum E, F, G A E B local minimum B F C absolute maximum [This object is a pull tab] G C D absolute minimum D H E slope is zero slope is undefined F Slide 21 (Answer) / 163 Slide 22 / 163 7 Using the given graph, which of the following are 8 On which interval(s) is the function decreasing? occurring at ? Answer A B, E A local maximum B B local minimum C C absolute maximum [This object is a pull tab] D D absolute minimum E E slope is zero F slope is undefined F G Slide 22 (Answer) / 163 Slide 23 / 163 8 On which interval(s) is the function decreasing? B, D, F Answer A emphasize the function B is decreasing because the slope is negative C [This object is a pull tab] D E F G
Slide 23 (Answer) / 163 Slide 24 / 163 Slide 24 (Answer) / 163 Slide 25 / 163 Slide 25 (Answer) / 163 Slide 26 / 163 12 On which interval(s) is the function increasing?
Slide 26 (Answer) / 163 Slide 27 / 163 13 If a function has a critical value at x=3, then there must 12 On which interval(s) is the function increasing? be a local or absolute extrema at that value. Answer emphasize that the function is increasing True because the slope is positive False [This object is a pull tab] Slide 27 (Answer) / 163 Slide 28 / 163 13 If a function has a critical value at x=3, then there must be a local or absolute extrema at that value. Answer False True False [This object is a pull tab] Slide 28 (Answer) / 163 Slide 29 / 163 1 st Derivative Test Return to Table of Contents
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