Unit 4 Applications of Derivatives (Part I)
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- Roger Wilkerson
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1 Unit 4 Applications of Derivatives (Part I) What is the same? 1. Same HW policy 2. Same expectations 3. Same level of difficulty but different format 4. Still understanding at a conceptual level but more What is different? 1. See #3 and #4 above 2. The test is not graded on the AP scale 3. Format of the test is very different More details later
2 Up First Today --- A few Definitions
3 Before we start There is a difference between a definition of something and the theorems and tools we use to find those somethings. The something Horizontal Asymptote Definition lim f ( x) or lim f ( x) x x Tool for finding Growth comparison of numerator and denominator Extrema Minimum or maximum Derivative tests coming up
4 Notation with new terminology For the interval [a,b]. Points a and b are the endpoints. All the points in between a and b are interior points. For the interval cd,. There are no endpoints. All the points in between c and d are interior points.
5 Experiment 1 Draw a graph with domain ab, that satisfies: for any x and x in a, b such that x x, f x f x
6 Experiment 2 Draw a graph with domain ab, that satisfies: for any x and x in a, b such that x x, f x f x
7 Definition Increasing/Decreasing Let f be a function defined on an interval I and let x and x be any two points in I f increases on I if x x f x f x f decreases on I if x x f x f x On the pretty colored paper!
8 Now a debate... Which is correct? b a c d Decreasing on bc, Decreasing on bc, An applied mathematician s perspective A theoretical mathematician s perspective
9 By definition (theoretical perspective), it is best to state the largest most inclusive interval that still fits the definition. Let f 1 2 be a function defined on an interval I and let x and x be any two points in I. 1. f increases on I if x x f x f x f decreases on I if x x f x f x Decreasing on bc, b a c d Decreasing on bc,
10 Conclusion If an endpoint of the interval is in the domain of the function, include it in the increasing/decreasing intervals. b a c d Increasing on a, b c, d Decreasing on bc,
11 Experiment 3 Plot a point (c, f(c)) in the middle of an empty coordinate plane. Draw a graph for f(x) such that f for ALL x in the domain of f(x). x f c Try to make one different from everyone else.
12 Experiment 4 Plot a point (c, f(c)) in the middle of an empty coordinate plane. f x f c Draw a graph for f(x) such that for ALL x in the domain of f(x). Try to make one different from everyone else.
13 Definition: Absolute Extreme Values Let f be a function with domain D. Then f(c) is the (a) absolute maximum value on D if and only if f x f c for all x in D (b) absolute minimum value on D if and only if On the pretty colored paper! f x f c for all x in D Notice there is nothing about derivatives or calculus in the definition!
14 Experiment 5 Plot a point (c, f(c)) in the middle of an empty coordinate plane. Draw a graph for f(x) such that f x f c for some open interval containing c, but NOT for all x in the domain of f(x). Try to make one different from everyone else.
15 Experiment 6 Plot a point (c, f(c)) in the middle of an empty coordinate plane. f x f c Draw a graph for f(x) such that for some open interval containing c, but NOT for all x in the domain of f(x). Try to make one different from everyone else.
16 Definition: Local Extreme Values Let c be an interior point of the domain of the function f. Then f(c) is (a) A local maximum value at c if and only if f x f c for all x in some open interval containing c. (b) A local minimum value at c if and only if for all x in some open interval containing c. f x f c On the pretty colored paper!
17 IMPORTANT! Where means the x-value or the ordered pair When means the x-value What means the y-value f x x For the above function, where is the minimum? x = 3 or (3,f(3)) For the above function, what is the minimum? f(3) = 2 The minimum is 2.
18 There is never more than one ABSOLUTE minimum or maximum but... An absolute minimum or maximum value can occur at many locations. Example: f(x)=sin(x) has an absolute maximum of one but it occurs an infinite number of locations.
19 QUESTIONS about Increasing/Decreasing, Absolute/Relative Extrema before we go on??
20 What are the derivatives at the indicated points? f ' 0 f ' DNE f ' 0 These are known as Critical Points.
21 Definition of a Critical Point A point in the interior of the domain of a function f where f = 0 or f = DNE is a critical point of f. Notes to Add: Critical points are candidates for minimums and maximums. Critical points are not guaranteed to be a min or max. If an x-value is not in the domain of the function, it cannot be a critical point.
22 All 3 are critical points. But only the top 2 are extrema. f ' 0 f ' DNE EXTREMA EXTREMA f ' 0 NOT AN EXTREMA
23 Definition Recap QUESTIONS? 4 Definitions Increasing/Decreasing Absolute (Global) Extrema Relative (Local) Extrema Critical Point
24 On to Theorems
25 Brainstorm with your group and answer the following question: How can we use the first derivative to find where a graph is increasing/decreasing? SHARE with the class.
26 Theorem Increasing/Decreasing Functions Let f be continuous on a, b and differentiable on a, b. 1. If f ' 0 at each point of a, b, then f increases on a, b. 2. If f ' 0 at each point of a, b, then f decreases on a, b.
27 Compare and Contrast Definition Nothing about derivatives or calculus. Let f be a function defined on an interval I and let x and x be any two points in I f increases on I if x x f x f x f decreases on I if x x f x f x Theorem The First Derivative is a tool to find where a function is increasing and decreasing. It is not the definition. Let f be continuous on a, b and differentiable on a, b. 1. If f ' 0 at each point of a, b, then f increases on a, b. 2. If f ' 0 at each point of a, b, then f decreases on a, b.
28 Theorem Local Extreme Values If a function f has a local maximum or a local minimum value at an interior point c of its domain, and if f exists at c, then f (c) = 0 The converse is NOT true. f (c)=0 does NOT guarantee the critical point is an extrema. It is only a candidate.
29 More Free Response Tips When using a definition, be sure to address all the elements of the definition. For example, when using the definition of continuity, you must use limits. When using a theorem, be sure to address all the elements of the hypothesis before drawing your conclusion. For example, if the theorem says If a function is continuous and differentiable then you must explicitly say the function is continuous and differentiable and why you know that.
30 Working with tools.
31 Interpret an f sign chart f ' 2 Function f is increasing on,2. Function f is decreasing on 2,. Function f has a maximum at x 2 because f ' is positive for x 2 and negative for x 2
32 Interpret the sign chart f ' 1 3 Function f is increasing on,1 3,. Function f is decreasing on 1,3. Function f has a maximum at x 1 because f ' is positive for x 1 and negative for x 1 Function f has a minimum at x 3 because f ' is negative for x 3 and positive for x 3
33 Interpret the sign chart f ' 4 2 Function f is increasing on 2,. Function f is decreasing on, 2. Function f has a minimum at x 2 because f ' is negative for x 2 and positive for x 2 x 4 is not an extrema because the derivative sign does not change therefore the graph does not change from increasing to decreasing or vice versa.
34 1 st Derivative Test--Summary Find the critical points (CPs) by determining where f (x)=0 AND where f (x)=dne Analyze sign behavior on either side Create a sign chart with the CPs Interpret the results Where is function increasing/decreasing Where are extrema What type of extrema
35 Key concepts -- Summary f (x)=0 or f (x)=dne - Critical point, which is a POSSIBLE extrema f (x)>0 - f(x) is increasing f (x)<0 - f(x) is decreasing Extrema occur at critical points where the graph changes from increasing to decreasing or vice versa.
36 Find the location of any extrema and indicate if the extrema is a maximum or minimum. Also find where the function is increasing and decreasing. x y xe x y ' xe e x y ' 0 y ' DNE
37 One more! y ( x 4) 2/3 y ' 2 3( x 4) 1 3
38 For Free Response Questions Sign charts are valuable tools and are allowed, BUT THEY ARE NEVER SUFFICIENT TO EARN POINT To earn the test points you must interpret the sign chart using WORDS!!! Examples of what to write: x = 5 is the location of a local maximum because f changes from positive to negative. OR x = 5 is the location of a local maximum because f changes from increasing to decreasing. OR x = 7 is the location of a local minimum because f changes from negative to positive.
39 Practice Use the first derivative to find where the graphs are increasing and decreasing and find and classify all extrema. 3 2 x 7x 1. y 10x x 7x 2. y 4x y x 4x 4x 9 4. y 4 2 x x e
It has neither a local maximum value nor an absolute maximum value
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