ABSOLUTE EXTREMA AND THE MEAN VALUE THEOREM

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1 61 LESSON 4-1 ABSOLUTE EXTREMA AND THE MEAN VALUE THEOREM Definitions (informal) The absolute maimum (global maimum) of a function is the -value that is greater than or equal to all other -values in the function. The absolute minimum (global minimum) of a function is the -value that is less than or equal to all other -values in the function. A relative maimum (local maimum) of a function is a -value that is greater than or equal to all nearb -values in the function. A relative minimum (local minimum) of a function is a -value that is less than or equal to all nearb -values in the function. Etrema (etreme values) are either maimum values (maima) or minimum values (minima). Critical Numbers are -values at which f () eists but f is either zero or undefined. THE CANDIDATE TEST PROCEDURE FOR FINDING ABSOLUTE (GLOBAL) EXTREMA: 1. Find all critical numbers of the function.. Find -values at each critical number and at each endpoint of the interval.. Choose the least and greatest -values as absolute etrema. Note: Absolute etrema can occur either at critical numbers or endpoints. Relative etrema can occur onl at critical numbers. We will not consider endpoint etrema to be relative etrema. Eamples: Use the figure of f at the right to answer these questions. 1. What is the absolute maimum of f?. At what -value does f have an absolute maimum?. What is the absolute maimum point on f? 4. What is the absolute minimum of f? 5. At what -value(s) does f have a relative minimum? 6. At what -value(s) does f have a relative maimum? Eamples: 1 7. Find the global etrema of f on the interval 1,.

2 6 8. Find the absolute maimum and minimum values of f on the interval 0,5. 9. Find the etrema of f on 1,. Discovering the Mean Value Theorem For Eamples draw these lines (if possible). (a) Draw the secant line between the two points (b) Draw all tangent lines parallel to the secant line a f a b f b, and,. a b a b a b a b a b a b For eamples (a) Draw the secant line between the two points f f, and 5, 5. (b) Draw all tangent lines parallel to the secant line at some point on the interval (,5). (c) Estimate the value of c where c, f c is a point of tangenc. 16. c 17. c 18. c

3 MEAN VALUE THEOREM: If f is continuous on ab, and differentiable on ab,, then there is a number c in ab, such that tangent slope (inst. rt. of ch.) f c f b f a b a. secant slope (avg. rt. of ch.) Informall: The Mean Value Theorem states that given the right conditions of continuit and differentiabilit, there will be at least one tangent line parallel to the secant line. In still other words: The instantaneous rate of change (slope of tangent) will equal the average rate of change (slope of secant) at least once. a a, f a c 6 b b, f b Eample 19. Given the interval,6. 6 f, find all c which satisf the Mean Value Theorem on ASSIGNMENT 4-1 Use the graph of f at the right for Problems What is the absolute maimum value of f?. At what point does f reach a global minimum?. At what -value(s) does f have a relative minimum? 4. At what -value(s) does f maimum? have a relative

4 64 Find the critical numbers of these functions f Without using a calculator, find the absolute maimum and absolute minimum values of f () on the given interval. 9. f on 1,5 10. f 6 on 1, f on,1 1. on 1, f f on,1 1. f on,0 14. on 1, 15. f Find the absolute minimum and maimum values of the functions graphed on 1, Does the Mean Value Theorem appl to the given function on the given interval? If it does, find the c-value. If it does not, eplain wh not. Do not use a calculator. f on 1, f on 1, f on 1,. f on, 1 4. f on 0,1 5. f on, f on 1,1 7. f on 0, Find the c-value guaranteed b the Mean Value Theorem for the given function on the given interval. You ma use a calculator f on 1, 9. f on, 1 0. The height, in feet, of an object at time t seconds is given b h 16t 00. a. Find the average velocit of the object during the first seconds. b. Use the Mean Value Theorem to find the time at which the object s instantaneous velocit equals this average velocit.

5 65 LESSON 4- INCREASING/DECREASING FUNCTIONS, FIRST DERIVATIVE TEST FOR RELATIVE EXTREMA PROCEDURE: (Increasing/ Decreasing and First Derivative Test) 1. Find domain restrictions.. Find all critical numbers (where f 0 or f is undefined but domain restrictions cannot be critical numbers).. Locate critical numbers and domain restrictions on an f number line. Label critical numbers CN. 4. Test the sign of f in each interval and label the signs on the number line. 5. List increasing/decreasing intervals and/or identif relative min/ma -values. If requested, find -values or points. Eamples: Find the intervals on which these functions are increasing and decreasing and find all local etrema points f 9. f. f The function in Eample is a strictl monotonic function. Strictl increasing or strictl decreasing functions are called monotonic. Eample illustrates two important points. 1. Not ever critical number produces a relative maimum or minimum.. Even though the slope at = 1 is zero, the function is alwas increasing.

6 66 ASSIGNMENT 4- Use the graphs shown with each function to answer these four questions. a. On which intervals is f increasing? b. On which intervals is f decreasing? c. At which point(s) does f have a relative minimum? d. At which point(s) does f have a relative maimum? 1 1. f. f. f f Use the f number lines shown to answer these four questions. a. On which intervals is f increasing? b. On which intervals is f decreasing? c. At which -values does f have a relative minimum? d. At which -values does f have a relative maimum? 5. f 6. Domain: 0-0 f CN CN - 0 CN CN Which of the functions in Problems 1-4 would have the following f number lines? 7. f 8. f CN CN CN CN CN

7 67 Without using a calculator, answer these four questions for Problems Show organized work and an f number line to support our answers. a. On which intervals is f increasing? b. On which intervals is f decreasing? c. At which point(s) does f have a relative minimum? d. At which point(s) does f have a relative maimum? 9. f f f 1 1. f f f f f 17. f Determine whether the following functions are strictl monotonic on the interval 0,. Do not use a calculator. 18. f 19. g 1 0. h For Problems 1- the height, in feet, of a ball is given b the position function s t 16t 64t 6. Assume 0t 4 seconds. Do not use a calculator. 1. On which interval of time is the ball moving upward?. What is the maimum height of the ball? An object has increasing velocit when its acceleration is greater than zero. It has increasing speed when the velocit and acceleration have the same sign.. For what times is a. the velocit of the ball increasing? b. the speed of the ball increasing? 4. If the function f a b has a relative minimum point at, 4 a and b. 5. Without using a calculator find the absolute etrema of interval 1,., solve for f on the 6. Without using a calculator find the absolute etrema of f on 7. Find all critical numbers of f. 8. Can the Mean Value Theorem be used for the function f 1,4. on the interval 0,? If it can be used, find the c-value. If it cannot, eplain wh not. 9. Find the c-value guaranteed b the Mean Value Theorem for the function f 1,1. You ma use a calculator. on the interval

8 68 LESSON 4- CONCAVITY AND POINTS OF INFLECTION, THE SECOND DERIVATIVE TEST FOR RELATIVE EXTREMA A graph with this shape is called concave upward. The tangent lines lie below the graph. The slopes of the tangent lines are increasing which means f 0. A graph with this shape is called concave downward. The tangent lines lie above the graph. The slopes of the tangent lines are decreasing which means f 0. Nonmathematical Memor Device: Concave upward positive smile face Concave downward negative frown face A point on a graph is a point of inflection if: 1. the graph has a tangent line at that point and. the graph changes concavit at that point. On the graph of shown, the point (0,0) is a point of inflection. Eamples: Use the graph at the right to answer these questions. Base our answers on appearances of the graph. 1. On which intervals is the graph concave upward?. On which intervals is the graph concave downward?. On which intervals does the graph have no concavit? 4. What are the points of inflection? Analticall we find concavit intervals and points of inflection b using a second derivative number line. The procedure is parallel to the procedure used in the last lesson to find increasing/decreasing intervals and relative etrema b using a first derivative number line.

9 69 Eamples: 5. Determine the points of inflection and discuss the concavit for the graph of f and 6. If f f , list the intervals where the graph of f is concave upward, concave downward, and list the points of inflection. THE SECOND DERIVATIVE TEST FOR RELATIVE EXTREMA This test does not require a second derivative number line. It does not find points of inflection. It is used to find relative etrema (ma/min). Procedure: 1. Use f to find critical numbers.. Plug critical numbers into f and analze concavit to determine if the function has a relative minimum or maimum. Note: The Second Derivative Test does not alwas give an answer (when f 0 ). Use it onl when the directions require it or when the given information requires it. Eamples: 7. Use the Second Derivative Test to find the relative minimum and relative maimum 4 points for the graph of f 6.

10 8. g is a function such that g g g g If g 4, g 1, g 0 0, g 1 5, and g 0, 1 0, 0 0, and 0., find the -values of the relative maimum and relative minimum values when possible. 70 ASSIGNMENT 4- Use the appearance of these graphs to answer these three questions. a. On which interval(s) is the graph of the function concave upward? b. On which interval(s) is the graph of the function concave downward? c. What are the points of inflection? Show organized steps and an f number line to answer the same three questions for these functions f 4 6. f 5 7. f Use the Second Derivative Test to find the relative etrema points (see Eample 7 on the previous page) f 6 9. f 5 For these problems, find the -values of relative minimum points and the -values of relative maimum points. 10., 1, and f f 0 f 4 0 are critical numbers of f and 11. f, f1 0, and f. f -1

11 71 Without using a calculator, find local maimum and minimum points and points of inflection. Then sketch a graph. It is not necessar to find the -intercepts. Show organized steps with f and f number lines to support our answers f f Use the following information to sketch a possible graph of f. f 1, f 0, f 1, f does not eist. f is a continuous function, f 15. Use a calculator to determine if the function f 4 sin 5 is concave upward or downward on an interval including =.5. Be sure to use radians mode. f 16. Find the absolute maimum and absolute minimum for the function 4 on the interval,1. Do not use a calculator. 17. Find the c-value guaranteed b the Mean Value Theorem for the function,6. f on the interval 18. If the graph of a b c d has a point of inflection at the point (0,) and a relative maimum at the point 1,4, find the values of a, b, c, and d. 19. a. Sketch a smooth curve whose slope is alwas positive and whose slope is increasing. b. Sketch a smooth curve whose slope is alwas positive and whose slope is decreasing. c. Sketch a smooth curve whose slope is alwas negative and whose slope is increasing. d. Sketch a smooth curve whose slope is alwas negative and whose slope is decreasing. 0. Given: a. Find the value of d d at the point d 1,. b. Find the value of d at 1,. c. Does the curve have a relative maimum, a relative minimum, or neither at 1, Use the f number line shown for Problems 1-. Fill in the blank with > or <. g f, then g If. If g f g. If g f g 4. -1, then , then 4 0. f - f. t t 7t 50 is the position function of a particle moving along a horizontal line. a. When is the velocit of the particle increasing? b. When is the speed of the particle increasing? c. When is the distance of the particle from the origin increasing? 5

12 7 LESSON 4-4 CURVE SKETCHING WITH EXTREMA AND POINTS OF INFLECTION You used a Precalculus Curve Sketching Recipe in Unit 1. The following incorporates the additional Calculus techniques ou have recentl learned. Curve Sketching Recipe: 1. Give the domain.. Reduce f ().. Find vertical asmptotes and holes. 4. Give - and -intercepts. 5. Find the end behavior (horizontal asmptotes or other). 6. (optional) Check for smmetr. 7. Find increasing/decreasing intervals and relative etrema (show an f number line). 8. Find concavit and points of inflection (show an f number line). 9. Graph. Eamples: 1. (a rational function) Do.: V.A.: Holes: -int.: -int.: E.B.: Smmetr: Rel. Ma.: Rel. Min.: P.I.: 4 f, f, f. (a rational function) Do.: V.A.: Holes: -int.: -int.: E.B.: Smmetr: Rel. Ma.: Rel. Min.: P.I.: f, f, f

13 7. (a radical function) Do.: V.A.: Holes: -int.: -int.: E.B.: Smmetr: Rel. Ma.: Rel. Min.: P.I.: 6 f, f, f 5 ASSIGNMENT 4-4 Without using a calculator, find local etrema, points of inflection, and sketch a graph. Show organized steps and justification. It is not necessar to find -intercepts and there are no domain restrictions or asmptotes. However, an end behavior analsis will be helpful Find intercepts and relative etrema and graph these functions without using a calculator. 4. f 8 5. g 8 Find relative etrema, points of inflection, Find the domain, relative etrema, and end behavior and graph without a asmptotes, and end behavior calculator. and graph without a calculator. 6 1 There are no points of inflection. 6., 1 7. f Find the domain, reduced function, hole, intercepts, relative etrema, and points of inflection. Then graph without using a calculator f, f, f , f Without a calculator, find the domain, -intercepts, and relative etrema. Then graph f. There are no points of inflection. f True or False? If f 0 for all real -values, then lim f Show a graph to illustrate our answer..

14 74 Find the discontinuities. Which are removable? Do not use a calculator. 4, f 1. f 1. f 1, 0 Find these limits without a calculator. 14. lim 15. lim lim Find an equation of the line tangent to the graph of You ma use a calculator. 4 at the point where = a. Find the points at which the graph of 4 6 has horizontal tangent lines. b. Determine whether each of these points is a local minimum or a local maimum. (Show organized work using the Second Derivative Test.) 19. Find the points at which the graph of 4 6 has vertical tangent lines. Match the graph of f in the top row with the appropriate graph of f in the bottom row A. B. C. D. 4. Find (or estimate) the slope of each graph at each lettered point. A B C D E F G H I J

15 75 LESSON 4-5 GRAPHING DERIVATIVES AND ANTIDERIVATIVES FROM GRAPHS Derivatives: f graph f graph (or f graph f graph) Find (or estimate) slopes and plot them as points. Eample: 1. Use the graph of f shown to sketch a graph of f. f f Antiderivatives: f graph f graph 1. Make an f number line b using the location or position of the points on the f graph. This does not involve the slopes of f.. Make an f number line b using the slopes of the f graph.. Combine information from both number lines to graph f. If no starting point is given, ou are free to shift the graph verticall. Eamples:. Use the graph of f shown to sketch a graph of f with a starting point of (0,1). f f f f. Use the graph of f shown to sketch a graph of f and a possible graph of f. f f f f f

16 76 ASSIGNMENT 4-5 Sketch a graph of the derivative of the function whose graph is shown Use the graph of f shown and the given starting point to graph f (the antiderivative) starting point (0,0) starting point (1,) starting point 1, starting point (0,0) Use the graph of f shown to sketch a graph of f and a possible graph of f Find the vertical asmptotes, end behavior, and relative etrema. Then graph 1 f without using a calculator If f 4, find relative etrema and points of inflection. Then graph f. without using a calculator. Hint: f 1 4

17 77 f Use the Second Derivative Test to find the relative etrema of 15. Use the following information to sketch a possible graph of f. f 0 f 4 0, f, 0 when f 0 when, f 0 when, f does not eist, f 16. Find the c-value guaranteed b the Mean Value Theorem for the interval 0,. 17. Find the absolute minimum and absolute maimum of the function f 1 0,4 without using a calculator. on the interval 18. Without using a calculator, sketch a graph of 19. The graph of f 6. f on a b c passes through the point 1,6 and has a tangent line at 0,16 which is parallel to the graph of 1. Find a, b, and c. 0. If the onl critical number of a function f () is, f f 6, and 4 7, does f have a local minimum or a local maimum at? Assume f is continuous. 1. If is a critical number of a function g () and g relative minimum or a relative maimum at? 6, does g have a True or False?. Ever fourth degree polnomial has three critical numbers.. Ever fourth degree polnomial has at most three critical numbers. 4. If a polnomial has two critical numbers, one must be at a relative maimum and the other must be at a relative minimum. f 0, then f is increasing at. 5. If 6. If a function f is increasing at, then f 0.

18 78 UNIT 4 SUMMARY Definition of Critical Numbers: -values where f is zero or undefined (cannot be domain restrictions) Absolute Etrema: Compare -values at endpoints and critical numbers. Relative (Local) Etrema: First Derivative Test (best wa): Find critical numbers, and make an f number line. Domain restrictions must be on all number lines but cannot be ma/min points. An f number line also gives increasing/decreasing intervals. Second Derivative Test (Use onl when ou have to.): Use f to find the critical numbers then plug them into f, and use concavit to see if the are at a maimum or a minimum. Points of Inflection and Concavit: Find possible points of inflection (where f is zero or undefined), and make an f number line. Remember, domain restrictions cannot be points of inflection but must be on all number lines. Curve Sketching; Precalculus: domain, intercepts, vert. asmptotes, holes, end behavior, smmetr Calculus: f number line inc./decr. and ma./min. f number line concavit, pts. of infl. Graph to Graph: f f Find slopes on f and plot them as points on f. f f Make an f number line using the location of points on the f graph. Make an f number line using the slope at points on the f graph. Use both number lines to sketch a graph of f. Mean Value Theorem: f must be continuous on, inst. rt. ch. = avg. rt. ch. IROC = AROC tangent slope = secant slope f b f a fc b a MVT is used to find the c-value. ab and differentiable on ab,. a a, f a c b b, f b

19 79 ASSIGNMENT 4-6 REVIEW 1. Without using a calculator, find the intercepts and relative etrema. Then draw a f 4 5. graph of 16 48, If f 16, f, and f find the domain, intercepts, relative etrema, and points of inflection, and then sketch a graph of the function f without using a calculator.. Without using a calculator, find the domain, vertical asmptote, hole, -intercept, and end behavior, and then graph f. 4. Find the intercepts, relative etrema, and points of inflection, and then graph f without using a calculator. 5. Without using a calculator, graph g. 6. Find the maimum and minimum points on the graph of Find the absolute etrema of f on the interval, a calculator. 8. Find the absolute etrema of g on the interval 8,1 calculator. without using without using a Can the Mean Value Theorem be applied for these functions on the given interval? If it applies, find the c-value. If it does not appl, eplain wh not f on 1, 10. f on 4,6 f on, Given: f g f f g 4, 4, 4 1,, and 4 5, a. If h f g, find h4. b. If c. If k f g, find k 4. j f, find j 4. a, f b, 1, Find: a and b 1. Given: is a continuous function

20 Find these limits. a. lim 9 b. lim 9 c. lim 9 d. lim 5 9 Use these graphs of f to graph f. 15. f 16. f 17. f 18. f 19 and 0. Use the graphs of f for Problems 15 and 16 above to sketch a possible graph of f. 1. Use the graph of f in Problem 17 to sketch a graph of f with the starting point,.. Use the graph of f in Problem 18 to sketch a continuous graph of f with the starting point 0,. Find Find the point where the tangent line to the graph of to the graph of 10. f 5 is parallel 7. The position (in cm) of an object moving on a horizontal line is given b s t t t t 8 (where time is measured in seconds). Answer the following questions. You ma use a calculator. a. What is the object s velocit equation? b. What is the object s initial velocit? c. What is the object s acceleration equation? d. What is the object s acceleration at t = seconds? e. What is the object s speed at t = 1 second? f. When is the object moving left? g. What is the object s displacement between zero and two seconds? h. What is the object s total distance traveled between zero and two seconds? i. When is the object s velocit decreasing? j. When is the object s speed decreasing.