NOTES 5: APPLICATIONS OF DIFFERENTIATION

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1 NOTES 5: APPLICATIONS OF DIFFERENTIATION Name: Date: Period: Mrs. Nguyen s Initial: LESSON 5.1 EXTREMA ON AN INTERVAL Definition of Etrema Let f be defined on an interval I containing c. 1. f () c is the MINIMUM of f on I if f () c f() in I. f () c is the MAXIMUM of f on I if f () c f () in I Figures on pg 314: - Etreme values or Etrema are the minimum and maimum of a function on an interval - On an interval: Maimum ABSOLUTE ma Mininum ABSOLUTE min Theorem 5.1: The Etreme Value Theorem If f is continuous on a closed interval [ ab,, ] then f has both a minimum and a maimum on the interval. Note: In order for f () c to be an etreme value, f () c must eist (or must be defined). In other words: If f is defined at every c between a and b, and at the endpoints, then f must have the lowest and the highest points on ab., the graph over [ ] Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 1

2 Definition of Relative Etrema Relative Etrema can occur in two ways: If there is an open interval containing c on which: 1. f () c is a maimum, then ( c, f( c )) is called a RELATIVE MAXIMUM of f.. f () c is a minimum, then ( c, f( c )) is called a RELATIVE MINIMUM of f. Smooth and rounded curve horizontal tangent line derivative is 0. In other words: Relative etrema are hills or valleys of the graph. Sharp and peaked turn not differentiable. f ( ) f ( ) Eample 1: f( ) + 1 Find the derivative at, 1 ( 0,1 ) and ( ) f '( ) f '(0) What can you conclude 0,1 and about points ( ) (, 1)? Evaluate graph: Note: 0 and are the CRITICAL NUMBERS f '() Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page

3 Definition of a Critical Number Let f be defined at c. If f '( c ) 0 or if f is not differentiable at c, then c is a critical number of f. Evaluate Eample 1: Theorem 5.: Relative Etrema Occur Only at Critical Numbers Guidelines for Finding ABSOLUTE EXTREMA on a Closed Interval If f has a relative minimum or relative maimum at c, then c is a critical number of f. Note: The critical numbers of a function need not produce relative etrema. To find the etrema of a continuous function f on a closed ab,, use the following steps. interval [ ] 1. Find the critical numbers of f in ( a, b ).. Evaluate f at each critical number in ( a, b ). 3. Evaluate f at each endpoint [ ab., ] 4. The least of these values is the minimum. The greatest is the maimum. Eample : Find the ABSOLUTE Etrema of 4 3 f ( ) 3 4 1, on [ ] Step 1: To find the Critical Numbers: find f '( ), set f '( ) 0 and solve for Step : Find f ( CN ) f '( ) Step 3: Find f ( a ) f ( 1) Step 4: Find f ( b ) f () Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 3

4 Practice Problem 1: Find the ABSOLUTE Etrema of g ( ) 3 on [ 1, 1] Practice Problem : Find the ABSOLUTE Etrema of g ( ) 3 1 on ( 1, 4 ] 4 Practice Problem 3: Find the Critical Numbers of f( ) + 1 Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 4

5 LESSON 5. ROLLE S THEOREM AND THE MEAN VALUE THEOREM Theorem 5.3: Rolle s Theorem Let f be continuous on the closed ab, and differentiable interval [ ] on the interval ( a, b ). If f ( a) f ( b), then there is at least a, b such that one number c in ( ) f '( c ) 0. Eample 1: Find the two - intercepts of the function f and show that f '( ) 0 at some point between the two intercepts given f( ) + 4 Consider the following conditions before using Rolle s theorem: a. Is the function a, b? continuous in ( ) b. Is f ( a) f( b)? Eample : Let 3 f( ). + Find all values c in [ 1, 3] such that f '( c ) 0. Consider the following conditions before using Rolle s theorem: a. Is the function a, b? continuous in ( ) b. Is f ( a) f ( b)? Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 5

6 Theorem 5.4: The Mean Value Theorem Let f be continuous on the closed ab, and differentiable interval [ ] on the interval ( a, b ), then there eists a number c in ( a, b ) such that f '( c) f ( b) f( a). b a f( b) f( a) y y1 Note: b a 1 Alternative form of Mean Value Theorem f ( b) f( a) f '( c) b a ( ) f '( c) b a f( b) f( a) ( ) f '( c) b a + f( a) f( b) ( ) f ( b) f( a) + b a f '( c) Eample 3: Use the Mean Value Theorem to find all values of c in ( a, b ) such that f ( b) f( a) f '( c) b a given f( ) ( ). 1, 1 in [ ] Consider the following conditions before using the Mean Value theorem: a. Is the function a, b? continuous in ( ) b. What is f ( b) f( a) f '( c)? b a Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 6

7 LESSON 5.3 INCREASING AND DECREASING FUNCTIONS AND THE FIRST DERIVATIVE TEST Definition of Increasing and Decreasing Functions Decreasing: (, a) Negative derivative: f '( ) < 0 A function f is increasing on an interval if for any two numbers 1 and in the interval, 1 < f( 1) < f( ) and decreasing when < f( ) > f( ) 1 1 Constant: ( a, b ) Zero derivative: f '( ) 0 Increasing: ( b, ) Positive derivative: f '( ) > 0 Theorem 5.5: Test for Increasing and Decreasing Functions Guidelines for Finding Intervals on Which a Function Is Increasing or Decreasing Let f be a function that is continuous on the closed interval [ ab, ] and differentiable on the interval ( a, b ) 1. f '( ) > 0 in ( a, b) f is increasing on [ ab., ]. f '( ) < 0 in ( a, b) f is decreasing on [ ab., ] 3. f '( ) 0 in ( a, b) f is constant on [ ab., ] Let f be a function that is continuous on the interval ( a, b. ) To find the open intervals on which f is increasing or decreasing. 1. Locate the critical numbers of f in ( a, b ) by setting the derivative of f equal to 0. Use these numbers to determine the test intervals.. Determine the sign of f '( ) at one test value in each of the intervals. f '( ) < 0 Decreasin g 3. If f '( ) 0 Constan t f '( ) > 0 Increasin g Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 7

8 Practice Problems: Identify the open intervals on which the function is increasing, decreasing, or constant. 1. f ( ) 7 3 Intervals Test Values (, ) (, ) (, ) (, ) < < < < < < < < f '( TV ) Conclusions. f( ) + 1 Intervals Test Values (, ) (, ) (, ) (, ) < < < < < < < < f '( TV ) Conclusions Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 8

9 A Strictly Monotonic Function Theorem 5.6: The First Derivative Test (to find RELATIVE Etrema, not Absolute Etrema) A function is either increasing or decreasing on the entire interval. Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, ecept possibly at c, then f () c can be classified as follows. 1. If f '( ) changes from negative to positive at c, then f () c is a RELATIVE MINIMUM of f.. If f '( ) changes from positive to negative at c, then f () c is a RELATIVE MAXIMUM of f. 3. If f '( ) does not change sign at c, then f () c is neither a relative minimum nor relative maimum. Eample: f ( ) Relative Minimum Relative Maimum 3 Neither a ma nor a min Practice Problems: Find the relative etrema of the following functions 3. f( ) ( ) Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 9

10 Intervals Test Values (, ) (, ) (, ) (, ) < < < < < < < < f '( TV ) Conclusions Identify Relative Etrema 4. f( ) 9 Intervals Test Values (, ) (, ) (, ) (, ) < < < < < < < < f '( TV ) Conclusions Identify Relative Etrema Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 10

11 LESSON 5.4 CONCAVITY AND THE SECOND DERIVATIVE TEST Definition of Concavity Theorem 5.7: Test for Concavity Let f be differentiable on an open interval I. The graph of f is CONCAVE UPWARD on I if f '( ) is increasing on the interval and CONCAVE DOWNWARD on I if f '( ) is decreasing on the interval. Concave upward graph f lies above all of its tangent lines Concave downward graph f lies below all of its tangent lines Let f be a function whose nd derivative eists on an open interval I 1. f ''( ) > 0 in I graph is concave upward.. f ''( ) < 0 in I graph is concave downward. 3. f ''( ) 0 in I graph is linear, no concavity. Figures on pg 338 To find the intervals: 1. Find f ''( ). Set f ''( ) 0, and solve for. 3. Set the denominator of f (if any) 0 and solve for. Practice Problem 1: Determine concavity for 4 f( ) Check for continuity:. f '( ) 3. f ''( ) 4. Solve for possible points of inflection Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 11

12 Intervals Test Values (, ) (, ) (, ) (, ) < < < < < < < < f ''( TV ) Conclusions Practice Problem : Determine concavity for 1 f( ) Check for continuity:. f '( ) 3. f ''( ) 4. Solve for possible points of inflection Intervals Test Values (, ) (, ) (, ) (, ) < < < < < < < < f ''( TV ) Conclusions Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 1

13 Definition of Point of Inflection A point where the concavity of f changes. Figures on pg 340 Theorem 5.8: Points of Inflection If ( c, f( c )) is a point of inflection of the graph f, then either f ''( ) 0 or f ''( ) does not eist at c. Practice Problem 3: Determine the points of inflection and discuss the concavity f( ) 4 ( ) 3 1. Check for continuity:. f '( ) 3. f ''( ) 4. Solve for possible points of inflection Intervals Test Values (, ) (, ) (, ) (, ) < < < < < < < < f ''( TV ) Conclusions Identify Points of Inflection Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 13

14 Theorem 5.9: Second Derivative Test Let f be a function such that f '( c ) 0 and the second derivative of f eists on an open interval containing c. 1. f ''( c ) > 0 RMin at ( c, f( c )). f ''( c ) < 0 RMa at ( c, f( c )) 3. f ''( c ) 0 No conclusion, need to use the First Derivative Test Practice Problem 4: Find the relative etrema for f ( ) Using 1 st Derivative Test 1. Determine continuity. f '( ) Using nd Derivative Test Repeat steps 1-3 in the First Derivative Test.. f ''( ) Graph: 3. Find CN s: 3. Test CN s in the nd Derivative 4. Test Table: 4. Conclusions: 5. Conclusions: Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 14

15 LESSON 5.5 LIMITS AT INFINITY Review: How to find horizontal, vertical and slant asymptotes n n 1 a n + an a + a 1 + a0 f( ) m m 1 b m + bm b + b 1 + b0 Vertical Set the denominator equal to zero then solve for Note: Vertical asymptotes are in the form of the zeros Horizontal i. n < m y 0 Note: Horizontal ii. n m an asymptotes are in the form y b of y... m iii. n > m no horizontal asymptotes Slant When n > m by 1 the slant asymptote is the quotient of the rational epression. (If n > m by or more, the function has neither horizontal nor slant asymptotes) Note: Slant asymptotes are in the form y m + b Note: These guidelines can be used to check for LIMITS at INFINITY of RATIONAL functions only. Eamples: Determine the VA, HA and SA for the following functions f( ) f( ) f( ) VA: HA: SA: VA: HA: SA: VA: HA: SA: Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 15

16 Limits at Infinity Everything for larger then M fits inside the given ε window around L. Definition of a Horizontal Asymptote Theorem 5.10: Limits at Infinity The line y L is a horizontal asymptote of the graph f is lim f ( ) L or lim f ( ) L. If r is a positive rational number and c is c any real number, then lim 0. r r Furthermore, if is defined when < 0, c then lim 0. r Remember: Given lim f ( ) L and c lim g ( ) K, then c lim f ( ) ± g( ) L± K c [ ] [ f g] lim ( ) ( ) c LK In other words: cons tant lim 0 or r cons tant lim 0 r Practice Problems: Evaluate the following limits and find their HA lim lim 5 HA: HA: Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 16

17 Practice Problems: Evaluate the following limits and find their HA. 3. lim 9 Note: when you encounter an indeterminate form, the suggested method is to DIVIDE the NUMERATOR and the DENOMINATOR by the HIGHEST POWER of in the DENOMINATOR. Practice Problems: Evaluate the following limits by finding the actual limits, then compare the answers to their HA s using the guidelines lim lim lim Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 17

18 Functions with two horizontal asymptotes and non-rational functions. a. lim Note: Eample 4: Find the following limits b. lim Graph: y Functions with Infinite Limits at Infinity a. lim Note: Eample 5: Find the following limits b. lim Graph: y Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 18

19 LESSON 5.6 A SUMMARY OF CURVE SKETCHING Absolute Etrema How to find Absolute Etrema on ab, [ ] An interval [ ab, ] is given AMa Highest point AMin Lowest point 1. Find the Critical Numbers: Find the 1 st derivative Set f '( ) 0 Solve for. Evaluate: f ( a ) f ( b ) f ( CN ) 3. The least of these values is the minimum. The greatest is the maimum. Eample: Find the ABSOLUTE Etrema of 1 hs () s on [ 0, ) Relative Etrema (hills and valleys) First Derivative Test 1. Find the CN: a, b, c. Set up the intervals using the CN 3. Choose a test value in each interval 4. Find f '( TV ) in each interval (, a) ( ab, ) (,) bc f '( TV ) > 0 Inc f '( TV ) < 0 f '( TV ) < 0 (, c ) f '( TV ) > 0 Dec Dec Inc a, f( a ) RMin( c, f( c )) RMa ( ) Second Derivative test 1. Find the CN: a, b, c. Find the nd derivative 3. Evaluate f ''( CN ' s ) 1. f ''( a ) > 0 RMin at ( a, f( a )). f ''( b ) < 0 RMa at ( b, f( b )) 3. f ''( c ) 0 No conclusion, need to use the First Derivative Test Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 19

20 Points of Inflection and Concavity 1. Find the 1 st derivative. Find the nd derivative 3. Set f ''( ) 0 4. Solve for to obtain POSSIBLE POINTS OF INFLECTION: ( a, f( a )), ( b, f( b )), ( c, f( c )) 5. Set up the intervals using the PP of I (, a) f ''( TV ) > 0 Upward ( ab, ) f ''( TV ) < 0 Downward, ( ) (,) bc f ''( TV ) < 0 Downward (, c ) f ''( TV ) > 0 Upward c, f( c ) P of I at ( a f a ) P of I ( ) Practice Problem: Use charts to show intervals of increasing and decreasing functions, label all relative etrema if applicable, find the intervals of concavity and points of inflection, find all vertical and horizontal asymptotes. Sketch the following graphs. (Find - and y-intercepts if they help you with your graph). Must show all work for full credit. Graphing calculator is just a checking device. 4 3 f ( ) f '( ) CN: Inc: Dec: RMin: RMa: f "( ) Label your grid carefully! Upward: Downward: P of I: -int: VA: HA: Mrs. Nguyen Honors PreCalculus Chapter 5 Notes Page 0