NOTES 4: APPLICATIONS OF DIFFERENTIATION Name: Date: Period: Mrs. Nguyen s Initial: WARM UP: Assume that f ( ) and g ( ) are differentiable functions: f ( ) f '( ) g ( ) g'( ) - 3 1-5 8-1 -9 7 4 1 0 5 9 9-3 1 3-3 6-5 3 8? 1. Let h ( ) e f( ). What is h '(0)?. Let j( ) 4 f( ) g( ). What is j '(1)? 3. Let f( ) k ( ). What is k '( )? g ( ) 4. Let 3 L ( ) g ( ). If L '() 48, find g '()? Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page 1
LESSON 4.1 MAXIMUM AND MINIMUM VALUES Definition of Absolute Etrema Let f be defined on an interval I containing c. 1. () f c is the Absolute Minimum of f on I if f () c f() in I f c is the Absolute Maimum of f on I if f () c f () in I. () Definition of Local (Relative) Etrema The Etreme Value Theorem - Etreme values or Etrema are the minimum and maimum of a function on an interval - On an interval: Maimum ABSOLUTE ma Mininum ABSOLUTE min 1. A function f has a Local Maimum at c if f () c f() when is near c.. A function f has a Local Maimum at c if f () c f() when is near c. If f is continuous on a closed interval ab,, then f has both a minimum and a maimum on the interval. 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 the graph over ab,. f c to be an f c must eist Note: In order for () etreme value, () (or must be defined). Note: Local (Relative) etrema are hills or valleys of the graph. Eamples: Sketch the graph of a function f that is continuous on 1, 5 and has the given properties. 1. AMin at 3, Ama at 3, LMin at 4. Ama at 5, Amin at, LMa at 3, LMin at & 4. Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page
Fermat s Theorem If f has a local maimum or minimum at c, and if f '( c ) eists, then f '( c) 0. Definition of a Critical Number A critical number of a function f is a number c in the domain of f such that either f '( c) 0 or f '( c ) does not eist. Caution: y f but 3 has '(0) 0 0, 0 is not a local etremum. f '( c) 0 refers to the points with horizontal tangent lines. The converse of Fermat s is false. If f has a local maimum or minimum at c, then c is a critical number of f. Guidelines for Finding ABSOLUTE EXTREMA on a Closed Interval To find the etrema of a continuous function f on a closed interval ab,, use the following steps. 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. Practice Problems: 4 3 1. f ( ) 3 4. Find the absolute maimum and absolute minimum of f ( ) on 1,. for 0. Find the absolute maimum of f ( ) on 0, 4.. Let f ( ) 1/ 1 3 3. Given that f ( ) a b has critical numbers at 1 and 3. Find a and b (a and b are constants). 4. Find the AMin, AMa, LMin, LMa for 4 3 f ( ) 3 16 18 1, 4. on Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page 3
5. Find the critical numbers for f ( ) 3/5 4. 6. 1, 0 g ( ). Graph g ( ) on 1,., 0 7. Find the ABSOLUTE Etrema of 1, 4. 3 g ( ) on 1 4 8. Find the Critical Numbers of f( ) 1. 9. Given f( ) 4, find f 1 '(1). 10. Given f, find 3 ( ) 1 f 1 '(3). Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page 4
LESSON 4. THE MEAN VALUE THEOREM Rolle s Theorem Let f be continuous on the closed interval and differentiable on the interval, a b. If f ( a) f ( b) a, b such that f '( c) 0. c in ab,, then there is at least one number The Mean Value Theorem Let f be continuous on the closed interval and differentiable on the interval, there eists a number c in, f ( b) f( a) f '( c). b a ab, a b, then a b such that f( b) f( a) y y1 Note: ba 1 Practice Problems: 1. f ( ) 4. Find all values of c in, Rolle s Theorem. by Graph: y 4. f( ) 5 Mean Value Theorem.. Find all values of c in 1, 4 by the Graph: y Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page 5
LESSON 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF GRAPH Definition of Increasing and Decreasing Functions A function f is increasing on an interval if for any two numbers 1 and in the interval, f( ) f( ) 1 1 and decreasing when f( ) f( ) 1 1 Decreasing:, a Negative derivative: f '( ) 0 Constant: a, b Zero derivative: f '( ) 0 Increasing: b, Positive derivative: f '( ) 0 Test for Increasing and Decreasing Functions 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,. f '( ) 0 in a, b f is decreasing on, 3. f '( ) 0 in a, b f is constant on ab,. ab. ab. The First Derivative Test (to find Local Etrema, not Absolute Etrema) 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 '( ) c, then f () c is a RELATIVE MINIMUM of f. changes from negative to positive at. If f '( ) c, then f () c is a RELATIVE MAXIMUM of f. changes from positive to negative at 3. If f '( ) does not change sign at c, then f () c is neither a relative minimum nor relative maimum. Relative Minimum Relative Maimum Neither a ma nor a min Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page 6
Practice Problems: Find the relative etrema of the following functions 1. f ( ) 4 4 3 Intervals Test Values,,,, f '( TV ) Conclusions Identify Relative Etrema. f( ) 9 Intervals Test Values,,,, f '( TV ) Conclusions Identify Relative Etrema Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page 7
Definition of Concavity Test for Concavity Definition of Point of Inflection Points of Inflection Second Derivative Test 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. A point where the concavity of f changes. If 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. c, f( c ) is a point of inflection of the graph f, then either f ''( ) 0 or f ''( ) does not eist at c. 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 3: Find the relative etrema for f ( ) 3 5 5 3 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. Conclusions: Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page 8
Practice Problem 4: Determine the points of inflection and discuss the concavity 3 f( ) 4 1. Check for continuity:. f '( ) 3. f ''( ) 4. Solve for possible points of inflection Intervals Test Values,,,, f ''( TV ) Conclusions Identify Points of Inflection Practice Problem 5: Find and classify Local Etrema and Points of Inflection for 3 f ( ) 3. Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page 9
WARM-UP: 1. Find and classify Local Etrema and Points of Inflection for f( ).. Find the intervals of increasing and decreasing, label all relative etrema if applicable, find the intervals of concavity and points of inflection: 5 f ( ) 4 Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page 10
LESSON 4. 5 SUMMARY OF CURVE SKETCHING Absolute Etrema An interval ab, is given AMa Highest point AMin Lowest point How to find Absolute Etrema on ab, 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 hs () 1 s on 0, Relative Etrema (hills and valleys) Points of Inflection and Concavity 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) f '( TV) 0 Inc RMa ( ab, ) f '( TV) 0 Dec (,) bc f '( TV) 0 Dec (, c ) f '( TV) 0 Inc a, f( a ) RMinc, f( c ) 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 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 (, a) ( ab, ) (,) bc (, c ) f ''( TV ) 0 f ''( TV ) 0 f ''( TV ) 0 f ''( TV ) 0 Upward P of I at Downward Downward Upward a, f( a ) P of I c, f( c ) Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page 11
1. Find the intervals of increasing and decreasing, label all relative etrema if applicable, find the intervals of concavity and points of inflection: f ( ) cos. Find the intervals of increasing and decreasing, label all relative etrema if applicable, find the intervals of 4 3 concavity and points of inflection: f ( ) 1 48 64. Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page 1
LESSON 4.7 OPTIMIZATION PROBLEMS Steps in Solving Optimization Problems First Derivative Test for Absolute etreme values 1. Understand the problem. Draw a diagram 3. Introduce notation: for instance Q as the quantity to be maimized or minimized 4. Write Q in terms of other symbols 5. Find relationships among symbols. Eliminate all variables but one: Q f( ) 6. Find absolute ma or min of f ( ) Suppose that c is a critical number of a continuous function f defined on an interval. a. If f '( ) 0 for all c and f '( ) 0 for all c, the f ( c ) is the absolute maimum value of f. b. If f '( ) 0 for all c and f '( ) 0 for all c, the f ( c ) is the absolute minimum value of f. Practice Problems: 1. Find the point on the parabola closest to 1,1. y 1 that is. Find the dimensions of the largest rectangle that can be inscribed between y 10 and the -ais. 3. An 8.5" 11" sheet of paper is going to be transformed into a bo with no top. We will cut squares out of each corner so that the sides of the bo can be formed Find out how big each square should be to make the largest bo. 4. Let s build a bo. The bo is to have a square base 3 and can contain 51 in of stuff. Find the dimensions of the bo made from the least amount of material (i.e.: smallest surface area). Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page 13
5. Now, we want to build a different bo, this time out of different materials. The base is still square, but now it has a top. The bo is made of steal, costing $1 / ft. The sides are pleiglass, at $8/ ft. The top is made from wood, costing $ / ft. The bo needs to contain a freshman and lots of boiling water, 3 so it needs to hold 36 ft. Find the cost of the cheapest bo. 6. Mr. Shay (of course, it s Mr. Shay and not Mrs. Nguyen) sells the hides of freshmen. Last month, he sold 15 skins at $600 / skin. Two month ago, he sold 18 skins at $400 / skin. a. Find the demand function (assuming it s linear). b. Find the revenue function. c. It costs Mr. Shay $00 to make each hide, what is the cost function? d. What is the profit equation? e. How much should each skin be to maimize the profit? Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page 14
WARM UP: 1. Find the local and absolute etreme values of 3 f ( ) on,1.. We can sell 10 widgets at $7 / widget. If the price is lowered by $, we can sell 3 more widgets. The cost to make widget is a flat fee of $3 and $1/ widget. How much should each widget cost to make the most money? 3. Sketch the given: - f '( ) f '(1) 1 - f '(0) is undefined f '( ) 0 on, & 0,1 - f '( ) 0 on,10 & 1, - f ''( ) 0 on, 3 - f ''( ) 0 on3, 0 & 0, - f( ) 4, f(1) 1, f(0) 0-3 4. A closed bo is to contain 50 in. The bo is to have a square base. The materials for the base (top & bottom) cost $ / in, while the sides cost $3/ in. Find the dimensions of the most affordable bo. Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page 15
LESSON 4.9 ANTIDERIVATIVES Table of Antidifferentiation Formulas cf ( ) d cf( ) C f ( ) g( ) d F( ) G( ) C n1 n d Cn, 1 n 1 1 d ln C ede C cos d sin C sin d cos C sec d tan C sec tan d sec C 1 1 d sin C 1 1 1 d tan C 1 Practice Problems: Find the antiderivative of the following problems. 1. f '( ) 5 7 sin. f '( ) e sec 3 3. f( ). Find F ( ), the antiderivative. 4. f '( ) 1 1, 1 f 1 5. f 3 ''( ) 0 1 4, f 0 8, f '1 5 6. An object is thrown upward form a height of 6 feet. The velocity after 1 second is 8 ft/sec. Use a(t) = 3. (a) How high will the ball go? (b) When will it hit the ground? Mrs. Nguyen AP Calculus AB Chapter 4 Notes Page 16