MA 131 Lecture Notes Chapter 4 Calculus by Stewart

Transcription

1 MA 131 Lecture Notes Chapter 4 Calculus by Stewart 4.1) Maimum and Minimum Values 4.3) How Derivatives Affect the Shape of a Graph A function is increasing if its graph moves up as moves to the right and is decreasing if its graph moves down as moves to the right. The formal definition is as follows: Definition of Increasing and Decreasing Functions A function f is increasing on an interval if for any tow numbers 1 and in the interval, 1 implies f ) f ( ). ( 1 A function f is decreasing on an interval if for any tow numbers 1 and in the interval, 1 implies f ) f ( ). ( 1 We use the derivative of a function to determine where a function is increasing or decreasing. The following information can be used for such a purpose. Test for Increasing and Decreasing Functions Let f be differentiable on the interval (a,b) 1. If f '( ) 0 for all in (a,b), then f is increasing on (a,b).. If f '( ) 0for all in (a,b), then f is decreasing on (a,b). 3. If f '( ) 0 for all in (a,b), then f is constant on (a,b). Draw a picture to see if you can see why this is true. Discuss why we only use open intervals to discuss the properties of increasing or decreasing. Prove that is decreasing on,0) ( and increasing on ( 0, ).

2 Definition of Critical Number If f is defined at c, then c is a critical number of f if f '( c) 0 or if f (c) does not eist (is undefined.) Find all critical numbers of the following functions: f 3 ( ) 1 3 Guidelines for Applying Increasing/Decreasing Test 1. Find the derivative of f.. Locate the critical numbers of f and use these numbers to determine test intervals. Tat is, find all for which f ()=0 or f () is undefined. 3. Test the sign of f () at an arbitrary number in each of the test intervals. 4. Use the test for increasing and decreasing functions to decide whether f() is increasing or decreasing on each interval.

3 Eample: Find the open intervals on which the function is increasing or decreasing. Test Value Sign of f () 1 Test Value Sign of f () 4 Test Value Sign of f ()

4 Be mindful of any discontinuities in f() these isolated values also need to partition your interval when testing for increasing and decreasing. Etrema and the First-Derivative Test Consider a function that is continuous and increasing then at a certain point when =c the function begins to decrease. At this point, the function has achieved a maimum value on that immediate interval. What happens when a continuous function is decreasing and then changes to increasing? This is eactly the focus of this section. When a function changes from increasing to decreasing or vice versa, we call this point a relative etremum (the plural of etremum is etrema.) The relative etrema of a function include the relative minima and relative maima of the function. Definition of Relative Etrema Let f be a function defined at c. 1. f(c) is a relative maimum of f if there eists an interval (a,b) containing c such that f ( c) for all in (a,b).. f(c) is a relative minimum of f if there eists an interval (a,b) containing c such that f ( c) for all in (a,b). Occurrence of Relative Etrema If f has a relative maimum or relative minimum when =c, then c is a critical number of f. That is f (c)=0 or f (c) does not eist. We can use the first derivative test to classify any critical numbers as relative maima or relative minima. First-Derivative Test for Relative Etrema Let f be continuous on the interval (a,b) in which c is the only critical number. If f is differentiable on the interval (ecept possibly at c), then f(c) can be classified as a relative minimum, a relative maimum, or neither, as shown. 1. On the interval (a,b), if f () is negative to the left of =c and positive to the right of =c, then f(c) is a relative minimum. (decreasing then increasing). On the interval (a,b), if f () is positive to the left of =c and negative to the right of =c, then f(c) is a relative maimum. (increasing then decreasing) 3. On the interval (a,b), if f () is negative on both sides of =c, or if f () is positive on both sides of =c, then f(c) is neither a relative minimum or a relative maimum.

5 Eample: Find all relative etremum of f Test value Sign of f () Find the relative etremum of f. f 4 3 ( ) 4 Find the relative etremum of f. 3 3 We use the terms relative etremum to describe the local behavior of a function. To describe the global behavior of the function on an entire interval we can use the terms absolute maimum and absolute minimum.

6 Definition of Absolute Etrema Let f be defined on an interval I containing c. 1. f(c) is an absolute minimum of f on I if f ( c) for every in I.. f(c) is an absolute maimum of f on I if f ( c) for every in I. The absolute minimum and absolute maimum values of a function on an interval are sometimes simply called the minimum and maimum of f on I. Do all functions have absolute maimums and absolute minimums? Under what conditions can we guarantee them? The following theorem answers these questions. Can you eplain why? Etreme Value Theorem If f is continuous on [a,b], then f has both a minimum value and a maimum value on [a,b]. Note it is possible that an etremum (the y-value) can occur at more than one value. Guidelines for Finding Etrema on a Closed To find the etrema of a continuous function f on a closed interval [a,b], use the steps below. 1. Evaluate f at each of its critical numbers in (a,b).. Evaluate f at each endpoint, a and b. 3. The least of these values is the minimum, and the greatest is the maimum. Eample: Find the minimum and maimum values of 5 on [-1,3]. (Note the chart below is for organization purposes and may have more rows than necessary.) Endpoint Endpoint Critical Number Critical Number Function value

7 Find the minimum and maimum values of on the interval [3,5]. Find the absolute maimum and minimum values of ( ) on [ ]. Eample: Coughing forces the trachea (windpipe) to contract, which in turn affects the velocity of the air through the trachea. The velocity of the air during the coughing can be modeled by v k( R r) r, 0 r R, where k is a constant, R is the normal radius of the trachea, and r is the radius during coughing. What radius r will produce the maimum air velocity? Net we begin our discussion about concavity of a graph, that is, the curving upward or curving downward property of a function. Begin by drawing the function ( ) and ( ). Discuss the placement of the tangent line and the second derivative. Definition of Concavity Let f be differentiable on an open interval I. The graph of f is 1. concave upward on I if f is increasing on the interval.. concave downward on I if f is decreasing on the interval.

8 Note: 1. A curve that is concave upward lies above its tangent line.. A curve that is concave downward lies below its tangent line. Test for Concavity Let f be a function whose second derivative eists on an open interval I. 1. If f ()>0 for all in I, then f is concave upward on I.. If f ()<0 for all in I, then f is concave downward on I. Guidelines for Applying Concavity Test 0. Always check domain and any discontinuities. 1. Locate the -values at which f ()=0 and f () does not eist.. Use these -values to determine test intervals. 3. Test the sign of f () in each test interval. Eample: Determine the open interval(s) on which the graph of the function is concave upward or concave downward. 6 3 Test Value Sign of f Eample: Determine the open intervals on which the graph of the function is concave upward or concave downward. f ) 3 ( Definition: Points of Inflection If the graph of a continuous function has a tangent line at a point where its concavity changes from upward to downward (or downward to upward), then the point is a point of inflection. Discuss inflection points of.

9 Property of Points of Inflection If (c,f(c)) is a point of inflection of the graph of f, then either f (c)=0 or f (c) is undefined. Eample: Discuss the concavity of the graph of f and find its points of inflection We can use the second derivative to find the relative maimum and relative minimum of a function. second derivative guide. Second Derivative Test (for finding relative maimums and minimums) Let f (c)=0, and let f eist on an open interval containing c. 1. Let f (c)>0, then f(c) is a relative minimum. Let f (c)<0, then f(c) is a relative maimum 3. If f (c)=0, then the test fails. In such cases, you can use the First Derivative Test to determine whether f(c) is a relative minimum/maimum or neither. Justify the results of the above test. Here is a chart to help you organize the above results. First Derivative Second Derivative f'(c)=0 f' (c)>0 c is a relative minimum f'(c)=0 f' (c)<0 c is a relative maimum f'(c)=0 f' (c)=0 Test is inconclusive Eample: Find the relative etrema of ( ).