ELEM CALC & ITS APPLICS
ELEM CALC & ITS APPLICS MA 123
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Date Created: 10/23/15
MA123 Chapter 6 Extreme values the Mean Value Theorem Date curve sketching and concavity pp 103123 Chapter Goals 0 Assignment 12 Due date Oct 20 Apply the Extreme Value Theorem to nd the global maximum and minimum values of a continuous function on a closed and bounded interval Understand the connection between critical points and local extreme values Understand the relationship between the sign of the derivative and the intervals on which a function is increasing and on which it is decreasing Understand the statement and consequences of the Mean Value Theorem Understand how the derivative can help you sketch the graph of a function Understand how to use the derivative to nd the global extreme values if any of a continuous function over an unbounded interval Understand the connection between the sign of the second derivative of a function and the concavities of the graph of the function Understand the meaning of in ection points and how to locate them Assignment 13 Due date Oct 27 Assignment 14 Due date Oct 30 Finding the largest pro t7 or the smallest possible cost7 or the shortest possible time for performing a given procedure or task7 or guring out how to perform a task most productively under a given budget and time schedule are some examples of practical real world applications of Calculus The basic mathematical question underlying such applied problems is how to nd if they exist the largest or smallest values of a given function on a given interval This procedure depends on the nature of the interval gt Global or absolute extreme values The largest value a function possibly attains on an interval is called its global or absolute maximum value The smallest value a function possibly attains on an interval is called its global or absolute minimum value Both maximum and minimum values if they exist are called global or absolute extreme values Example 1 Find the maximum and minimum values for the function fxx12 e 3 if they exist Example 2 Find the maximum and minimum values for the function Example 3 Find the maximum and minimum values for the function 1 7 95 7 2i 37 1035 m2 1 35 6 712 if they exist 34 if they exist 24 We rst focus on continuous functions on a closed and bounded interval The question of largest and smallest values of a continuous function f on an interval that is not closed and bounded requires us to pay more attention to the behavior of the graph of f and speci cally to where the graph is rising and where it is falling Closed and bounded intervals An interval is closed and bounded if it has nite length and contains its endpoints For example7 the interval 727 5 is closed and bounded gt The Extreme Value Theorem EVT If a function f is continuous on a closed7 bounded interval 11 then the function f attains a maximum and a minimum value on 11 2 m ifmgt0 Example 4 Let x 2 fjm ifmlt0 Does fz have a maximum and a minimum value on 734 How does this example illustrate the Extreme Value Theorem The following two examples show that the assumptions on f and the interval 11 are essential ingredients in the statement of the Extreme Value Theorem Example 5 1 i i i Let gz 7 Does gz have a maximum value and a minimum value on 727 3 Does this example contradict the Extreme Value Theorem Why or why not Example 6 Let hz 4 7 2x2 1 Does hz have a maximum value and a minimum value on 7125715 Does this example contradict the Extreme Value Theorem Why or why not The EVT is an existence statement it doesn t tell you how to locate the maximum and minimum values of f The following results tell you how to narrow down the list of possible points on the given interval where the function 1 might have an extreme value to usually just a few possibilities You can then evaluate f at these few possibilities and pick out the smallest and largest value gt Fermat7s Theorem Let x be a continuous function on the interval 11 If f has an extreme value at a point c strictly between a and b7 and if f is differentiable at x c then fc O gt Let x be a continuous function on the closed7 bounded interval 11 If f has an extreme value at x c in the interval7 then either 0 c a or c b o altcltbandf c0 o a lt c lt b and f is not differentiable at x c so that f is not de ned at x 0 Example 7 Find the maximum and minimum values of x x3 7 3x2 7 9x 5 on the interval 07 4 For which values x are the maximum and minimum values attained 2 1 Example 8 Find the maximum and minimum values of Fs 87 on the interval 717 5 For 3 7 which values 3 are the maximum and minimum values attained Example 9 Find the maximum and minimum values of x x23 on the interval 717 8 For which values 3 are the maximum and minimum values attained Example 10 Find the 25 values on the interval 71010 where gt lt 7 4 7 takes its maximum and minimum values What are the maximum and minimum values Example 1 1 2 A lt Find the maximum and minimum values of m 2m 1 If m 1 73 7 if m gt 1 0n the interval 723 Example 12 Find the maximum and minimum values of 713 7s 7 252 7 353 7 454 on the interval 07 2 For which values 3 are the maximum and minimum values attained Example 13 Find the maximum and minimum values of gz 1 z 2 3 on the interval 07 2 For which values z are the maximum and minimum values attained gt Local or relative extreme points In addition to the points where a function might have a maximum or minimum value there are other points that are important for the behavior of the function and the shape of its graph global max local max local max local min local max V as global min local min If you thought of the graph of the function as the pro le of a landscape the global maximum could represent the highest hill in the landscape While the minimum could represent the deepest valley The other points indicated in the graph which look like tops of hills although not the highest hills and bottom of valleys although not the deepest valleys are called local or relative extreme values More precisely A function f has a local or relative maximum at a point c fc if there is some interval about 0 such that fc 2 x for all z in that interval A function f has a local or relative minimum at a point c fc if there is some interval about 0 such that fc x for all z in that interval If f has a local extreme value at c fc and is differentiable at that point c then fc 0 Let f be a function If f is de ned at the point z c and either fc 0 or fc is unde ned then the point c is called a critical point of f gt Increasing and decreasing functions A function f is said to be increasing when its graph rises and decreasing when its graph falls More precisely we say that f is increasing on an interval I if fz1 lt zz Whenever 1 lt 2 in I f is decreasing on an interval I if fz1 gt zz Whenever ml lt 2 in I y y f 951 1 HM m 0 mil 2 m f is increasing on the intervals 11 and 0 d f is decreasing on the interval 130 1 is increasing 1 is decreasing 56 Example 14 Find the intervals over which the function in the graph is increasing and decreasing gt The Mean Value Theorem MVT If f is continuous on 11 and differentiable at every point strictly between a and b7 then there exists some point z c and maybe more than one strictly between a and I such that N 7 1 7 b 7 a 7 f c Geometric interpretation of the MVT For some non necessarily unique point 0 between a and b the tangent line to the graph of f at Pc7 fc has the same slope as the secant line connecting the points 1417 fa and 1317 fb on the graph of f A rewording 0f the MVT If f is continuous on 11 and differentiable at every point strictly between a and b7 then there exists some point z c and maybe more than one strictly between a and I such that the average rate of change the instantaneous rate of change offon 11 offatmc Example 16 in Chapter 2 is an illustration of the MVT Two additional examples are proposed next Example 15 Let Qt 252 Find a value A 7 1 such that the average rate of change of Qt from 1 to A equals the instantaneous rate of change of Qt at t 2 Example 16 Let x x 7 x3 Verify that the function satis es the hypotheses of the Mean Value Theorem on the interval 72 0 Then nd all numbers 0 that satisfy the conclusion of the Mean value Theorem Here are three consequences of the Mean Value Theorem If f is differentiable on an interval I and the derivative f x 0 for all x E I then f is constant on I Corollary If f and g are differentiable on an interval I and f x g x for all x E I then f 7 g is constant on I that is x gx c where c is a constant Theorem First derivative test for increasing and decreasing functions If f is differentiable on an interval I and f x gt 0 for all points x E I then f is increasing on I If f is differentiable on an interval I and f x lt 0 for all points x E I then f is decreasing on I 4 Example 17 Let x m i 7 Find the intervals over which the function is increasing 1 Example 18 Find the smallest value of A such that the function t t4 7 10t2 9 is increasing for all t in the interval A 0 7 1 is increasing for all 7 8 7 94 Example 19 Find the largest value of A such that the function 715 3 in the interval 7007 A First derivative test for local maxima and minima If f has a critical value at z c then 0 f has a local maximum at z c if the sign of 1quot around 0 is o f has a local minimum at z c if the sign of 1quot around 0 is Example 20 Find the local and global extrema7 if any7 of x mze m for foo lt z lt 00 Find the largest value of A such that the function Example 21 Suppose k t t 7 5t 1t 7 3 kt is decreasing for all t in the interval 37 A Example 22 Suppose u m m2 1m 7 3m 7 1m 5 Find the z value in the interval 753 where takes its maximum value Example 23 Suppose g m 1 2 m4 Find the x values in the interval 737 4 where gz takes its minimum Example 24 xm 7 3 Suppose gz Find the value of z in the interval 37 00 where gz takes its maximum 13 gt Curve sketching Information on the rst derivative can be used to help us sketch the graph of a function For example7 the rst derivative can be used to determine where a function is increasing and where it is decreasing Example 25 Find the intervals where the function x x3 7 3x2 1 is increasing and the ones where it is decreasing Use this information to sketch the graph of x x3 7 3x2 1 gt The second derivative can also be used to help sketch the graph of a function We will see that the second derivative can be used to determine when the graph of a function is concave upward or concave downward The graph of a function y x is concave upward on an interval 071 if the graph lies above each of the tangent lines at every point in the interval 11 The graph of a function y x is concave downward on an interval 071 if the graph lies below each of the tangent lines at every point in the interval 11 y y a b m a b m graph of function concave upward on 071 graph of function concave downward on 11 Second derivative test for concavity Consider a function lf f x gt 0 over an interval 11 then the derivative f x is increasing on the interval 11 That means the slopes of the tangent lines to the graph of y x are increasing on the interval 11 From this it can be seen that the graph of the function y x is concave upward lf f x lt 0 over an interval 11 Then the derivative f x is decreasing on the interval 11 That means the slopes of the tangent lines to the graph of y x are decreasing on the interval 11 From this it can be seen that the graph of the function y x is concave downward 61 A point c c on the graph is called a point of in ection if the graph of y x changes concavity at x c That is if the graph goes from concave up to concave down or from concave down to concave up lf 0 c is a point of in ection on the graph of y x and if the second derivative is de ned at this point then f c 0 Thus points of in ection on the graph of y x are found where either f x 0 or the second derivative is not de ned However if either f x 0 or the second derivative is not de ned at a point it is not necessarily the case that the point is a point of in ection Care must be taken In this course however points of in ection will tend to occur precisely at those points where f x 0 Example 26 Find the intervals over which the function x x4 7 6x3 12x2 3x 7 1 is concave upward and the ones over which it is concave downward Locate the x coordinate of the in ection points of 1 Example 27 If the derivative of the function gx is given by g x 4x2 12x 15 determine the intervals where gx is concave upward and the ones where it is concave downward Find the x coordinate of the in ection points Example 28 Find the x coordinate of the in ection points of the function gx 5amp2 Example 29 Let hz m 3 lnm 7 5 for z gt 5 Find the interval over which hz is concave upward Example 30 Let hz ze m Find the interval over which hz is concave downward Example 31 Find the coordinates of the in ection point of the function gz zxzz 3 4 Example 32 The graph of the derivative 1quot of a function f is shown 7 a On what intervals is 1 increasing or decreasing b At what values of z does 1 have a local maximum or minimum c On what intervals is f concave upward or downward 1 State the z coordinate of the in ection points of f
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