Module 3 Graphing and Optimization

Transcription

1 Module 3 Graphing and Optimization One of the most important applications of calculus to real-world problems is in the area of optimization. We will utilize the knowledge gained in the previous chapter, in conjunction with the upcoming new material to learn how to analze, discuss and sketch the graphs of various functions. (These functions ma be specified b equations, tables, verbal descriptions or other graphs.) Finall, in the last section, we will use these graphing tools to solve optimization problems. First Derivative and Graphs Learning Objectives: Find intervals on which a function is increasing, decreasing, and/or constant Define and find local etrema Create and/or use sign charts for graphs of functions Use the first derivative as needed to sketch graphs of functions Solve applications involving the graph of a function s first derivative The derivative tells us a great deal about the graph of a function. In this section, we will learn how to use this knowledge to determine the intervals where a graph rises and falls. This will lead to methods that will let us determine maimum and minimum values for functions without first obtaining the graph of a function. This is a method b which companies can determine production levels that will minimize cost or maimize profit. Where it s Used Business: Profit Analsis The graph in the figure approimates the total profit, P in dollars, from the sale of laser-guided levels. Write a brief description of the graph of the marginal profit function P (). Include a discussion of an -intercepts. R() Page 1 of 11

2 R. 1 Algebra skills review Factor out the greatest common factor and simplif. A) t 7t 1 t 7t 8 B) 54 b 2 4 b 3 74 b R.2 For the graph given below, find the intervals where the graph is rising (function is increasing), graph is falling (function is decreasing), and where the graph is constant (function is constant). Let s recall: 1) We read graphs from left to right; 2) We identif intervals b the values on the horizontal ais; and 3) The customar notation for naming intervals is interval notation. We use ( or ) if the endpoint is NOT included and [ or ] if the endpoint IS included. A) The function is increasing on the intervals and. B) The function is decreasing on the intervals and. C) If we drew tangent lines to an point of the graph where the function is increasing, the slope of each tangent line is. D) If we drew tangent lines to an point of the graph where the function is decreasing, the slope of each tangent line is. E) There are three (3) turning points on this graph. The tangent line at each of the turning points is and the slope of the tangent line at each of the points is. F) What is the relationship between intervals where the function is increasing and the slope of the tangent line to the graph on that interval? Page 2 of 11

3 Increasing and Decreasing Intervals of Functions 3 E. 1 Let s eamine the graph of f ( ) 3 and a sign chart for f (). Discuss the relationship between the graph of f and the sign chart of f over each interval where the sign of f () is constant. A) Graph of the function is given below. B) Make a sign chart for f. C) Discuss relationship. Increasing and Decreasing Intervals of Functions For the open interval a, b: When the Sign of f is the function f () and the Graph of f Eamples Increases Decreases Rises Falls Page 3 of 11

4 2 E. 2 Consider the function: f ( ) A) Which values of correspond to horizontal tangent lines? B) Determine the intervals of for which f () is increasing and f () is decreasing. C) The graph of f is provided. Add an horizontal tangent lines. Page 4 of 11

5 Critical Values of f We know that if f ( c) 0, the graph of f () will have a horizontal tangent line at c. However, the partition numbers for f also include the numbers c where f (c) does not eist. The values of in the domain of f where f ( ) 0 or f () does not eist are the critical values of f. The critical values of f are alwas partition numbers for f, but f ma have partition numbers that are not critical values. E. 3 For each function, find the partition numbers for f, the critical values for f, and determine the intervals where f is increasing and those where f is decreasing. A) f ( ) B) f ( ) (1 ) Page 5 of 11

6 C) f ( ) 1 E.3C illustrates two important ideas: 1) Do not assume that all partition numbers for the derivative f are critical values of the function f. To be a critical value, a partition number must also be in the domain of f. 2) The values where a function is increasing or decreasing must alwas be epressed in terms of open intervals that are subsets of the domain of the function. Local/Relative Etrema Theorem: Eistence of Local/Relative Etrema If f is continuous on the interval a, b, k is a number in a, b and f (k) is a local etremum, then either f ( k) 0 or f (k) does not eist (is not defined). E. 4 Let s look at the graph from E. 1 and identif the local/relative etrema. The local/relative etrema are the turning points on this graph. Page 6 of 11

7 First Derivative Test for Local/Relative Etrema If f () eists on both sides of a critical value, k, the sign of f () can be used to determine whether the point 1) If f k 0 k, f k is a relative maimum, a relative minimum, or neither., then a horizontal tangent eists at kand kis either a turning point or a resting point. 2) If f k is not defined, but f k is defined, then First-Derivative Test for Local/Relative Etrema Let k be a critical value of f f ( k) is definedand either f ( k) 0 or f ( k) isnot defined. Construct a sign chart for f () on both sides of k. Sign Chart f ( k ) f () f (k) is a local minimum. If f () changes from negative to positive at k, f () then f (k) is a local minimum. f () f (k) is a local maimum. If f () changes from positive to negative at k, f () then f (k) is a local maimum. Page 7 of 11

8 3 2 E. 5 Given f ( ) : A) Find the critical values of f. B) Find the local/relative etrema. C) Mark all etrema for the provided graph of f. Page 8 of 11

9 Theorem: Intercepts and Local/Relative Etrema for Polnomial Functions n n1 th If f ( ) an an a1 a0, a n 0, is an n -degree polnomial, then f has no more than n -intercepts and no more than n 1 relative etrema. Recall the following propert from college algebra. Notice a ver strong similarit between it and the theorem stated above. The graph of a polnomial function of positive degree n can have most n 1 turning points and at most n real zeros (-intercepts). E. 6 Suppose (),. Use the given information to sketch the graph of f. A) f f is continuous on f () B) f ( 3) 1, f (0) 1, f (2) 3; f ( 3) 0, f (2) 0; f () 0 on, 3, 3, 2,and 2, Page 9 of 11

10 E. 7 Use the graph of () f shown below to sketch a possible graph of f. E. 8 Use the graph of f shown below to sketch a possible graph for f. Page 10 of 11

11 ANALYZING GRAPHS E. 9 Profit Analsis. The graph in the figure approimates the total profit, the sale of laser-guided levels. R() P in dollars, from A) Write a brief description of the graph of the marginal profit function P (). Include a discussion of an -intercepts. B) Sketch a possible graph of P (). Eplain wh the graph of the derivative looks the wa it does. Page 11 of 11