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# Class Note for MATH 1314 with Professor Heeth at UH 2

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Date Created: 02/06/15

Math 1314 Lesson 11 Applications ofthe Second Derivative Concavity Earlier in the course we saw that the second derivative is the rate of change of the rst derivative The second derivative can tell us if the rate of change of the function is increasing or decreasing In business for example the first derivative might tell us that our profits are increasing but the second derivative will tell us if the pace of the increase is increasing or decreasing Example 1 y From these graphs you can see that the shape of the curve change differs depending on Whether the slopes of tangent lines are increasing or decreasing This is the idea of concavity De nition Let the functionfbe differentiable on an interval a b Thenfis concave upward on a b if f39 is increasing on a b andfis concave downward on a b if f39 is decreasing on a b Determining Where a Function is Concave Upward and Where it is Concave Downward By Analyzing the Sign ofthe Second Derivative Algebraically We can also determine concavity algebraically The procedure for doing this should look pretty familiar Find the second derivative of the function Determine all values ofx for Whichf39x 0 or is undefined Use the values found in step 2 to divide the number line into open intervals Choose a test value c in each open interval Substitute each test value 0 into the second derivative to determine the sign of f quot0 Apply the following theorem QMerJNf Theorem If f 39x gt 0 for each value of x in a b then f is concave upward on a b If f 39x lt 0 for each value of x in a b then f is concave downward on a b Example 1 Determine where the function fx x3 7 3x2 7 24x 32 is concave upward and where it is concave downward z Example 2 Determine where the function fx xe x IS concave upward and where it IS concave downward Example 3 Determine where the function x x4 6x3 is concave upward and where it is concave downward Inl lec on Points The point where a function changes from being concave upward to concave downward or from being concave downward to concave upward is called a point ol inl lec on or an inflection point We ll show the signi cance ofthis point by an example Example 4 5000 4000 3000 2000 i000 lO 20 0 40 50 00 70 20 90 l Note that at the beginning the slopes of the lines tangent to the graph are increasing slowly Then the slopes of the tangent lines increase rapidly But after a certain point the rate of increase slows down Suppose you are in business and this is a graph of your total sales where x represents the amount spent in millions of dollars on advertising and y represents total receipts in millions of dollars When your company begins an advertising campaign the rate of increase in sales speeds up But this can t go on forever There comes a point where the advertising campaign won t have as much of an effect on sales This is the point of in ection You may think of it as the point of diminishing returns In business it might be the time to start a new ad campaign De nition A point on the graph for a differentiable function f at which the concavity changes is called an in ection point Finding In ection Points To find the in ection points of a function 1 Compute f 39x 2 Find all points in the domain of f for which f 39x 0 3 Determine the sign of f 39x to the left and to the right of each point x 0 found in step 2 If there is a change in the sign of f quotx as we move across the point x c from left to right then the point c c is an in ection point off Note To find the ycoordinate of an in ection point substitute the x coordinate into the original function Example 5 Determine any points of in ection if fx x4 7 2x3 6 Example 6 Determine any points of in ection if fx xezx Example 7 Determine any points of in ection if x x4 7 5x3 2x 3 The Second Derivative Test Recall that we use the rst derivative test to determine if a critical point is a relative extremum There is also a second derivative test to nd relative extrema It is sometimes convenient to use however it can be inconclusive The Second Derivative Test 1 Find f x and f 39x 2 Find all critical points 3 Compute f 39c for each critical point c a If f 39c gt 0 then f has a relative minimum at c b If f 39c lt 0 then f has a relative maximum at c c If f 39c 0 then the test fails It is inconclusive Try the First Derivative Test Note To nd the ycoordinate of a relative maximum or relative minimum substitute the value you found for x into the original function Example 8 Find any relative extrema using the Second Derivative Test if fx x3 2x2 5x 10 For the next example we ll need to review factoring by grouping Factor x3 3x2 7 16x 7 48 using FBG Example 9 Find any relative extrema using the Second Derivative Test if fxx4 x3 2x2 8x4 From this lesson you should be able to Explain what we mean by concavity and in ection point Find intervals where a function is concave upward concave downward for a polynomial exponential or logarithmic function Find in ection points for a polynomial exponential or logarithmic function Use the second derivative test to determine relative extrema of a function

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