MATH 149 LABORATORY ASSIGNMENT 3 DERIVATIVES AND PROPERTIES OF GRAPHS The important characteristics of the graph of a function f x can be established by studying its first and second derivatives These characteristics include the location of any local maxima local minima and points of inflection and intervals in which the graph is increasing or decreasing or is concave upward or concave downward As an example we will study the function f x x2 3sin 2x x restart f x x 2 3 sin 2 x f x x2 3 sin 2 x First we plot the graph of the function plot f Pi Pi There appear to be two local minima a local maximum close to x 1 and possibly another local maximum near x 3 At points where there is a local maximum or minimum the derivative is 0 We next compute the derivative and draw the graphs of f x and f x on the same set of axes D f x 2 x 6 cos 2 x plot f D f Pi Pi color red green To find the exact locations of the local maxima and minima we solve the equation f x 0 solve D f x 0 x 1 RootOf Z 6 cos Z 2 Apparently MAPLE does not know a general solution so we will find the solutions using the fsolve command X 1 fsolve D f x 0 x 1 0 X 2 fsolve D f x 0 x 0 1 X 3 fsolve D f x 0 x 1 2 X1 0 6723755227 X2 0 9457599482 X3 1 992913103 To see if there is another solution near 3 we zoom in on the graph of f x plot D f Pi 2 8 1 1 color blue Since the graph of f x does not touch the X axis there is not an additional solution Note that from the graph of f x f x is negative in the intervals X1 and X2 X3 and thus f x is decreasing on these intervals f x is positive in X1 X2 and X3 and therefore f x is increasing there We now apply the First Derivative Test Since f x is negative to the left of X1 and positive to the right there is a local minimum at X1 similarly there is a local minimum at X3 Since f x is positive to the left of X2 and negative to the right there is a local maximum at X2 The second derivative is used to find intervals of concavity and points of inflection We will compute f x and plot it and f x on the same set of axes D D f x 2 12 sin 2 x plot f D D f Pi Pi color red blue Points of inflection occur at points where f x 0 and the second derivative changes sign The graph is concave down on intervals where f x 0 and concave upward when f x 0 thus the points of inflection are the points where the concavity changes Here there are four such points Z 1 fsolve D D f x 0 x Pi 3 Z 2 fsolve D D f x 0 x 2 1 Z 3 fsolve D D f x 0 x 0 1 Z 4 fsolve D D f x 0 x 1 2 Z1 3 057868614 Z2 1 654520366 Z3 0 08372403961 Z4 1 487072287 The graph of f x is concave up for x in Z1 Z2 Z3 and Z4 It is concave down in Z1 Z2 and Z3 Z4
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