Math 149 Lab 2 Using derivatives to find properties of functions Introduction Differentiation is a process that for a broad class of functions may be performed using only a small set of rules However even for relatively simple functions such as those in the examples and exercises that follow the results may become rather complicated and unwieldy As such differentiation of elementary functions is very well suited to being performed by computer software This laboratory will explore how to compute derivatives and use them to find global absolute maxima and minima local maxima and minima intervals of increase or decrease intervals of concavity and points of inflection Computing derivatives The function f x x2 Sec x may be defined with the assignment operator as follows In 43 f x x 2 Sec x To see the results of the definition we may compute the value of the function by simplying typing the name of the function and enclosing the argument in square brackets In 44 f x Out 44 x2 3 Sin 2 x The first and second derivatives may be computed using the D function In 45 D f x x Out 45 2 x 6 Cos 2 x In 46 D D f x x x Out 46 2 12 Sin 2 x To find the n th derivative we use the D f x n form of the command For example the fourth derivative may be computed as follows In 47 D f x x 4 Out 47 48 Sin 2 x This is much more pleasant than repeatedly applying the differentiation rules for our function by hand four times 2 Using derivatives to find properties of functions nb Finding absolute maxima and minima For a continuous function defined on a closed interval from the Extreme Value Theorem we know that the function assumes its absolute maximum and absolute minimum values somewhere within the interval Moreover thanks to Fermat s Theorem we know that we only need to consider the values of the function at the endpoints and and at the critical numbers where the derivative is zero or does not exist As an example consider the function f x x Cos x2 Sin x on the interval 0 p In 48 Clear f In 49 f x Sin x x Cos x 2 In 50 f x Out 50 In 51 x Cos x2 Sin x Plot f x x 0 Pi 3 2 1 Out 51 0 5 1 0 1 5 2 0 2 5 3 0 1 2 3 Place the cursor within the plot region and right click twice to display a pull down menu from with the option Get Coordinates may be selected in order to find approximate values of the coordinates for the absolute maximum and minimum Apparently they occur near 2 507 3 11 and 2 087 3 023 Since this function has a derivative everywhere the only critical points occur where the derivative vanishes In 52 Solve D f x x 0 x Solve nsmet This system cannot be solved with the methods available to Solve Out 52 Solve Cos x Cos x2 2 x2 Sin x2 0 x Since the Solve command cannot provide a solution will will need to approximate the zeroes of the derivative numerically with the FindRoot command In 53 X 1 x FindRoot D f x x x 0 9 Out 53 0 92011 In 54 X 2 x FindRoot D f x x x 1 6 Out 54 1 82428 Using derivatives to find properties of functions nb In 55 X 3 x FindRoot D f x x x 2 5 WorkingPrecision 20 Out 55 2 5096823655698523433 In 56 X 4 x FindRoot D f x x x 3 1 WorkingPrecision 20 Out 56 3 0869953834970000053 In 57 f X 1 Out 57 1 40527 In 58 Out 58 In 59 3 f X 2 f X 3 f X 4 0 824633 3 1000750940745536964 3 015500542533563523 f 0 f Pi Out 59 0 Cos 2 In 60 N f Pi 10 Out 60 2 835869702 Therefore the absolute maximum is 3 015500542533563523 f X 3 3 1000750940745536964 and the absolute miminum is f X 4 Derivatives and the shapes of graphs Derivatives may be used to tell where the graph of a function is increasing or decreasing because the derivative gives the slope of the tangents to the graph Therefore the first derivative may be used to find where the function has local maxima where the graph goes from increasing to decreasing and local minima where the graph goes from decreasing to increasing The second derivative gives information on how graphs bend if a curve always lies above its tangents throughout an interval then it is said to be concave upward if below then concave downward As an example we study the function f x x2 3 Sin 2 x on the closed interval p p Clear f In 64 f x x 2 3 Sin 2 x f x Out 64 x2 3 Sin 2 x First we plot a graph of the function 4 Using derivatives to find properties of functions nb In 65 Plot f x x Pi Pi 10 8 6 4 Out 65 2 3 2 1 1 2 3 2 There appear to be two local minima near 0 7 and 0 9 respectively and one local maximum near 1 0 and possibly another local maximum near 3 0 At points where there is a local extremum 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 In 66 D f x x Out 66 2 x 6 Cos 2 x In 68 g x 2 x 6 Cos 2 x g x Out 68 2 x 6 Cos 2 x In 69 Plot f x g x x Pi Pi PlotStyle Red Green Thick 10 5 Out 69 3 2 1 1 2 3 5 In 71 X 1 x FindRoot D f x x x 0 67 WorkingPrecision 10 Out 71 0 6723755227 In 73 X 2 x FindRoot D f x x x 0 90 WorkingPrecision 10 Out 73 0 9457599482 Using derivatives to find properties of functions nb In 74 X 3 x FindRoot D f x x x 2 0 WorkingPrecision 10 Out 74 1 992913103 5 To see if there is a solution near 3 0 we may zoom in on the graph of the derivative In 75 Plot g x x Pi 2 9 PlotStyle Green Thick 0 20 0 25 0 30 Out 75 0 35 0 40 3 05 3 00 2 95 2 90 Since the graph of the derivative does not reach up to the X axis where y 0 there is no other local extremum The graph of the derivative is negative from p to X 1 and from X 2 to X 3 so f is decreasing on those intervals The derivative is positive from X 1 to X 2 and from X 3 to p so the function in increasing there By applying the First Derivative …
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