[PDF] Antiderivatives

14402_2Antiderivative.pdf

Antiderivatives

ObjectiveIn this lab you will develop your understanding and manipulative abilities with antiderivatives. BackgroundAn antiderivative of a functionfis any functionFwhich satisfies the condition F ?(x) =f(x). Recall that antiderivatives are not unique. IfFis an antiderivative off, then the general antiderivative offisF(x) +C The basic Maple command for antiderivatives isint. The syntax for?f(x)dx isint( f(x), x );. Note, however, that Maple reportsanantiderivative. If you need the general antiderivative it will be necessary for you to include an appropriate arbitrary constant. TheAntiderivativemaplet is an interface to the visual relationship between a function and its antiderivatives. TheIntegrationmaplet is a calculator-like interface with buttons corresponding to each of the primary rules for evaluating integrals. DiscussionEnter, and execute, the following Maple commands in a Maple worksheet. Example 1: TheintCommand>restart;# clear Maple"s memory >f := x -> x^2 - 3 + 8*sin(x);# define integrand >F := int( f(x), x );# an antiderivative >diff( F, x );# verify thatF?(x) =f(x) >G := int( f(x), x ) + C;# general antiderivative >diff( G, x );# verify thatF?(x) =f(x) Example 2: TheAntiderivativeMaplet•launch theAntiderivativemaplet •in theFunctionbox, enterx^2 - 3 + 8*sin(x) •in thea =andb =boxes, enter-2*Piand2*Pi, respectively •pressPlot •in theValuebox, enter[ 0, 0 ]; pressPlot •place a check in theShow class of antiderivativescheckbox; pressPlot Observe that the local extrema of each antiderivative (in blue or green) occur at the points where the function (in red) is zero. Example 3: TheIntegrationMaplet•launch theIntegrationmaplet •in theFunctionbox, enterx^2 - 3 + 8*sin(x) •in theVariablebox, enterx •to start the evaluation of this antiderivative, pressStart •to apply theSum Rule(twice), pressSum(once) •to evaluate the first integral using thePower Rule, pressPower •to evaluate the second integral using theConstant Multiple Rule, pressCon- stant Multiple •in theFunction Rulesarea of the interface, pressSelect a Function, click onsin, pressApplyMaple Lab for Calculus IFall 2003 Notes (1) To create a Maple function corresponding to an antiderivative, use theunapplycommand.

For example,

>F := unapply( int( f(x), x ), x ); (2) The plots displayed in theAntiderivativemaplet are created with theAntiderivativePlot command, from theStudent[Calculus1]package. The basic syntax is >AntiderivativePlot( f(x), x=a..b );. To obtain the antiderivative that passes through a specific point (x0,y0), include the optional argumentvalue=[x0,y0]. To see a family of antiderivatives, include the optional argument showclass=true. (3) To perform aGeneralized Power Rulewith the functiong(x), enteru=g(x)in the large box in theIntegration Rules with Argumentsregion and press theChangebutton. Then, at the end of the problem, pressRevertto return to the original independent variable. (4) In the last step for the example with theIntegrationmaplet, it is also possible to type the name of the function, e.g.,sin, in the box instead of working through theSelect a

Functionmenu.

(5) TheUnderstood Rulesmenu in theIntegrationmaplet can be used to identify rules to be applied automatically whenever possible. (6) TheStudent[Calculus1]package contains commands that correspond to many of the buttons on theIntegrationmaplet. There is no reason to explain these here as there is no reason for you to use these commands.

Questions

(1) Use theIntegrationmaplet to evaluate?sin2x dx. Use theAll Stepsbutton to obtain a full listing of the steps in the evaluation of this indefinite integral. Summarize this evaluation in your lab report. Note: If you have trouble with the formatting of mathematical expressions, use Maple notation for integrals. For example, write?sin2x dxasint( sin(x)^2, x ). (2) Use theIntegrationmaplet to evaluate each of the following indefinite integrals. Note: It is not necessary to show all steps, just report the answer. (a)? x⎷x+ 1dx (b)? x⎷x2+ 1dx (c)? x

2sinx3+ 1dx

(d)? sin

3((x2+ 1)4)cos((x2+ 1)4)(x2+ 1)3x dx

Hint: For (d), do not use theAll Stepsbutton. Think! (See also, Note (3).) (3) Let F(x) = (sinx+ cosx)4andG(x) = 2sin(2x)-4cos4x+ 4cos2x+ 5. Show thatFandGare antiderivatives of the same function. Explain why two functions that appear so different can be antiderivatives of the same function. Find the functionf withF?(x) =f(x) andG?(x) =f(x).Maple Lab for Calculus I2 Antiderivatives