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# rule x lim 1 1 1x lim ex ln1 x x x

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Unformatted text preview: gle Variable) This subpackage helps you step through diﬀerentiation, integration, and limit computations. The package contains visualization and interactive routines, and single-step computation. For details about the Student[Calculus1] package, refer to the ?Student[Calculus1] help page. 88 • Chapter 4: Maple Organization Finding the Derivative of a Function You can ﬁnd the derivative of a function by using the DiffTutor Maplet application, an interactive routine in the Student[Calculus1] package. For example, the following code invokes the interactive DiffTutor. &gt; &gt; &gt; &gt; with(Student[Calculus1]); DiffTutor(); DiffTutor(x*cos(x)); DiffTutor(x*cos(x),x); See the ?Student[Calculus1][DiffTutor] help page and copy the example into you worksheet. Worksheet Examples The following examples are generated using the Student[Calculus1] subpackage. Derivative Given the function 4*x^2, ﬁnd its derivative. First, activate the short forms of all the command names in the package by using the with command. &gt; with(Student[Calculus1]): &gt; infolevel[Student[Calculus1]] := 1: 4.2 The Maple Packages • 89 To view a list of all the commands that Maple is loading, replace the colon at the end of the command with a semicolon. &gt; Diff(4*x^2, x); d (4 x2 ) dx Use the constantmultiple rule. &gt; Rule[constantmultiple](%); Creating problem #1 d d (4 x2 ) = 4 ( (x2 )) dx dx Use the power rule. &gt; Rule[power](%); d (4 x2 ) = 8 x dx Integration Consider the following integration example. Integrate x ∗ cos(x) + x from x = 0 to x = π . &gt; Int(x*cos(x) + x, x=0..Pi); π x cos(x) + x dx 0 Use the sum rule. &gt; Rule[sum](%); Creating problem #2 90 • Chapter 4: Maple Organization π π π x cos(x) + x dx = 0 0 x cos(x) dx + 0 x dx Use the power rule. &gt; Rule[power](%); π π x cos(x) + x dx = 0 0 x cos(x) dx + 12 π 2 Use the Hint command to determine a possible next step for the problem. &gt; Hint(%); [parts , x, sin(x)] Use the hint with the Rule command. &gt; Rule[%](%%); π π x cos(x) + x dx = − 0 0 sin(x) dx + 12 π 2 Use the sin rule to complete this computation. &gt; Rule[sin](%); π x cos(x) + x dx = −2 + 0 12 π 2 Calculating Limits Calculate the limit of (1+1/x)x . Use the Understand command to use rules for calculating the Limit without explicitly applying them. To add the constant, constant multiple, power, and sum Limit rules to the list of understood rules for the following example, use the Understand command. &gt; Understand(Limit, constant, ‘c*‘, power, sum); Limit = [constant , constantmultiple , power , sum ] 4.2 The Maple Packages &gt; Limit((1 + 1/x)^x, x=infinity); • 91 x→∞ lim (1 + 1x ) x Request a hint for the next step of the computation. &gt; Hint(%); Creating problem #3 Rewrite the expression as an exponential to prepare for using l‘Hopital’s rule 1 1x ) = e(x ln(1+ x )) ] x [rewrite , (1 + Use the rule that is returned by Hint. &gt; Rule[%](%%); x→∞ lim (1 + 1 1x ) = lim e(x ln(1+ x )) x→ x &gt; Hint(%); [exp] &gt; Rule[%](%%); x...
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