Limits, differentiating in Maple using the D Operator and diff Command and Finding the Equations of Tangent LinesMaple can differentiate quantities entered as functions and expressions. Before demonstrating, we make sure all variables are unassigned and memory is cleared by enteringrestart;
<Text-field style="Heading 1" layout="Heading 1">Limits</Text-field>The limit command is used to calculate the limit of a function. Suppose we want to find the 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 Limit command is designed to compute the limit of a function. To set up the limit, we typeLimit(sin(x)/x, x = 0);The value command will calculate the answer for the previous result.value(%);We can also calculate limits at infinity. The following commands demonstrate how to calculate 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Limit(exp(-x)*cos(x), x = infinity);value(%);
<Text-field style="_cstyle46" layout="_pstyle42">Differentiating Maple Functions using the D operator</Text-field>Recall that a function in Maple is entered using the := x -> notation. Suppose we enter the functionf(x) = 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in Maple.f := x -> x^7*(x^4 - 8)^9;The D operator is used to differentiate functions in Maple. To compute the derivative of the function we just entered, typeD(f);To evaluate the derivative at a point, say f '(2), we enterD(f)(2);Higher order derivatives can be computed in two ways, either by enclosing the D operator with another D operator, or by typing (D@@2)(f) for the second derivative, (D@@@2)(f) for the third, etc. The following commands will calculate f ''(x) and f ''(2) for the following functions.D(D(f)); (D@@2)(f);D(D(f))(2); (D@@2)(f)(2);To end this section, we display the first and second derivatives for the functions g(x) = 3cos(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYkUSIyRicvRjtRJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRkFGK0ZB - 1) and h(t) = 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. Note for a trigonometric function that the angle is always enclosed in parenthesis.g := x -> 3*cos(x^2+1); h := t -> t^2*exp(t^3);D(g);D(D(g));D(h);D(D(h));
<Text-field style="_cstyle46" layout="_pstyle42">Differentiating Functions as Expressions</Text-field>Recall a Maple expression is just an assignment statement made to a variable using the := statement. To differentiate an expression, we use the diff command with the syntax diff(expression, x), which assumes the function will be differentiated with respect to the variable x. For example, the differentiate the expression 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 with respect to x, we enter the following commands:f := 3*x^4 + 2*x^3 + 5*x^2 + x - 5;df := diff(f, x);To find f ' (2) for this example, we use the subs command.subs(x = 2, df);To calculate higher order derivatives, we place a $ sign plus a number after the x in the diff command indicating the order of the derivative we want. The next command calculates the second derivative f '' for the above function.dff := diff(f, x$2);The next sequence of commands finds the first and second derivatives of the functions g(x) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNiZGKy1JJW1zdXBHRiQ2JS1GIzYnLUYsNiVRI2xuRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnRj4vJSZmZW5jZUdGPS8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZVLUYjNiYtRkI2LVEiKEYnRj4vRkZRJXRydWVGJ0ZHL0ZKRmhuRktGTUZPRlEvRlRRLDAuMTY2NjY2N2VtRicvRldGW28tRiM2JC1GLDYlUSJ4RicvRjxGaG4vRj9RJ2l0YWxpY0YnRj4tRkI2LVEiKUYnRj5GZ25GR0ZpbkZLRk1GT0ZRRmpuL0ZXUSwwLjExMTExMTFlbUYnRj5GK0Y+LUkjbW5HRiQ2JFEiM0YnRj4vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPi1GQjYtUSIrRidGPkZFRkdGSUZLRk1GT0ZRL0ZUUSwwLjIyMjIyMjJlbUYnL0ZXRmVwLUYjNidGOEZBLUYjNiZGWi1GIzYmRistRiM2JkYrLUY0NiVGX29Gam9GXnBGK0Y+RitGPkZlb0Y+RitGPkYrRj5GK0Y+g := (ln(x))^3 + ln(x^3);diff(g, x);diff(g, x$2);The next command sequence of commands finds the first and second derivatives of the functions 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 t + 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. Note how LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1cEdGJDYlLUYsNiVRJGNvc0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtbkdGJDYkUSI0RidGOi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0Y6RitGOg== t is entered, with the entire function cos(t) placed in parenthesis and the exponent on the outside.h := (cos(t))^4 + cos(t^4);diff(h,t);diff(h, t$2);
<Text-field style="Heading 1" layout="Heading 1">Tangent lines</Text-field>Recall that the derivative f '(x) of a function f provides a formula for finding the slope of the tangent line at a particular point. That is, f '(a) represents the slope of the tangent line to the graph of the function f (x) at the point (a, f (a)). We can use Maple to find the equation of the tangent line at any point for a given function and plot the function and its tangent line on the same graph. We demonstrate using the following examples.
<Text-field style="Heading 2" layout="Heading 2">Example 1</Text-field>Consider the function 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 and supposed with want to find the equation of the tangent line at the value x = 1. We store this function asf := x -> 1/x^2;The derivative provides a formula for getting the slope of the tangent line at any point where it is defined. The next command uses the limit definition to compute the derivative.df := D(f);To find the equation of the tangent line in Maple, it is easiest to use the point slope equation for a line which is given 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,where m is the slope and ( 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) is the point on the line. For a tangent line to a function f (x), the tangent line to the point 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 hasslope m = f ' (a). Hence, the point slope equationcan be rewritten 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 = f ' (a) (x - a)Using the solve command, we can find the equation of the tangent line for the function f given above by enteringt := solve(y - f(1) = df(1)*(x - 1), y);The following plot command will graph the function and its tangent line on the same graph on the interval [-5..5].plot([f(x), t], x = -5..5, color = [red, blue], view = [-5..5, -5..10], thickness = [2,1], title = "Graph of function f(x) = 1/x^2 and tangent line at x = 1" );
<Text-field style="Heading 2" layout="Heading 2">Example 2</Text-field>Suppose we want to find the equation of the tangent line tangent to the graph of 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 at the value x = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiZGKy1JJm1zcXJ0R0YkNiMtSSNtbkdGJDYkUSIyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGOy1GIzYkRjdGOy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRS8lKWJldmVsbGVkR1EmZmFsc2VGJ0YrRjs=. We first enter this function and compute its derivative, using unapply to store the result as a function.g := x -> sin(x^2);dg := D(g);The next equation finds the equation of the tangent line.t := solve(y - g(sqrt(2)/2) = dg(sqrt(2)/2)*(x - sqrt(2)/2), y);The next command plots the function and its tangent line on the same graph on the interval [LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLUYsNiVRI1BpRicvJSdpdGFsaWNHRjpGNUY1RitGNQ==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjUGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI=].plot([g(x), t], x = -Pi..Pi, color = [red, blue], view = [-Pi..Pi, -2..2], thickness = [2,1], title = "Graph of function g(x) = sin(x^2) and tangent line at x = sqrt(2)/2" );
<Text-field style="_cstyle46" layout="_pstyle42">Exercises</Text-field>1. Calculuate the following limits. a. 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 b. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYuUSRsaW1GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZ1bnNldEYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dRJXRydWVGJy8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjE2NjY2NjdlbUYnLUYjNictSSNtaUdGJDYoUSJ4RicvJSdpdGFsaWNHRkUvJStmb3JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJ0YyLyUscGxhY2Vob2xkZXJHRkUvRjZRJ2l0YWxpY0YnLUYvNi1RLSZyaWdodGFycm93O0YnRjUvRjlRJmZhbHNlRicvRjxGW28vRj5GRS9GQEZbby9GQkZbby9GREZbby9GR0Zbby9GSVEsMC4yNzc3Nzc4ZW1GJy9GTEZjby1JI21uR0YkNiRRIjJGJ0Y1LyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnRjUvJSxhY2NlbnR1bmRlckdGW28tRi82LVEifkYnRjVGam5GXG8vRj5GW29GXm9GX29GYG9GYW9GSC9GTEZKLUkmbWZyYWNHRiQ2KC1GIzYnLUZRNiVGU0ZURmVuLUYvNi1RKCZtaW51cztGJ0Y1RmpuRlxvRmFwRl5vRl9vRmBvRmFvL0ZJUSwwLjIyMjIyMjJlbUYnL0ZMRl5xRmVvRjJGNS1GIzYnLUklbXN1cEdGJDYlRmhwLUYjNiUtRmZvNiRRIjNGJ0Y1RlRGZW4vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRmpwLUZmbzYkUSI4RidGNUZURmVuLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zlci8lKWJldmVsbGVkR0Zbb0Zpb0Y1 c. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYuUSRsaW1GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZ1bnNldEYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dRJXRydWVGJy8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjE2NjY2NjdlbUYnLUYjNiYtSSNtaUdGJDYoUSJ4RicvJSdpdGFsaWNHRkUvJStmb3JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJ0YyLyUscGxhY2Vob2xkZXJHRkUvRjZRJ2l0YWxpY0YnLUYvNi1RLSZyaWdodGFycm93O0YnRjUvRjlRJmZhbHNlRicvRjxGW28vRj5GRS9GQEZbby9GQkZbby9GREZbby9GR0Zbby9GSVEsMC4yNzc3Nzc4ZW1GJy9GTEZjby1GLzYtUSgmaW5maW47RidGNUZqbkZcby9GPkZbb0Zeb0Zfb0Zgb0Zhb0ZIL0ZMRkpGNS8lLGFjY2VudHVuZGVyR0Zbby1GLzYtUSJ+RidGNUZqbkZcb0Zob0Zeb0Zfb0Zgb0Zhb0ZIRmlvLUkjbW5HRiQ2JFEiNEYnRjVGXHAtRi82LVEiK0YnRjVGam5GXG9GaG9GXm9GX29GYG9GYW8vRklRLDAuMjIyMjIyMmVtRicvRkxGZ3BGXHAtRmBwNiRRIjVGJ0Y1LUklbXN1cEdGJDYlLUZRNiVRImVGJ0ZURmVuLUYjNiYtRi82LVEqJnVtaW51czA7RidGNUZqbkZcb0Zob0Zeb0Zfb0Zgb0Zhb0ZmcEZocC1GUTYlRlNGVEZlbkZURmVuLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YyRjU= d. 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2. The following commands have an error or errors that needs to be corrected. Correct the error so that the proper output is produced. a. To differentiate the function 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 using the D operator, the following is entered:f := x^3 + x; D(f); b. To differentiate the function 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 as an expression using the diff command, the following is entered:f := x -> sin(x^2); diff(f, x); c. To differentiate the function 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 as an expression using the D operator, the following is entered:f := t -> sin(x^2); D(f); d. To differentiate the function 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 as an expression using the diff command, the following is entered:g := cos*(x^3+1) + e^(x^3+1); diff(g, x); e. To differentiate the function 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(x) + 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 as an expression using the diff command, the following is entered:h := sin^5(x) + exp^(x^3+1); diff(h,x);3. Using the D operator, find the first and second order derivatives of the following as Maple functions.
a. 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
b. 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 c. 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
4. Using the diff command, find the first and second order derivatives of the following as Maple expressions.
a. 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
b. 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 . c. 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(t) + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSRzaW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tRiM2Ji1GOzYtUSIoRidGNy9GP1EldHJ1ZUYnRkAvRkNGV0ZERkZGSEZKL0ZNUSwwLjE2NjY2NjdlbUYnL0ZQRlotRiM2JkYrLUYjNiZGKy1JJW1zdXBHRiQ2JS1GLDYlUSJ0RicvRjVGVy9GOFEnaXRhbGljRictSSNtbkdGJDYkUSI1RidGNy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0Y3RitGNy1GOzYtUSIpRidGN0ZWRkBGWEZERkZGSEZKRlkvRlBRLDAuMTExMTExMWVtRidGN0YrRjdGK0Y35. For the following functions, find the equation of the line tangent to the graph at the given value of x. Graph the function and its tangent line at the given point on the same graph. a. 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 at x = 2. b. 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 at x = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy.