Maple Lab #2 Introduction Using Maple to Work With Derivatives So far, we have used the notation f '(x) to represent the derivative of the function f(x). (Unfortunately, Maple does not recognize the notation f'). Example 1: Suppose that we want to find the derivative of the function NiMvLSUiZkc2IyUieEcsKComIiIlIiIiKiRGJyIiJEYrRisqJkYtRisqJEYnIiIjRishIiIqJiIiKEYrRidGK0Yr using Maple. First, we will enter the function f( x ). (Remember that the syntax in Maple is a little different than what you would write by hand). f := x -> 4*x^3-3*x^2+7*x; NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJDkkIiIkIiIlKiRGLiIiIyEiJEYuIiIoRiVGJUYl Maple has a limit command, so we can type in the definition of the derivative and have Maple evaluate it. There are two limit commands: Limit, and limit. With an uppercase L, Maple will not evaluate the limit. This gives you a chance to check your work. Limit(f(x+h)-f(x)/h,h=0); NiMtSSZMaW1pdEc2JEkqcHJvdGVjdGVkR0YmSShfc3lzbGliRzYiNiQsLCokLCZJInhHRigiIiJJImhHRihGLiIiJCIiJSokRiwiIiMhIiRGLSIiKEYvRjUqJiwoKiRGLUYwRjEqJEYtRjNGNEYtRjVGLkYvISIiRjovRi8iIiE= limit((f(x+h)-f(x))/h,h=0); NiMsKCokSSJ4RzYiIiIjIiM3IiIoIiIiRiUhIic= fprime := x->12*x^2-6*x+7; NiM+SSdmcHJpbWVHNiJmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsKCokOSQiIiMiIzdGLiEiJyIiKCIiIkYlRiVGJQ== Maple also has a built-in command to calculate the derivatives of functions. Unfortunately, Maple does not recognize the notation f'(x). The Maple command for derivatives is D . Now, we will use the D command to find f'(x): D(f); NiNmKjYjSSJ4RzYiRiY2JEkpb3BlcmF0b3JHRiZJJmFycm93R0YmRiYsKCokOSQiIiMiIzdGLCEiJyIiKCIiIkYmRiZGJg== So, the derivative of f(x) is NiMsKComIiM3IiIiKiQlInhHIiIjRiZGJiomIiInRiZGKEYmISIiIiIoRiY= . Notice that it comes out as a function (you can tell it is a function because of the "x->" ). We can't use the notation f ' on Maple, but we can name it "fprime". fprime := D(f); NiM+SSdmcHJpbWVHNiJmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsKCokOSQiIiMiIzdGLiEiJyIiKCIiIkYlRiVGJQ== Suppose you wanted to graph the function f(x) with its derivative. You could use the following plot command. plot([f(x),fprime(x)],x=-3..8,y=-100..100); 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 Next, say we want to find the slope of the tangent line when x = 3. To do this, we just need to evaluate the derivative at x=3: fprime(3); NiMiIygq When x = 3, the tangent line has slope 97. To find the y-coordinate of the point when x = 3, we would substitue x=3 into f(x): f(3); NiMiJC0i This tells us that the tangent line at x=3 goes through the point (3, 102), and its slope is 97. So the equation of the tangent line is NiMvLCYlInlHIiIiIiQtIiEiIi0iIygqNiMsJiUieEdGJiIiJEYo . You can solve for y by hand, but you can also ask Maple to do this: solve(y-102=97*(x-3),y); NiMsJiEkKj0iIiJJInhHNiIiIygq The equation of the tangent line at x = 3 is y = -189 + 97x. Now, graph the function and line to see if this answer is reasonable. plot([f(x), -189+97*x],x=0..5); 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 plot([f(x), -189+97*x],x=2.5..3.5); 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 It looks good. Notice that for values of x close to 3, the function and tangent line are very close together. Since the equation of the tangent line is simpler than the equation of the orignial function, the tangent line is often useful for approximating function values. Let's look at some examples for this function. First, we'll define the tangent line to be the function h(x) . h := x->-189+97*x; NiM+SSJoRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYhJCo9IiIiOSQiIygqRiVGJUYl If we compare the function value and the tangent line value at x=3, we should get the same thing, since the tangent line should go through one point on the graph of the function: f(3); NiMiJC0i h(3); NiMiJC0i Good--they are the same. What about at 3.5? f(3.5); NiMkIiddI2YiISIk h(3.5); NiMkIiUwOiEiIg== The y-values are still relatively close. Now let's try at x = 6: f(6); NiMiJCl6 h(6); NiMiJCRS The tangent line is no longer a good approximation of the function. It is only close to the function the the point were it is tangent. Example 2. We would like to find the derivative of the function NiMvLSUiZkc2IyUieEcqJiwmRiciIiIiIiNGKkYqLCYqJEYnRitGKkYqRiohIiI= . Since we are giving f(x) a new definition, first use the restart command to clear Maple's memory and then define the function: restart; f := x->(x+2)/(x^2+1); NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYsJjkkIiIiIiIjRi9GLywmKiRGLkYwRi9GL0YvISIiRiVGJUYl fprime := D(f); NiM+SSdmcHJpbWVHNiJmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJiokLCYqJDkkIiIjIiIiRjJGMiEiIkYyKigsJkYwRjJGMUYyRjJGLiEiI0YwRjJGNkYlRiVGJQ== Now, to find the slope of the tangent line at x = 4: fprime(4); NiMjISNKIiQqRw== There is the exact value of the slope. If we want a decimal approximation to see how big it is, we can use the command evalf (short for "eval uate f loating point"), or you could enter 4 as 4.0 or even just 4. to make Maple think you are already using numbers that are rounded: fprime(4.0); NiMkIStnVm1zNSEjNQ== fprime(4.); NiMkIStnVm1zNSEjNQ== evalf(fprime(4)); NiMkIStnVm1zNSEjNQ== The equation of the tangent line when x = 4 will be y-f(4) = f ' (4) (x-4). We can enter this into Maple and ask it to solve for y: solve(y-f(4) = fprime(4)*(x-4),y); NiMsJiMiJEUjIiQqRyIiIkkieEc2IiMhI0pGJg== Here is a graph of f(x) with the tangent line at x = 4: plot({f(x), 226/289-31/289*x},x=-2..6); 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