Calculus I Maple Lab #3 Introduction and ExamplesGraphing with Calculus and MaplePart I: Extreme Value TheoremThe Extreme Value Theorem tells us that whenever we have a function that is continuous on a closed interal, that function will have both an absolute maximum and an absolute minimum somewhere on that interval. There are two types of x-values where these extreme values can occur: either at critical numbers inside the interval, or at an endpoint of the interval.1. Graph each of the functions below on the specified interval. If necessary, adjust the range for y so that you can see a good picture of the graph. First, state whether the absolute maximum and absolute minimum occur at a critical numbers or the interval endpoints. Then use Maple to find the exact value of the absolute max and min, and where they occur. State the answer clearly.EXAMPLE: NiMvLUkiZkc2IjYjSSJ4R0YmKiQsJiokRigiIiMiIiJGLSEiIiIiJA==on the interval [-1, 2]plot((x^2-1)^3,x=-1..2,y=-3..5);

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

The absolute maximum on this interval occurs at x=2, the right hand endpoint of the interval. The absolute minimum occurs at a critical number inside the interval.First to find the critical numbers, we need to find the derivative of the function and set it equal to zero.f := x -> (x^2-1)^3;

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQsJiokOSQiIiMiIiIhIiJGMSIiJEYlRiVGJQ==

fp := D(f);

NiM+SSNmcEc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwkKiYsJiokOSQiIiMiIiIhIiJGMkYxRjBGMiIiJ0YlRiVGJQ==

solve(fp(x)=0);

NiciIiEiIiIhIiJGJEYl

The critical numbers of the function are x = -1,0,1. It looks like the absolute minimum occurs at x = 0. To confirm this and to find the extreme values, we substitute the critical numbers AND the endpoints into the original function.f(-1);

NiMiIiE=

f(0);

NiMhIiI=

f(1);

NiMiIiE=

f(2);

NiMiI0Y=

The absolute maximum is 27, and it occurs at x = 2. The absolute minimum is -1, and it occurs at x = 0. (Another way to say this is that the absolute maximum is f(2) = 27, and the absolute minimum is f(0) = -1).Part II: Using Calculus to get Better Graphs of Functions.When you graph a function, you would like to be able to see all the interesting features of the function from your graph. For example, where the derivative is zero, where the graph crosses the x or y axis, and where the function has local extreme values. Knowing what the derivative tells us about the function can help us pick appropriate intervals for the graph of a function.For each of the following functions:a) Use the plot command to graph the function.b) Use Maple to find the first derivative. Graph f', and use the graph to estimate the critical numbers. Also, state whether the function will have a local maximum, a local minimum, or neither at each critical number. c) Use Maple to find the second derivative. Find the points of inflection (you can estimate them from the graph) of the function and also graph f''. State on which intervals f is concave up and on which intervals f is concave down. d) Graph the original function again, choosing intervals that demonstrate all the aspects of the function you discovered above. After you print out your graphs (remember to make them smaller first), label the x and y-intercepts, extreme values (local and global), and inflection points. EXAMPLE: NiMvLUkiZkc2IjYjSSJ4R0YmLCoqJEYoIiImIiIiKiYiIzdGLCokRigiIiVGLEYsKiRGKCIiIyEiIkYoRiw=plot(x^5+12*x^4 - x^2 +x,x=-40..40);

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

restart:f := x->x^5 + 12*x^4 - x^2 +x;

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCoqJDkkIiImIiIiKiRGLiIiJSIjNyokRi4iIiMhIiJGLkYwRiVGJUYl

fp := D(f);

NiM+SSNmcEc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwqKiQ5JCIiJSIiJiokRi4iIiQiI1tGLiEiIyIiIkY1RiVGJUYl

solve(fp(x) = 0);

NiYtSSdSb290T2ZHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkLCoqJEkjX1pHRigiIiUiIiYqJEYsIiIkIiNbRiwhIiMiIiJGMy9JJmluZGV4R0YlRjMtRiQ2JEYqL0Y1IiIjLUYkNiRGKi9GNUYwLUYkNiRGKi9GNUYt

Maple cannot come up with a solution to this equation, but we can use the graph of the derivative to estimate the critical numbers.plot(fp(x),x=-20..20,y=-6000..3000);

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

f(-9.5);

NiMkIitEMUhFPyEiJg==

The critical numbers are at approximately x = 0 and x = -9.5. At x = -9.5, there is a local ____. At x = 0, there is a local ______. fpp := D(fp);

NiM+SSRmcHBHNiJmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsKCokOSQiIiQiIz8qJEYuIiIjIiRXIiEiIyIiIkYlRiVGJQ==

plot(fpp(x),x=-10..2);

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

The second derivative is zero around x = -7 and x = 0. At x = -7, the concavity of f switches from concave down to concave up. At x = 0, the concavity of f does not change--f remains concave up on both sides of zero.Let's see how big we need to let y get so that we can see the values of the local max's and min's:f(0);

NiMiIiE=

f(-9.5);

NiMkIitEMUhFPyEiJg==

We can also change the range for x, because all the critical numbers are between -14 and 8. plot(f(x),x=-14..8,y=-3000..21000);

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