Name: _

Calculus I

Maple Lab #3 Exercises

Part I: Extreme Value Theorem

The 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.

a. g(x) = sin(x) + cos(x) on [0, NiMqJkkjcGlHNiIiIiIiIiQhIiI=]

b. f(x) = NiMsKiokSSJ4RzYiIiIkIiIiKiYiIidGKCokRiUiIiNGKCEiIiomIiIqRihGJUYoRihGLEYo on [1/2, 7/2]

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.

Do the following for each of the functions given below. Label each part clearly.

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. If possible, also use Maple to find the exact values of the critical numbers. State whether the function will have a local maximum, a local minimum, or neither at each critical number, and find the extreme value using Maple.

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''. If possible, find the points of inflection exactly, using Maple. State on which intervals f is concave up and on which intervals f is concave down.

You may need to type in the values for a-c into a text box for each problem.

d) Graph the original function again, carefully choosing intervals that demonstrate all the aspects of the function you discovered above.

1. f(x) = NiMsJiomIiIkIiIiKiRJInhHNiIiIiZGJiEiIiomRipGJiokRihGJUYmRiY=

2. h(x) = NiMqJkkieEc2IiIiIy1JI2xuR0YlNiNGJEYm

3. g(x) = NiMqJiwmKiYiIiMiIiJJInhHNiJGJ0YnRichIiJGJy1JJXJvb3RHNiRJKnByb3RlY3RlZEdGLkkoX3N5c2xpYkdGKTYkLCgqJEYoIiIlRidGKEYnRidGJ0YzRio= (the denominator is the fourth root of x^4 + x + 1)