Calculus I Exam 3 Examples

CALCULUS I EXAM 3 EXAMPLES

For the function f(x) = x³ - 12x, find the absolute and relative maximum and minimum values if any exist over each indicated interval.

[-3,3], [-1,4], (-1,4), [-1,3]

Answers

[-3,3]

abs min: -16

rel min: -16

abs max: 16

rel max: 16

[-1,4]

abs min: -16

rel min: -16

abs max: 16

rel max: none

(-1,4)

abs min: -16

rel min: -16

abs max: none

rel max: none

[-1,3]

abs min: -16

rel min: -16

abs max: 11

rel max: none

This Maple Worksheet investigates some aspects of the two functions listed below. A potential source of confusion regarding real output in Maple is demonstrated in looking at f(x). We investigate f3(x) to look at a function that is the absolute value of the function in the example above.

Finding a polynomial function satisfying given data points and then finding its relative maximum and relative minimum points.

Your task is to find the fourth degree polynomial function that gives the path followed by the tip of the beak of the bird in the animation. In the animation the coordinates of the end of the orange branch at the far left are (1,27). The coordinates of the end of the green branch are (10,20), the coordinates of the end of the red branch are (18,31), and the coordinates of the end of the purple branch are (21,28). The coordinates of the end of the orange branch at the far right are (26,39). Distance is in feet. The tip of the beak of the bird is at (1,27), (the end of the orange branch at the left), when the bird falls off the branch. The bird begins to fly and its path takes the tip of its beak through points exactly 2 feet above the tips of the green, red, purple, and orange (on the right) branches. It is autumn. Click here to see the animation without scales on the x- and y-axis.

This example illustrates using a TI graphing calculator to find f(x) = ax⁴ + bx³ + cx² + dx + e to best fit given data points.

The data points in this example would be (1,27), (10,22), (18,33), (21,30), and (26,41). Form two lists, one for the x-values and one for the y-values. In the instructions below "->" means "hit the store key" and "[ENTER]" means "hit the enter key". Input into your TI as follows:

{1,10,18,21,26}->L1 [ENTER]

{27,22,33,30,41}->L2 [ENTER]

Then enter these commands if using a TI-89 or a TI-92.

QuartReg L1, L2 [ENTER]

ShowStat [ENTER]

This will give you the coefficients a, b, c, d, and e. The regression equation is stored in Regeq. If you would like to see the graph along with a plot of the data points hit [ENTER] again and then

Regeq(x)->y1(x) [ENTER]

NewPlot 1, 1, L1, L2 [ENTER]

Then ask your calculator to graph after putting in appropriate window settings. You can find more information on page 462 in your TI-89 Guidebook and on page 430 in your TI-92 Guidebook.

Here are the commands to use on your TI-86 after you have input the two lists just as above.

P4Reg L1, L2, y1 [ENTER]

The coefficients will be displayed in a list along with other information. To see the graph of the function along with a plot of the data points do this:

Plot1(1, L1, L2) [ENTER]

ZData [ENTER]

More details can be found on page 332 in your TI-86 Guidebook.

The function you should get is

f(x) = 0.002199x⁴ - 0.126409x³ + 2.387646x² - 15.230995x + 39.96756.

After finding the equation of the bird's path (above), approximate any relative maximum and relative minimum points using your TI graphing calculator. The path is pictured on the right.

Here is a nice online polynomial regression utility.

Section 3.2 #46 Mean Value Theorem Example

Determine whether the Mean Value Theorem can be applied to f on the closed interval [a,b]. If the Mean Value Theorem can be applied, find all values of c in the open interval such that

Pictured above is the graph of f over the interval in question in the problem. Pictured below is the graph of f over a larger interval. Note that although the derivative of f at point B is zero, point B is not a relative maximum point and is not a relative minimum point. The coordinates of points C and B are

Mean Value Theorem Example: Find all values of c in the open interval (-1,2) guaranteed by the Mean Value Theorem applied to the function f(x) given below over the closed interval [-1,2]. Click on the picture below to see a larger picture.

Finding relative maximum and relative minimum points on the graph of a polynomial function analytically.

The number line below indicates the intervals where the first derivative is positive and the intervals where the first derivative is negative.

Animation of a synthetic division example QT Powerpoint Powerpoint2

The function is graphed in red and its derivative in blue above.

Maple Worksheet

A graph with NO inflection point where f"(x) = 0.

Note: (1,-3) is also an absolute minimum point.

In the graph on the right above, f is graphed in red, f' in green, and f" in blue.

If I zoom in on the point (0,0) on the graph above, a point that is not an inflection point even though f"(0) = 0, should I expect to see some sort of strange behavior around (0,0)?

A Graph with an Inflection Point
Find the inflection point and any relative maximum points and any relative minimum points on the graph of the function whose equation is given.

Concave down in red Concave up in blue

Here is a Maple Analysis of

Maple Worksheet

Section 3.4 Problem Number 63

The plane would be descending at the most rapid rate at the point on the path where f'(x) (which is negative) is a minimum, i.e., where the absolute value of f'(x) is a maximum. This would occur at the point where the derivative of f'(x) is zero, i.e., at the point where f"(x) = 0 which is the inflection point (-2,1/2). The article "How Not to Land at Lake Tahoe!" can be read by following the Matharticles link below and looking under chapter three.
Here are some animations and a link to learn more:

The tangents to the plane's path

The plane pointing in the direction of the tangent

The plane level

The plane level, no graph

Matharticles.com

Limits at Infinity--Examples

Click on the graph for a larger version.

Maple Worksheet with critical points and inflection points

Click on the graph for a larger version.

Here is the example I said half of you would miss.

At the right you will find the function graphed in red and the slant asymptote graphed in blue.
Below is the typical error in graphing.

This one is better.

A graph with vertical and horizontal asymptotes and a removable (point) discontinuity.

The vertical asymptotes are in blue and the horizontal asymptote is in green. The "missing point" is at the red dot.

Another "dangerous" example

Note that the top graph on the right is "incomplete" due to the inadequate window. The bottom graph on the right would be complete. The "asymptotic parabola" is shown in blue in the graph below.

Maple Worksheet

Newton's Method Example
Use Newton's Method to approximate a zero of f(x) = cos(x) using an initial guess of 0.5.

Newton's Method--Powerpoint Presentation Quicktime Movie Maple Worksheet

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Lane Vosbury, Mathematics, Seminole State College email: vosburyl@seminolestate.edu