Math 1205 Matlab Worksheet 6: Rates of Change

Introduction: Background: (Refer to the text, Section 3.3)

In your text homework, you have used the derivative to find the rate at which one variable changes with respect to another. One application involves the motion of a particle along a coordinate line. Problem 1 expands the work in Section 3.3 to include speed and acceleration functions. Problem 2 considers rates of change for a function defined by a table of values.

Matlab Hints:

·  Review the section “Using the Symbolic Toolbox” in Matlab Basics before attempting Problem 1.

·  You will need to use the diff command from the Symbolic Toolbox to differentiate functions. For example, suppose the position of an object moving along a line has position s(t) at time t. If t is defined symbolically, then the velocity of the object is v = diff(s).

·  When using the Symbolic Toolbox, the ezplot command is used for graphing.

·  You may also want to review the use of the fzero command. This is not a symbolic function. Recall that to use fzero, you must use the variable x.

·  If you wish to view more than one figure window on your screen (see Problem 1a), you can simply number each window. For example, type the following commands. (You may need to move your second graph if the first one is directly behind it.)

»syms t

»s=(1/4)*t^4-t^3+t^2;

»v=diff(s);

»ezplot(s,0,2)

»figure(2)

»ezplot(v,0,2)

Instructions:

·  Submit your work as a printed Word document.

·  Number each problem clearly and circle answers.

·  Use Matlab-generated graphs pasted into the document.

·  Show the correct Matlab input and corresponding output for each problem.

Problems:

Problem 1: Motion along a coordinate line

Recall from the text, page 197, that the velocity of a particle moving on a coordinate line is found by differentiating the position function. In symbols, where is the position of the particle on the line at time . In other words, the velocity of the particle at time is the rate of change of position with respect to time. Similarly, we can define the acceleration of the particle to be the rate of change of the velocity with respect to time. In terms of derivatives, the acceleration function is the derivative of the velocity function, . We can also use the velocity function to find the speed of the particle at time . The velocity indicates both “how fast” the particle is moving (speed) and the “direction” of the motion (+ or -). Therefore, the speed is the non-negative numerical value for the velocity. Using absolute value, . In summary:

______

Position function: Velocity function:

Acceleration function: Speed function:

The position of a particle moving along a horizontal coordinate line is given by the

formula where is in seconds.

(a) Plot the graphs of position, velocity, acceleration, and speed separately on the interval . These graphs will be useful in assisting you to visualize your answers to parts (b-e). Paste them into your Word document and label each one.

(b) Describe a process for determining when a particle in motion is momentarily at rest. To four decimal place accuracy, at what time(s) in the interval is the particle of this exercise momentarily at rest?

(c) Describe a process for determining when a particle in motion along a coordinate axis is moving right or left (or up or down). To four decimal place accuracy, on what time interval(s) (in the interval ) is the particle of this exercise moving to the right? To the left?

(d) To four decimal place accuracy, at what time(s) is the acceleration zero?

(e) Use your graph of the speed of the particle to visually estimate the time intervals over which the speed of the particle increasing. Repeat for decreasing speed.

(f) Graph the position, velocity, and acceleration functions together on the same coordinate axes on the interval . Use ezplot with the “hold on” command.

Paste this graph into your Word document.

After printing your document, do the following by hand.

(i) Draw in vertical lines to show the intervals on which the object is moving to the right and the intervals on which the object is moving to the left.

(ii) Draw in vertical lines to show the intervals on which the object is slowing down and the intervals on which the object is speeding up.

(iii) Write in “speeds up”, “slows down”, “moves to the right”, “moves to the left” in the appropriate places on the graph.

An Example: Your finished graph should look similar to the following, which, as an example, shows this information provided for the function .

(g) Look carefully at your graph and at the example. (You may need to label the graphs as s(t), v(t), and a(t) on your plot.) Notice that a positive value for the acceleration does not always guarantee that the object is speeding up. However, the relationship between the signs (+ or -) of the acceleration and velocity will determine when the particle is speeding up and slowing down. Describe this relationship.

h) Use an appropriate graph to examine where . What does your answer tell you about the motion of this particle?

Problem 2: Other Rates of Change

The following data shows the fare per person of a charter bus for a particular trip. In the table, p is the number of passengers and F(p) (in dollars) is the fare per passenger. (For instance, on a trip with 30 passengers, each passenger pays a fare of $20.50.)

p / 20 / 25 / 30 / 35 / 40
F(p) / 26 / 23 / 20.50 / 18.50 / 17

(a) Find the average rate at which the fare per person changes on each of the following intervals. Give units for your answers.

(i) (ii) (iii) (iv)

(b) Use the information found in part (a) to approximate the value of . What are the units? Interpret your answer.

(c) The total revenue per trip is R(p) = pF(p). Use the product rule, the data from the table, and your answer from (b) to find R'(30). What are the units? Interpret your answer.