Math 131 A,B,C,D Lab 6: Derivatives of the Trigonometric ...



Math 131 A,B,C,D Lab 6: Derivatives of the Sine and Cosine Functions 2/21/06

I. Graphical and Numerical View of the Derivative of the Sine Function:

Objective: To explore the graph of y = sin x and its derivative.

1. GRAPH the following functions in the viewing window [-2(, 2(]z, [-2,2]y. Compare the two graphs. Explain what you see.

a) y1 = sin x

b) y2 = d(sin x, x) [Note: d is [2nd] [8:d] ] or y2 = d(y1(x),x)

2. Fill in the following table (with h = 0.001) and for cos x in radian mode. [Note: you can use the seq command; e.g. seq((sin(x+0.001) – sin(x))/0.001, x, 0, 2(, (/4) or TABLE with tb1Start = 0; (tb1 = [pic]]

[pic]

3. Compare the corresponding entries in the two tables. Explain the results. From your graphical & numerical explorations, guess the derivative of the sine.

[pic]

II. Repeat (I) for the function y = cos x. Try to identify the graph of [pic] with a graph we already know.

III. Confirm your observations with the definition of the derivative.

1. Use your calculator to evaluate the following limits: [pic] [pic]

2. Find the derivative of y = sin x (x in radians) algebraically.

[pic]

Use the limits in part (1) to get the result. (Recall from trig identities what sin(x + h) equals.)

3. Repeat part (2) for the function y = cos x (x in radians)

(Recall from trig identities what cos(x + h) equals.)

IV. Deer Population and Optimal Rate of Increase

Objective: To investigate cyclic phenomenon graphically to determine points where the rate of change is maximum or minimum.

The population of a herd of deer is modeled by:[pic]where t is measured in years.

GRAPH the two functions:[pic] and [pic]

For the questions below, use the viewing window [0,1]x ( 1 year cycle) by [-3500, 5000]y (population). Use [(] [GRAPH] [F5:Math] [3:Minimum], [4:Maximum] and [8:Inflection] with the population and derivative graphs to help find the answers to the questions. You should also use [(] [GRAPH] [F3:Trace] to explore both graphs and their relationship to each other.

Answer the questions posed in a) to d) using your best English! Be clear and concise. Make complete (English) sentences in your answers.

a) Sketch a graph of P(t) for one year. Describe how does this population varies with time.

b) Use the graph to determine when during the year the population is maximized. What is that maximum? Is there a minimum? If so, when?

c) Use the graph to determine when the population is growing the fastest. When does it decrease the fastest?

d) Estimate roughly how fast the population is changing on the first of July.

Lab Report: Submit a group report for Parts I, II, III, & IV. Discuss your observations and your results. Be sure to give suitable explanations of what you did to arrive at your results. Also reflect on the results of the lab and what you have learned about nature of and the uses of derivatives.

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