Lab Day 2

Mathematics 131 -- Calculus for Physical and Life Sciences 1

Lab 2: Geometry of Derivatives

October 4, 2004

Goals

Today, we will again be using Maple to draw graphs of functions of one variable (given by symbolic formulas) and their derivatives. On the mathematical side, we will study the relation between functions and their derivative functions.

Recall that we have introduced the derivative function f'(x) for a function f(x). In Maple, the derivative function of f is written D(f). (In case you're wondering, Maple ``knows'' shortcut rules for computing derivatives that we will also begin to study in class later this week!)

Lab Questions

A) In this question, we will study the function

f(x) = sin(x) - x cos(2 x).
and its derivatives. To plot this function in Maple, we will first translate the mathematical formula into a Maple expression, and define a function. Do this command first:

f := x -> sin(x) - x*cos(2*x);
This is slightly different from what we did last time, but it allows us to evaluate the function, compute its derivative(s), etc. in a natural way.
1. Plot the portion of the graph y = f(x) on the x-range -4 <= x <= 4, using
  
  plot(f(x),x=-4..4);
2. Now, plot the portion of the graph y = f'(x) for -4 <= x <= 4, on a separate set of axes, using
  
  plot(D(f)(x),x=-4..4);
3. Answer the following questions in a text region in the worksheet:
  - a. Over which intervals in the range -4 <= x <= 4 is f'(x) positive?
  - b. What is true about f(x) on each of the intervals from part a? Explain why this is true and why the same relationship between f and f' should be true whenever f' is positive on an interval.
  - c. Over which intervals in the range -4 <= x <= 4 is f'(x) negative?
  - d. What is true about f(x) on each of the intervals from part c? Explain why this is true and why the same relationship between f and f' should be true whenever f' is negative on an interval.
  - e. What is true about f(x) at each point where f'(x) = 0?
4. As we said in class last time, the derivative of the derivative of a function f(x) is called the second derivative of f and written f''(x). In Maple, the second derivative function of f is (D@@2)(f) (note the extra set of parentheses around the D@@2 -- they must be included). Plot the second derivative of f(x) = sin(x) - x cos(2 x) on the same interval as in the previous questions.
5. Answer the following questions in a second text region in your worksheet:
  - a. Over which intervals in the range -4 <= x <= 4 is f''(x) positive?
  - b. What is true about f'(x) on each of the intervals from part a? What is true about f(x)?
  - c. Over which intervals in the range -4 <= x <= 4 is f''(x) negative?
  - d. What is true about f'(x) on each of the intervals from part c? What is true about f(x)?
B) Repeat parts 1-5 of question A for the new function

g(x) = |x - 1|^1/5 sin(x^2)
on the interval -3 <= x <= 3 -- that is, answer the same questions, but do everything on the new, smaller, interval of x-values.
1. First define the function as follows
  
  g := x -> abs(x-1)^(1/5)*sin(x^2);
  (abs is Maple's notation for the absolute value function). Then plot as before.
2. Try plotting the derivative ``straight'' first. What do you think is happening at x = 1 on the graph y = g'(x)? Then try options like this in the plot command to restrict to a smaller window in the xy-plane and tell Maple that you think this function might have a discontinuity at x = 1.
  
  plot(D(g)(x),x=-3..3,y=-10..10,discont=true);
3. Answer the parts of question 3 as before, and in 3e, mention any points x = c where g'(c) does not exist.
4. You will need to do something similar to what you did in part 2 above for the second derivative function. Be sure you have adjusted the y-range so that you are sure you are seeing all of the graph y = g''(x)(!)
5. Answer the parts of question 5 as before.

Assignment

The lab write-up (a paper print-out of your worksheet, including all graphs and explanations in text regions) is due on Wednesday, October 13 (after Columbus Day break). Resize all of your plots smaller than the default size to conserve paper when you print out your work.