Mathematics 36, Section 1 -- AP Analysis

Lab 2: Differentiability = "Local Linearity"

September 12, 1997

Goals

If there is a unique, finite limiting value as h -> 0 (from both the positive and negative directions), we say f is differentiable at x = a and we call the number obtained in the limit f'(a), the derivative of f at a. Also, if f is differentiable at x = a, then the {\it tangent line\/} at that point is the line with equation

(it's the line through (a,f(a)) with slope f'(a)). Today, we will use Maple graphing to understand the meaning of the differentiability property for a function at a point in another,

Lab Questions

• A) In this question, we will study the graph
  
  y = f(x) = arctan(e^x - x⁵) - cos(x).
  

  To plot this function in Maple, you will need to translate the mathematical formula into a Maple expression. Some useful information: the function e^x is called exp(x) in Maple (the name stands for "exponential"!). Also, Maple knows the usual arctan function as one of its built-in functions.

  • 1) Plot the portion of the graph y = f(x), for -10 <= x <= 10.
  • Our function f(x) satisfies f(-1) = .39922 (approximate value). We want to use the graph to understand whether f(x) is differentiable at x = -1. The lim_{h -> 0} in the definition f'(-1) = lim_{h -> 0} (f(-1 + h) - f(-1))/h can be seen visually if we "zoom in" on x = -1. Here "zooming in" will mean plotting smaller and smaller pieces of the graph taking x in smaller and smaller intervals containing x=-1.
  • 2) For instance, try plotting the function on the new intervals -1.5 <= x <= -.5, then -1.05 <= x <= -.95, then -1.005 <= x <= -.995. What do you see as we ``zoom in'' this way? Keep only your final graph in the worksheet (the others are not necessary -- and long worksheets take longer to print out). Explain what happened in complete sentences, in your own words. (Be careful: the zoomed graph is a part of y = arctan(e^x - x⁵) - cos(x); can it actually be part of a straight line?)
  • 3) A "zoomed" graph like the one on the interval [-1.005, -.995] can be used to compute an approximation to f'(-1). If you place the cursor over the graph in your Maple worksheet and click the left mouse button, the approximate coordinates of the point at the head of the cursor arrow will be displayed in a box under the toolbar in the Maple V Release 4 window. Clicking on two different points will give you the information you need to compute a slope. Maple can also be used as a numerical calculator if you want to compute the slope value. For example, try entering a Maple command like (6.2-5.3)/(.443-.323); When you press ENTER the value will be computed and displayed. Use these ideas to find an approximate value for f'(-1) and for the equation of the tangent line to y = f(x) at x = -1.
  • 4) Repeat the "zooming process" near the point (2,f(2)), describe what happens, include your final "zoomed" graph in your worksheet, and use it to approximate f'(2), and find the (approximate) equation of the tangent line to the graph at that point.
• B) In this question, we will look at a new graph:
  
  y = g(x) = |x - 1| ^2/3 e^-sin(x2)

  In Maple format, this function is:
  

  (abs(x-1))^(2/3)*exp(-sin(x^2))

  (abs is the absolute value function).

  • 1) Plot this function on the interval -2 <= x <= 2.
  • 2) "Zoom" and find an approximate equation for the tangent line to y = g(x) at (1.4,g(1.4)).
  • 3) Now, "zoom in" on the point (1,0). Is this different from the other ones we have done? How? What does this mean about the possibility of defining a value for g'(1)? What happens to the slopes of secant lines through (1,0) and (1+h,g(1+h)) as h -> 0?