Mathematics 135, Section 1 -- Calculus 1

Lab 2: Differentiability, "Local Linearity," and what f' says about f

October 4, 2013

Goals

We have introduced the derivative of a function by considering instantaneous velocities and slopes of tangents. We can attempt to measure the "instantaneous" rate of change of a function f at x = a by studying

lim_{h -> 0} (f(a+h) - f(a))/h

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 tangent line at that point is the line with equation

y = f(a) + f'(a)(x - a)

(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, more graphical way. We will also start to see how properties of f' can be used to deduce properties of f.

Lab Questions

A) In this question, we will study the graph

y = f(x) = sin(x - x⁵) - cos(x).

To plot this function in Maple, we will need to translate the mathematical formula into a Maple expression. Maple knows the usual sin, cos functions as built-in functions.
- 1) Plot the portion of the graph y = f(x), for -2 <= x <= 2 using the command
  
  plot(sin(x - x^5) - cos(x),x=-1..1,scaling=constrained)
  
  We want to use the graph to understand whether f(x) is differentiable at x = .5. The lim_{h -> 0} in the definition f'(.5) = lim_{h -> 0} (f(.5 + h) - f(.5))/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 .25 <= x <= .75 (just change the interval of x values in the plot command above) then .4 <= x <= .6, then .49 <= x <= -.51, etc. What do you see as we ``zoom in'' this way? Explain what happened in complete sentences, in your own words. (Be careful: the zoomed graph is a part of y = sin(x - x⁵) - cos(x); can it actually be part of a straight line?)
- 3) A "zoomed" graph like the one on the interval [.499, .501] can be used to compute an approximation to f'(-1). Estimate this using the coordinates of two points on the graph. You can use Maple as a calculator to compute the estimated slope just by entering the correct slope formula and evaluating. Also find the (approximate) equation of the tangent line. See if you can figure out how to plot the graph together with the tangent line (Maple's online help has the information you need).
- 4) Repeat the "zooming process" near the point (-.75,f(-.75)), describe what happens, include your final "zoomed" graph in your worksheet, and use it to approximate f'(-.75), and find the (approximate) equation of the tangent line to the graph at that point.
B) Next, plot our function f(x) = sin(x - x^5) - cos(x) for x in the interval -1.5 <= x <= 1.5. Take out the scaling=constrained option from before. Maple knows various techniques for computing derivative functions (that we'll learn too starting next week). The derivative f'(x) can be computed with diff(sin(exp(x) - x^5) - cos(x),x). The formula itself is not so interesting though. What we want to look at is the relation between the graphs y = f(x) and y = f'(x). So, keeping your first graph in this part up, also plot the derivative like this:

plot(diff(sin(x - x^5) - cos(x),x),x=-1.5..1.5);
(being sure all parentheses match!) Look carefully at your two plots and think about the answers to these questions:
- On which intervals is f'(x) > 0? What is the graph of f doing on those intervals?
- On which intervals is f'(x) < 0? What is the graph of f doing on those intervals?
- What is true about f'(x) at points where the graph of f reaches the ``top of a hill'' or the ``bottom of a valley''?
- (Harder) What is the graph of f doing when the graph of f' reaches the ``top of a hill'' or the ``bottom of a valley''?