Mathematics 136, section 4 -- AP Calculus

Lab Day 3: Families of Curves and Critical Points

October 1, 2003

Background

In class on Monday we introduced the language of critical points, local maxima and local minima. Today we will apply those to study a new topic: families of curves. In many circumstances, we might need to understand the behavior of a collection of functions that are defined by a formula f(x,a) that contains both the variable x and some other constant parameter a. The graphs form a family of curves in the plane. As a varies, the positions and even the types of critical points of f(x,a) can change. We will investigate some interesting examples of this using Maple today. In the process, we will also start to see some of the power of Maple for performing symbolic and numerical calculation (in addition to graphics).

Lab Activities

• A) A ``warm-up''. One family of curves is the collection of the graphs of the cubics y = x³ - ax. Plot several of these, including some with both positive and negative a-values. Describe how the critical points of the curves in this family vary with a.
• B) Next, we will study the family of functions defined by
  
  f(x,a) = 1/((1-x²)² + 2 a x²)
  
  This family comes from the study of resonance phenomena in forced oscillations in physics. We want to study the functions in this family for values of a with 0 <= a <= 1. To work with these functions in Maple, we can define a function of two variables like this:
  
  f := (x,a) -> 1/((1-x^2)^2 + 2*a*x^2);
  
  
  1. Use the following plotting command to plot three functions from the family together:
     
     plot([f(x,0.2),f(x,0.5),f(x,0.8)],x=0..5,color=[red,green,blue]);
     
     (This plots the graph y = f(x,0.2) (i.e. a = 0.2) in red, y = f(x,0.5) in green and so forth.) Try some other values too and formulate a conjecture (that is, an educated guess) about how the location of the critical point of the function f(x,a) depends on the value of a. Also try to formulate a conjecture about how the y-coordinate at the local maximum depends on a.
  2. One of the really powerful features of Maple is that it contains a number of features for computing symbolically and numerically as well as plotting. For instance, we can compute the derivative of f(x,a) as a function of x with the following command (the D[1](f) means compute the derivative with respect to the first variable, i.e. x):
     
     df:=D[1](f)(x,a);
     
     Do this and verify that Maple's answer is correct by computing the same derivative by hand.
  3. Next, we can take df, set it equal to zero, and solve for x using:
     
     critpoints:=solve(df=0,x);
     
     Execute this now. The output should list three values, but only one will be (strictly) positive. This is the x-value of the critical point. How does this correlate with what you said in part 1 above?
  4. Finally, to understand the y-coordinate of the local maximum (the ``critical value''), we can substitute the critical point into f(x,a). Do that by executing
     
     critval:=f(sqrt(1-a),a);
     
     You can study this by plotting critval as a function of a (for example. How does this match what you said about the height of the peak depends on a? (Caution: The function critval has a vertical asymptote at a = 0. If you plot critval, you probably want to ``cut it off'' to the right of the asymptote to avoid getting huge numbers out!)
• C) Finally, we will study the family of functions defined by
  
  g(x,a) = e^-axsin(x)
  
  This family comes from the study of damped oscillations in physics. We want to study the functions in this family for values of a with a >= 0.
  1. Plot the curves of the family with a = .3, .4, .5, 1 for x from 0 to 4*Pi (pick a new color for the fourth one -- Maple knows most ``standard colors'') and formulate a conjecture about how the locations of the critical points of the function g(x,a) depend on the value of a. Also try to formulate a conjecture about how the y-coordinate at the local maximum depends on a.
  2. Compute the derivative of g(x,a) as a function of x with the following command (the D[1](g) means compute the derivative with respect to the first variable, i.e. x):
     
     dg:=D[1](g)(x,a);
     
     Do this and verify that Maple's answer is correct by computing the same derivative by hand.
  3. Take dg, set it equal to zero, and solve for x using:
     
     critpoints:=solve(dg=0,x);
     
     Execute this now. Check the output by solving the equation by hand. Why do you get just one answer when it looks like there should be several critical points?
  4. Where are the first positive critical points for the curves in the family with a = .3 and a = 1? What are the y-coordinates there (the critical values)? How does this match up with what you said above?

Lab Assignment

Prepare a Maple worksheet showing all your computations, graphs, and answers for questions A,B,C above. Due: Monday, October 6.