WHEN IS A FUNCTION NOT DIFFERENTIABLE?

Objective: In this lesson, we will determine when a function is not differentiable.
Materials: TI-92 graphing calculator.
Instructions: While working through this discovery lesson, you will find hints for how to type calculator functions in bold typing. It is assumed you have used the TI-92 previously for basic function graphing and algebraic manipulation.

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1. Based on your knowledge of the derivative, identify what properties you think a function might need in order to be differentiable at a point.






2. Let's look at the function f(x) = |x|.
(a) Put this function in y1(x) and graph using ZoomDec (F2, 4 ). y1(x) = abs(x)
(b) Set your Zoom factors to 10 and 10 (F2, C ) and Zoom In (F2, 2 ) with a
center at (0, 0) several times.

• What consistencies appear in the successive viewing rectangles?
  
  
  
  
  

(c) Looking at the function again in the ZoomDec viewing rectangle, you will construct a few limiting secant lines around x=0.
Go to the Home screen and Define f(x) = abs(x) (F4, 1 ).
Then Define secant(x1, x2) = (f(x2) - f(x1))/(x2 - x1)*(x - x1) + f(x1).
Now Define y2(x) = secant(0, 0+h)|h=2. (| is 2nd k )
To see and use y2(x) you need to type it in, copy the output, and paste it
into y2(x) on the Y= Editor. (F1, 5 & 6 )
Now to get a sequence of secant lines, type seq(secant(0, 0+h), h, 2, 0.4, -0.4).
Copy this output into y3(x) and then look at your graph.

• Tell what you think the limit of the slope of the secant slopes is to the right
  of zero. Explain your answer.
  
  

• To the left of zero? (If necessary you can execute the above commands
  again from a text file or just scroll up to each one on your Home screen. In
  either case you will need to change the secant line to secant(0, 0 - h).
  (d) Defend your answers to part (c) by finding the one sided limits of the
  difference quotient on the Home screen. (F3, 3 ) Show your work.
  
  
  
  
  
  
  
• Does the limit of secant lines as you approach x = 0 exist? Why or why not?
  
  
  
  
  

(e) If the limit does not exist, then we say the function is not differentiable at
the point x = 0. If it does exist, then the function is differentiable at the
point x = 0. Using your conclusion from part (d), formulate a theory
about the differentiability of a function at a sharp peak or corner of a
graph.










3. Now let's look at the function g(x) = (x2 + 0.0001)1/2.
(a) How would you test whether g(x) is differentiable at the point (0, 0)?




(b) Compare the graphs of f(x) and g(x) on a very small viewing rectangle
centered at (0, 0). As you Zoom In on g(x) try to keep the center near the
minimum point of the function.






(c) What does this comparison lead you to conclude about the
differentiability of g(x) at the point (0, 0)?

4. Finally, let's take a look at the function h(x) = x1/3.
(a) Look at the graph to propose an alternative place where a function, such
as this one, might not be differentiable.




(b) Zoom with a center of (0, 0) to a very small viewing rectangle and describe
what you see.








(c) Return to the ZoomDec window. Repeat the procedure for drawing in
successive secant lines to this curve close to zero. Sketch your viewing
rectangle.














(d) The slope of the curve at the point (0, 0) is .

(e) Using the calculator's algebra system, how else would you show that the
function is not differentiable at the point (0, 0)? Show your work.








5. The function f(x) = sin(x) is differentiable everywhere, while the function
g(x) = tan(x) is not differentiable for certain values of x.
(a) Decide which of the remaining trig functions you think are differentiable
everywhere and defend your choices.




(b) Compose an argument for the differentiability of continuous/non-
continuous functions.










6. Using all of your results from above, state the three characteristics of a graph
which tell you if a function is not differentiable at a particular point.

(1)


(2)


(3)













Kara Harmon
Champaign Central High School
harmonka@knight.cmi.k12.il.us