CONTINUITY AND INTERMEDIATE VALUE THEOREM

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CONTINUITY
Goal: In this lab, we will distinguish between different types of discontinuities. We will
study a family of piecewise functions, and investigate if there is a value of the parameter
that guarantees continuity at a point.
Introduction: We study the following family of functions:
 x  5, 2  x  1

4.05, 1  x  0

f m ( x)  2.5, x  0
4.05, 0  x  1

 x3  mx,1  x  2
Here m is a parameter, i.e., different values assigned to it will determine different curves.
This is called a family of curves.
In DERIVE, we can use chi function to author piece-wise defined functions. This
function can also be authored using nested if statements as
(IF(-∞ < x < -2, 0, IF(-2 ≤ x ≤ -1, x + 5, IF(-1 < x < 0, 4.05, IF(x = 0, 2.5, IF(0 < x ≤ 1,
4.05, IF(1 < x ≤ 2, - x^3 + m·x, 0)))))))
We can use VECTOR function to determine the functions for different values of m, and
then plot them.
Lab Description: Author ( x  5)chi(2, x, 1) , 4.05chi(1, x, 0) , [0,2.5] ,
4.05chi (0, x,1) , ( x3  mx)chi(1, x, 2) . Now you have the various pieces of the
function on the lines #1-#5 of the stack, and you can gather them by authoring
[#1,#2,#3,#4,#5]  . Try plotting the expression above. DERIVE complains that there are
too many variables for a 2-D plotting window. This is because m is also considered a
variable.
Use the SUB icon to substitute m = 4. Now you have a function of x. Call this function
f_4. Now graph this function. The size of the points can be changed by Options Points
and then choose Large or Small. The scale may not be satisfactory. You may have to
zoom in for a closer look. Make sure the entire domain is visible. Press plot a few more
times to see different color schemes.
1. Annotate and print this graph (for a black and white print, in the plotting window,
go to Options, Printing, and select Black and White only), and embed in your
worksheet.
2. Use pencil to mark the graph with open or closed circles at the end points of the
pieces of the function.
3. State the x-values of the domain of f where it is discontinuous, and classify these
as “holes” or “jumps”.
4. Use the definition of continuity to prove that the function is discontinuous where
you claim. Do this by explaining what part of the definition doesn’t hold: does the
limit fail to exist, is the function not defined at that point or does the function
value disagree with the limit value.
5. If we can redefine f(c) so that lim xc f ( x)  f (c) , then we say that the
discontinuity is removable at x = c. State the x-values where the discontinuity of
f is removable.
We now graph a few members of the family of curves defined above. To do this, author
VECTOR( f m ( x) , m, initial value of m, ending value of m, increment). You can hit the
F4 key to get f m ( x) above. I suggest starting value 3, ending value 6, and increment 1.
Now graph these functions. Make sure the entire domain is visible. Press plot a few more
times to see different color schemes.
6. Which member of the family corresponds to which value of the parameter?
Annotate and print this graph. Embed the graph in your worksheet.
7. You have seen that different values of the parameter m give us different functions
(all belonging to the same family of functions). It appears that there is a value of
m that will guarantee continuity at x = 1. Estimate it, and then confirm your
estimate using algebra (you want the left hand limit = right hand limit = value).
8. Author the member of this family that corresponds to the value of m found in #7,
and then author its variation in which the removable discontinuities found in #5
are removed. You may do this in two steps, or in a single step. Plot and print this
function.
9. In the conclusion of your lab, include the definition of continuity and state when a
function is discontinuous. State when a discontinuity is removable, and what you
understand by a family of curves defined by a parameter.
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