MEAN VALUE THEOREM
with Desmos
GETTING STARTED
0. Open a blank graph: https://www.desmos.com/calculator.
1. Plot the function, and define the three variables:
f (x) = x3 + 3x2
a = −2.5
b = 0.5
c = −2
SECANT LINE
2. (a) Plot the points:
(a, f (a))
(b, f (b))
(b) Determine the slope of the line between the two points (in general; not just for the given
values):
mab =
(c) Plot the line, with the domain restricted {a ≤ x ≤ b}.
y=
3. Put the last four expressions in a folder called “Secant Line”.
TANGENT LINE
3. Define the derivative function:
fprime (x) =
d
f (x)
dx
4. Hide fprime (x) by clicking on the
to see it on the graph.
. We need this function for our calcultions, but don’t need
5. (a) Plot the point:
(c, f (c))
(b) Determine the slope of the tangent line at
(c, f (c)):
mc =
(c) Plot the line, with the domain restricted
to {a ≤ x ≤ b}.
y=
6. Put the last four expressions in a folder called
“Tangent Line”.
1
FINDING A SOLUTION
7. Define a new variable:
yes = {c < 0 : 1; 0}
8. (a) Plot the equation of the tangent line again, but make this copy green.
y=
The lines will be overlapping so the color may appear brown.
(b) We’re going to edit the slopes of the two tangent lines slightly. Change whatever you
currently have for the slope of the red line to be
mc
1 − yes
and the slope of the green line to be
mc
yes
(c) You should only see one of the two lines now. Which one, and why?
(d) Slide the point (c, f (c)) until the line changes color. Why did it change?
9. We want the line to be green when mab = mc . For computational reasons, we’ll actually let
the line be green when mab ≈ mc .
(a) Define a new variable:
epsilon = 0.1
(b) Change the definition of yes so that:
yes = {|mab − mc | < epsilon : 1; 0}
10. You’re done! Use your graph to determine (approximately) the value(s) of c between a and b
which are given by the Mean Value Theorem. The smaller you make epsilon , the more precise
your approximation will be:
c1 =
c2 =
2