Download AP Calculus AB Hands-On Activity: Rolle`s and Mean Value

Name:
_
AP Calculus AB
Hands-On Activity: Rolle's and Mean Value Theorem
The following activity will help you uncover two of the most important theorems of
differential calculus. Rolle's Theorem is one of those rare calculus theorems that makes
a lot of sense right away. The Mean Value Theorem is not really difficult either, and it
usually appears on the AP test frequently.
1. Given f(x) = x 3
6x 2 + 12x - 5. Draw the portion of the graph indicated by the
-
axes below such that
°
~
x ~4.
5
4
3
2
2
3
4
2.
Draw the secant line that connects x=1 to x=3 and calculate the slope of the secant
line.
4-2
m=--=1
3-1
3. Are there any places on the graph where the tangent line to the graph appears to
be parallel to the secant line you've drawn? How many times does this happen on
[0,3]? Draw these tangent lines that appear to be parallel to the secant line.
Yes, 2 places. Once between x=1 and x=2 and again between x=2 and x=3
4. What does it mean geometrically if those tangent lines are parallel to the secant
line?
They share the same slope.
5. Write your conclusion to problem 4 as a mathematical formula, If you solve this
for x (you'll have to use your calculator), you will find the x-values for the
tangent lines. What are they?
This should sgtate that the secant and tangent slopes are the same:
f(3) - f(l)
3-1
= j'(x)
6. Time to be generic. Given a continuous, differentiable function g(x) on the
interval [a,b], complete the following:
• Give the slope of the secant line from x=a to x=b.
g(b) - g(a)
b-a
• Give the slope of the tangent line to g(x) at any point on its domain.
g'(x)
Use your answers above to fill in the blanks and complete the Mean Value
Theorem:
The Mean Value Theorem: Given a function g(x) that is continuous and
differentiable on a closed interval [a,b] for which g(b) - g(a) , the slope of the
b-a
secant line, equals g'(x) , the slope of the tangent line.
7. Illustrate the Mean Value Theorem graphically using the graph below of g on
[a,b] .
8. Translate the Mean Value Theorem into a statement about rates of change.
At least once on the interval [a,b] the instantaneous rate of change quals the average rate
of change for that interval.
9. Rolle's Theorem is a specific case of the MVT, which applies whenever
g(a)=g(b). What is the slope ofthe secant line for such a function? What is
guaranteed by the MVT as a result, and what does that m.ean geometrically?
M=O
Slope of tangent = 0 so there is a horizontal tangent. Geometrically it guarantees the
existence of a max or a min.
10. Fill in the blanks to complete the theorem:
Rolle's Theorem: Given a function f(x) that is continuous and differentiable on the
closed interval [a,b] and _f(a)_ = _f(b)_, then there exists at least one x on [a,b]
_
such that _I' (x) = 0
11. Draw a function g(x) on the axes below that satisfies Rolle's Theorem on [a,b] but
satisfies it more than one time.
5
3
-2
3
The Mean Value and Rolle's Theorems
AveYCJ\~e slopes ::::- i.",sh:.. .",+- slopes
" IfJ(x) is continuous and differentiable on the interval [a,b] and a < c < b,
calculate the average rate of change ofJ(x) over [a,b] and the instantaneous rate
of change ofJ(x) at X= c.
.
2. Explain the geometric implications of the mean value theorem.
-4
~3
-2
-1
··Fig4felil
4jU71cti¢n g(x) th(i~ i~(ontinyous and
differ~1itzable
on{-5,5i"
.... '. .'
.4", How Ipany times does the function h(x) graphed in Figure 15-3 satisfy the mean
value theorem on the interval ['::'6,6]?
5. At what value(s) of x does f(x)
on the interval [O,1J?
= :x!' - 2x 2 - 4x + 1 satisfy the mean value theorem
..
6. At what value(s) of x does the continuous and differentiable function
g(x) = ax" + bx+ csatisfy the mean value theorem on the interval [O,h)?
Assume that a, b, and care nonzero rea] numbers .
. c.': ~
- .7, . A'P8licerrt~ncl()cks~cdm1Uut~r'Sc$Jle~(Lat~6il1Ph:asheentersf,l tpnl,l~Lwb:o~e
length is exactly 0:75 miles; Asectind~fficefmeasU:teslhetonnilUter'S'$R~~dc
at4?mph<¥S])e'e~it~thetupnTL~i3second~Jater~nd tickets the driver.for ,'.•
exceeclingtheposteclspeedliinit9f oOmph'.: Use the mean valuetheorem to.
justify the speeding charge levied byihe officer, even though the driver was
neither exceediIlg the posted speed limit while entering nor while exiting the
tunnel..
.'.
.
.
/
8~
q
I
Describe the difference between the mean value theorem and R
.
.'
Verifythatj(x)
.
. . '. . '.'
.
....
.
0
nest
'. h:eorem,
= 40x" + .22:x!' -;91x2 .: 58x+ 15 has at least one horizontal tan
·'ineortllie\ri~~lf~~~J:';'·
...