Download Activities - WVU Math Department

Mat
h
154
Lab
1. Recall the Extreme Value Theorem states that if f is continuous on a
closed interval [a,b ] then f attains an absolute maximum value f (c )
and an absolute minimum value f (d ) at some numbers c and d in [a,b ].
Consider the piecewise function on lab page 1, also shown in the
eCampus lab quiz. Hit the graph button and you can move the horizontal
line to intersect the graph of the function. Drag a to indicate the
function's global maximum. What is the maximum value of f on [-10,10]?
Answer:
2. What is the minimum value of f on [-10,10]?
Answer:
3. What can you say about the point (1,2) on the graph of f ?
a. f has an absolute maximum of 2 at x=1.
b. f has a relative maximum of 2 at x=1.
c. f has an absolute minimum of 2 at x=1.
d. f has a relative minimum of 2 at x=1.
e. f has an absolute maximum of 1.
f. f has an absolute minimum of 1.
Answer: _____
4. The function graphed in question 1 is a piecewise function. Indicate which equations might
make up "the pieces."
a. x + 9
b. x + 3
c. 9 - x
d. 3-x2
e. x2-2x+3
f. x2+3x+5
Answers: ____________________
5. Look at lab page 2, or the new function graphed in the eCampus quiz
page above. It is the graph of f (x)=x - x 3 in the standard window.
Use the +/- buttons to change the window to the interval [-1,1], and
change the limits on the slider for a to go from -1 to 1 as well. Then
you can see a displayed to two decimal places. What is a correct to two
decimal places? (If you can't quite decide from the a slider alone, the
black trace point that you can move with the arrow keys has its
coordinates displayed at the top left of the grapher, and it has more
accuracy.)
Answer:
6. You might have guessed that the maximum value happens symmetrically between the zeros
of the function, which in this case would be at x = 0.5. Do a calculation to find the true x value
directly, by applying Fermat's theorem that the local maximum in this case should happen where
f'(x)=0.
a.
3/2
b.
3 /3
c.
2 /3
2/2
d.
e. 0.6
Answer: _____
7. And what is the exact maximum value of f (x) ?
a.
3 /3
b.
2 3/3
c.
3 2/4
2 3/9
e. 3 2 / 8
d.
Answer: _____
8. Consider the piecewise graph f on lab page 3, or graphed in the eCampus quiz page above.
Mark all of the critical numbers of f.
a. -10
b. -6
c. -5
d. -3
e. -2
f. -1
g. 0
h. 2
i. 3
j. 4
k. 5
l. 10
Answers: ____________________
9. Look at page 4, or the applet above this question. A cubic function is a polynomial of degree
3. That is, it has the form ax3+bx2+cx+d, where a is not 0. Experiment with various cubic
functions in the graphing utility to decide possibly how many critical numbers a cubic function can
have.
a. 0
b. 1
c. 2
d. 3
e. 4
f. 5
Answers: ____________________
10. Experiment with the grapher to decide which of the following are true. Mark the true ones.
a. If f'(c)=0 then f has a local maximum or local minimum at c.
b. A function defined on a closed interval might have a maximum value but no minimum
value.
c. The absolute maximum of a function might be a negative number.
d. A linear function on a closed interval attains its extreme values at the endpoints of
the interval.
e. An even function has to have an even number of critical numbers.
Answers: ____________________