Stephani Eckelkamp
ÒGraphing Fun with CalculusÓ
Optimization is an
idea that most students in Calculus classes become very familiar with! Here is an exploration of optimization
using GSP and Excel to help give a visual representation for exactly what optimization
looks like.
For this investigation
I will be looking at the following problem:
Explore problems of
maximization such as the lidless box formed from a 5Ó x 8Ó sheet with a square
removed from each corner.
A good place to start
is by constructing a model in GSP.
Here is the one I came
up with
Rectangle ABCD is the 5Óx 8Ó piece of paper I started with. The squares cut out are hÓ x h.Ó
l is the length and w is the width of
the lidless box formed when hÓ x hÓ squares are cut out.
Question:
What h will maximize the volume of the
lidless box?
Make a guessÉwhat do
you think it is?
Click this GSP
link to manipulate.
**Note the chart in
the top right corner**
What does the chart represent? What formula is l x w x h ?
What do you think the graph of the
volumes in relation to the height will look like? Why?
Here is a picture of the graph done in
GSP
Link
to GSP graph and spreadsheet
Here is the graph done in excel
Link to excel graph and spreadsheet
For this problem the maximum h is approximately
1.5.
Notice in both graphs that the start
and end at 0. Why is this true for
this problem? How does this relate
to closed intervals? What are the
closed intervals for this problem?
Another exploration
Question:
A rectangle is enclosed in a parabola
with the equation f(x) = 4 - x^{2 }.
The vertices of the rectangle are
(-x,0), (-x,y), (x,0), and (x,y).
Here is a picture of the sketch of this
graph
Click here
to open the GSP file to manipulate this graph
**Note the table to at
the top right
What do you think the area graphed will
look like?
What might the graph look like in
relation to the length and the width of the rectangle?
Here are both
of them graphed on the same plane.
Which color goes with width and
length? Hint: Look back at the original graph.
Explain why one graph has a greater
domain than the other.
Here is the chart used for these graphs
w (in) |
l (in) |
Area |
7.59 |
0.01 |
0.06 |
7.07 |
0.52 |
3.69 |
6.6 |
0.96 |
6.37 |
6.26 |
1.27 |
7.96 |
5.82 |
1.64 |
9.55 |
5.54 |
1.86 |
10.32 |
5.25 |
2.08 |
10.91 |
4.85 |
2.36 |
11.45 |
4.33 |
2.69 |
11.66 |
3.98 |
2.89 |
11.52 |
3.4 |
3.19 |
10.86 |
2.69 |
3.49 |
9.4 |
2.19 |
3.66 |
8.03 |
1.62 |
3.81 |
6.19 |
1.09 |
3.91 |
4.27 |
0.65 |
3.97 |
2.58 |
0.17 |
3.99 |
0.68 |
0.02 |
3.99 |
0.09 |
The maximum for this problem is between
4 and 4.8.
Choose an optimization word problem,
create a picture in GSP and graph to find an approximation of the maximum.
Return