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You will need to download
the free DPGraph Viewer to view the
DPGraph animations on this page.
| A) Find the point on the graph of y =
sin(x) that is
closest to the point (2,-1). Give your answer accurate to three places to
the right of the decimal point. Click on My
Example to see an animation showing the graph of y = sin(x) in blue, the
distance from the graph of y = sin(x) to the point (2,-1) in green, the function
giving the distance in red along with the distance from the red distance
function to the x-axis in green. Click
here for a smaller version of the same animation (loads a little faster).
Here is a Quicktime animation for
this example. Here is a
Winplot demonstration for this problem. You may need to download the file to your
desktop and then use the freeware
Winplot to open the file (by opening Winplot, clicking on Window,
clicking on 2-dim, clicking on File, clicking on Open, and then opening
MaxMin1 from your desktop. You can use the slider to vary the value of
A. The A-value gives you the x-value at a point on the graph of y =
sin(x). In this demonstration the line tangent to the graph of y =
sin(x) is also shown. Notice the perpendicular relationship when the
distance d is minimized. |
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d = [(x - 2)2 + (sin(x) + 1)2]1/2
d2 = f(x) = (x - 2)2 + (sin(x) + 1)2
f '(x) = 2(x - 2) + 2(sin(x) + 1)cos(x) = 0
if x is approximately
0.815, 0.978, or 3.069
What is
the answer to the question?
Take a look at
the graph of
d(x) = [(x - 2)2 + (sin(x) + 1)2]1/2
using the
window xMin = 0.75, xMax = 1.05,
yMin = 2.095,
and yMax = 2.0956.
The graph
is shown at the right.
Answer:
approximately (3.069,0.073)
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B) Find the
dimensions and area of the largest rectangle that can be constructed in the
region bounded by the x-axis and the graph of the parabola y = 9 - x2
with one side of the rectangle on the x-axis and the vertices of the opposite
side lying on the graph of the parabola. Click
here to see an animation showing the graph of the parabola in blue, a
sequence of the inscribed rectangles in red, and the area function in red
(shifted four units to the right) with an animated point moving along the graph
of the area function that corresponds to the area of each rectangle as it is
drawn. Quicktime animation.
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C) A
total of 8 meters of fencing are going to be used to fence in a rectangular cage
for pets and divide it into three smaller cages as shown in the animation.
Determine the overall dimensions that will yield the maximum total enclosed
area. In the animation the total area function is graphed in red.
The blue point moving along the area function corresponds to the changing size
of the rectangular cage. The perimeter of the cage is shown in blue and
the added dividers in green. Let x stand for the length of one of the
sides (with 2 sides needed of length x) and y stand for the length of the other
side (with 4 sections of fencing needed of length y).
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2x + 4y = 8 so y = 2 - (1/2)x
A(x) = xy = x[2 - (1/2)x] = 2x - (1/2)x2
A'(x) = 2 - x = 0 if x = 2 and y = 1
The total area will be maximized with dimensions of 2 meters by 1 meter.
Quicktime animation.
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| D) A rectangular piece of
material measuring 4 ft by 3 ft is to be formed into an open topped box by
cutting equal sized squares out of each corner and folding up the
sides. Determine the size of the squares to be cut out if the
resulting box is to have the maximum possible volume. Click
here to see an animation with scales or on the figure at the right
below to see an animation without scales. Click
here for a 3-D animation without scales and click
here for a 3-D animation with scales. In the animation we see
the changing shape of the material after various sized squares are cut out
of the corners along with the volume function in red shifted to
the right. The 3-D animations show the changing shape of the box
after the sides have been folded up. The animated point moving along the graph of the volume
function corresponds to the changing box construction. Quicktime
animation 3D
Quicktime animation DPGraph
animation Here is a more psychedelic
DPGraph animation. Here are two DPGraph pictures of the
box. These can be looked at for entertainment for 15 minutes or so
if you have no life. Picture
of the box one Picture
of the box two (this is picture one with shading) |
x
= the length of a side of each of the squares
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E) (See a
picture under Section Project in the exercises for section 12.9 in your
text.--This is a Calculus I problem.) An oil company wishes to construct a
pipeline from its offshore facility A to its refinery B. The offshore
facility is 2 miles from the shore, and the refinery is 1 mile inland. A
and B are 5 miles apart as indicated in the figure in your text. The cost
of building the pipeline is $3 million per mile in the water, and $4 million per
mile on land. The total cost depends on the location of the point P in the
textbook figure, P being the point where the pipeline meets the shore. Let
x be as indicated in the figure and determine the value for x that will minimize
the total cost and approximate this minimum total cost. Click
here to see my animation for this problem. Quicktime
Version In the animation A is at
(0,17), B is at (4,14), and the distance between A and B (5 miles) is indicated
in black. Blue indicates the pipeline distance through water and orange
the pipeline distance on land. The distance from the point (0,15) to the
animated blue point on the green line is x. The total cost function, C(x),
is in red and the animated blue point moving along the total cost function
corresponds to the animated blue point (P) on the green shoreline. The
cost function is given below. Can you see how it was developed?
Finish the problem.
F) This
relates to Example 4 on page 214 in your textbook. Two posts, one 12 feet high and the other 28 feet high, stand 30
feet apart. They are to be stayed by two wires, attached to a single
stake, running from ground level to the top of each post. Where should the
stake be placed to use the least wire. This is a textbook example so the
solution is in your text. Click here to
see an animation of the possibilities along with the total length of wire
function. Quicktime animation
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G)
This relates to Example 5 on page 222 in your textbook.
Four feet of wire is to be used to form a square and a circle. How
much of the wire should be used for the square and how much should be used
for the circle to enclose the maximum total area? This is a textbook
example so the solution is in your text. Click
here or on the picture at the left to see an animation of the
possibilities. The blue graph is of the total area function.
The animated point moving along the blue graph corresponds to the changing
sizes of the circle and square. Quicktime
animation |
H)
In a problem similar to the one above, four feet of wire is to be used to form a square and
an equilateral triangle. How
much of the wire should be used for the square and how much should be used
for the triangle to enclose the maximum total area? How much for
each to enclose the minimum total area?
We can see from the work on the right that we
will maximize the total area if we use all four feet of wire to form a
square and the maximum total area will be 1 square foot. We can also
see that if we use approximately 1.73986 feet of the wire in forming the
square and the rest in forming the equilateral triangle we will minimize
the total area at about 0.4349645 square feet. Click
here or on the picture above to see an animation of the various
possibilities. The curve in red is a graph of the total area
function. The length of the vertical red line segment connecting the
two animated blue points represents the total area. Click
here to see the animation (without the vertical red line segment) in a
scaled axis setting. Quicktime
animation |
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| I) A blue boat is 30 nautical
miles due east of point A and traveling due west at 12 nautical miles per
hour. A green boat is 20 nautical miles due north of point A and
traveling due south at 15 nautical miles per hour. How long until
the two boats are closest together and how close do they get? Click
here or on the picture below to see an animation of the next
three hours of the boats' movement. The boats are not drawn to
scale. The endpoints of the red line segment connecting the two
boats represent the location of the boats. The purple graph is
representing the distance between the boats as a function of time. Quicktime
animation Second Quicktime
animation (This one requires the user to open it using Quicktime
Player but gives more options.)
Consider point A to be at the center of a
rectangular coordinate system. Let x represent the x coordinate of
the blue boat's position and let y represent the y-coordinate of the green
boat's position. Here is a
link to a
video presentation of this problem that includes audio of me
describing the problem.
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x = 30 - 12t, y = 20
- 15t
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| L) Section 3.7 #26
Here is a solution finding the area as a function
of x.
Area in terms of y
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Click
here for an animation, no scales.
Quicktime
animation
Click
here for an animation with scales. In
the animations the area function (of x) derived on the left is shown in red with
an animated blue point moving along the area function corresponding to the
area of the animated triangle.
Area in terms of angle a
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| M) Section 3.7 #30
Thus the least amount of paper is used if the
printed material is on a 6 inch by 6 inch square and therefore the
dimensions of the paper that would minimize the area of the paper would be
9 in. by 9 in. |
Click here
or on the picture above to see an animation. The red graph is a
shifted, stretched graph of the area of the paper as a function of
x. The blue point on the red graph corresponds to the area of the
paper (area of the green rectangle). When the blue point reaches the
low point on the red graph then the amount of paper is minimized.
The blue rectangle corresponds to the printed material.
Quicktime
animation
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| N) Find the point on the graph
of the parabola whose equation is y = x2 + 3 closest to the
point (2,2). State your answer accurate to 8 digits to the right of
the decimal. Click on the picture at the right to see an
animation. The parabola is graphed in blue and the function giving
the distance from the parabola to the point (2,2) is graphed in red.
The green line segments represent the changing distance between the
parabola and the point.
EXTRA CREDIT: Click
here to see a blow-up of the animation relating to the picture at the
right with the addition of the line (drawn in blue) tangent to the graph
of the parabola at the blue animated point. The EC is to tell
me what I am trying to show you by drawing this tangent. If you can
figure this out then you can receive more EC by proving it. Quicktime
animation |
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EXTRA CREDIT 2: Find the measure of a base
angle that would produce the isosceles triangle of maximum area constructed in a
circle of radius 78.549 pixels with one vertex at the center of the circle and
the other two vertices on the circle. Click
here to see a Geometer's Sketchpad java applet demo for this problem. Grab point C
with your mouse/cursor and drag it around the circle to see how the area changes
as the measure of angle ACB changes. You can grab and change other things
with your mouse too. Press the letter "R" on your keyboard to
revert to the original state. The extra credit is to construct an
area function and use calculus to prove the result indicated by the applet
linked to here.
Click
here to see a Geometer's Sketchpad java applet demonstrating inscribing a
rectangle in a circle and the corresponding graph of the function giving the
area of the rectangle (ABCD) as a function of the length of one of the sides
(AB). In the initial state of the applet grab the point on the right (D) with your mouse and cursor and move
it about the circle to form the rectangle. Points A, C, and D are the same
in the initial state. The point being dragged will eventually become point C
when a rectangle is formed. Point A will separate from point D when a
triangle or rectangle is formed. What type of rectangle results in the
largest area? You will see that you can also investigate finding the
triangle of largest area that can be inscribed in a semicircle. Points C
and D are the same point in the triangle application. The center point and
the point on the circle not on a blue line can be dragged to change the location
of the center and change the size of the circle. Press the letter "R" on your keyboard to
revert to the original state.
Click
here to see a Geometer's Sketchpad java applet demonstrating inscribing a
triangle in a semicircle. You can grab either point A or point B with your
cursor and move the point around the circle. You will be able to see the
values for the area and perimeter of triangle ABC change as you move point A (or
B) around the circle. You will also see points moving along the graphs of
the area and perimeter functions (functions of the length of AB).
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