If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
007. `query 7
Question:
`q **** Query class notes #07 ****
Explain how we obtain the tangent line to a y = k x^3 function at a point
on its graph, and explain why this tangent line gives a good approximation to
the function near that point.
Your solution:
Confidence Rating:
Given Solution:
`a If we know that y=kx^3, as in the sandpile model, we can
find the derivative as y = 3kx^2.
This derivative will tell us the rate at which the volume
changes with respect to the diameter of the pile.
On a graph of the y = k x^3 curve the slope of the tangent
line is equal to the derivative.
Through the given point we can sketch a line with the
calculated slope; this will be the tangent line.
Knowing the slope and the change in x we easily find the
corresponding rise of the tangent line, which is the approximate change in the y
= k x^3 function.
In short you use y' = 3 k x^2 to calculate the slope, which
you combine with the change `dx in x to get a good estimate of the change `dy in
y. **
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Question:
`qQuery class notes #08
**** What equation do we get from
the statement 'the rate of temperature change is proportional to the difference
between the temperature and the 20 degree room temperature'? What sort of graph do we get from this
equation and why?
Your solution:
Confidence Rating:
Given Solution:
`a y proportional to x means that for some k we have y = k x.
The rate of change of the temperature is the derivative
dT/dt.
The difference between temp and room temp is T – 20.
So the statement says that
dT/dt = k (T – 20).
Whenever the rate dT/dt is proportional to a quantity like T
- Troom, which is a linear function of T, the result is that T-Troom is an
exponential function of clock time. **
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Question:
`q1.2.10 (was 1.2.08
graph matching y = `sqrt(9-x^2)
Describe the graph that matches this function and explain how
you know this is the graph
Your solution:
Confidence Rating:
Given Solution:
`a It turns out that this graph is in fact the upper half of
a circle of radius 3 centered at the origin.
We can show that this graph is part of a circle:
If y = `sqrt(9 - x^2) then y^2 = 9 - x^2 so x^2 + y^2 = 9.
This is the Pythagorean Theorem for a right triangle defined
by center (0,0) and legs x and y; we see that the square of the hypotenuse is 9
so the hypotenuse is 3.
The hypotenuse represents the radius of the circle. **
PRETTY GOOD BUT ERRONEOUS STUDENT ANSWER
If you take the square root of 9-x^2 you get y = –x + 9 which
is linear.
INSTRUCTOR COMMENTS
Good try, and well expressed. However sqrt(9 - x^2) is
not 9 - x. It's not 3 - x either.
Be careful of the fallacy that sqrt(a^2 + b^2) = a + b. Seems like it
ought to be so, but it's not.
For example, sqrt(3^2 + 4^2) = sqrt( 9 + 16) = sqrt(25) = 5.
We see that sqrt(3^2 + 4^2) is 5.
So sqrt(3^2 + 4^2) is not 3 + 4 = 7.
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Question:
`qExtra problem: (was problem
1.2.10) If air freshener initially
contains 30 grams, what is the formula for the number of grams present if 12% of
the amount present evaporate per day?
Your solution:
Confidence Rating:
Given Solution:
`a If 12% evaporates per day then 88% remains at the end of
each day.
That is, the growth rate is -.12 so the growth factor would
be 1 + (-.12) = .88 and the function would be
Q = 30 gram * .88^t. **
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Question:
`qExtra Problem (was problem 1.2.18): What is the formula for exponential function
whose graph passes through the points (1,6) and (2,18)?
Your solution:
Confidence Rating:
Given Solution:
`a STUDENT ANSWER AND INSTRUCTOR COMMENT: I did not use simultaneous equation to solve.
I just tried different number for the given original and the number which
would be raised to the x power. I then plugged in the two points and
found an equation.
}INSTRUCTOR COMMENT: Trial and error might work for this
problem but only simultaneous equations will work if the numbers are less
obvious, so you need to understand that
procedure.
Using P = P0 * a^t we plug in the coordinates of the first
point to get
6 = P0 * a^1.
For the second point we get
18 = P0 * a^2.
Dividing the second equation by the first we obtain
18/6 = (P0 a^2) / (P0 a^1) or
3 = a,
so we know that a = 3.
Substituting this into the first equation we find that
6 = P0 * 3^1, which we easily solve for P0 to obtain
P0 = 2.
So our model P = P0 a^t becomes
P = 2 * 3^t. **
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Question:
`q1.2.32 (was 1.2.28
graph of y=`sqrt(x+1)
Describe your graph, including coordinates of intercepts,
whether increasing or decreasing (if both, where it does each), and concavity.
Your solution:
Confidence Rating:
Given Solution:
`a This graph intercepts the x axis where y = 0, which occurs
when x+1 = 0 or x = -1.
As x increases the square roots increase, but more and more
slowly (just consider the square roots for x = 0, 1, 2, 3 and you'll see how the
values increase by less and less each time). So the graph will be increasing at a
decreasing rate, which means it is concave downward.
The square root of a negative number is not a real number, so
this function is undefined when x + 1 < 0, which happens when x < -1.
So the function is undefined, and there is no graph at all, for x < -1. **
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Question:
`q1.2.56 (was 1.2.52
pts of intersection of x+y=7 and 3x-2y=11
What are the point(s) of intersection?
Your solution:
Confidence Rating:
Given Solution:
`a This system could be solved by elimination but that
solution is confined to linear equations (for which it is very appropriate) and
won't be demonstrated here. The methods
used here can be used with nonlinear equations.
We can solve both equations for y and then set the two
results equal:
The first equation is x + y = 7. Subtract x from both sides to get
y = 7 - x.
The second equation is 3x - 2y = 11. Subtract 3x from both sides:
-2y = 11 - 3 x.
Divide both sides by -2:
y = (11 - 3x) / (-2) so
y = -11/2 + 3 x / 2.
Now set both expressions for y equal to one another:
7 - x = -11/2 + 3 x /
2. Add x and 11 /2 to both sides:
7 - x + x + 11/2 = -11/2 + 3 x / 2 + x + 11/2.
7 + 11/2 = 3 x / 2 + x.
Put each side over common denominator:
14 / 2 + 11 / 2 = 3 x / 2 + 2 x / 2. Add:
25 / 2 = 5 x / 2.
Multiply both sides by 2/5:
2/5 * 25 / 2 = 2/5 * 5x / 2.
Simplify
5 = x.
So at the point of intersection x = 5. Thus, substituting this result into the first
equation, y = 7 - x = 7 - 5 = 2.
Alternatively we could have substituted into the second
equation to get
y = -11/2 + 3 x / 2 = -11/ 2 + 3 * 5 / 2 = -11/2 + 16 / 2 = 4
/ 2 = 2.
We get the same y value either way, which must be the case at
a point of intersection.
So the intersection point is at x = 5, y = 2, i.e., the point
(5, 2).
ALTERNATIVE SOLUTION:
Solve first equation for y then substitute into the second.
You could have solved your first equation for y, obtaining y
= 7 - x.
Substituting into the second equation you would have obtained
3x - 2(7-x) = 11 so 5x - 14 = 11 and x = 5.
Then substituting this value in either equation would have
given you y = 2. **
**** Query
Add comments on any surprises or insights you experienced as a result of
this assignment.
Self-critique (if necessary):
Self-critique Rating: