013. `query 13
Question: `qQuery last asst before test 1,
problem 1.Give your solution to x ^ 3 / 17 = 58
Your solution:
Confidence Assessment:
Given Solution:
`a** The solution to x^3 / 17 = 58 is found by first multiplying
both sides by 17 to get
x^3 = 58 * 17
then taking the 1/3 power of both sides, obtaining
(x^3)^(1/3) = (58 * 17)^(1/3) or
x = 9.95, approx..
COMMON ERROR:
If you interpret the equation as x^(3/17)
= 58 you will get solution x = 58^(17/3) = 9834643694. However this is not
the solution to the given equation
To interpret x ^ 3 / 17 you have to follow the order of
operations. This means that x is first cubed (exponentiation
precedes multiplication or division) then divided by 17. If
you introduce the grouping x^(3/17) you are changing
the
meaning of the expression, causing 3 to be
divided by 17 before exponentiation. **
Self-critique (if necessary):
Self-critique rating:
Question: `qGive your solution to (3 x) ^
-2 = 19
Your solution:
Confidence Assessment:
Given Solution:
`a** (3x)^-2 = 19 is solved by
taking the -1/2 power of both sides, or the negative of the result:
((3x)^-2)^(-1/2))
= 19^(-1/2) gives us
3x = 19^(-1/2) so that
x = [ 19^(-1/2) ] / 3 = .0765 or
-.0765. **
STUDENT QUESTION:
I'm not sure how I was supposed to arrive at two solutions.
INSTRUCTOR RESPONSE
If an even power of a number has a given value, so does that
power of the negative of that number.
It's fairly easy to understand why. For example:
5^2 = 25 and (-5)^2 = 25
3^4 = 81 and (-3)^4 = 81
2^(-4) = 1/16 and (-2)^(-4) = 1/16.
So when solving an equation of the form (c)^n = d, where n is
an even integer (whether positive or negative) and d is a positive quantity,
there are two solutions: c = d^(1/n) and c = - d^(1/n).
(note that - d^(1/n) is the negative of d^(1/n), not (-d)^(1/n), which would not
be defined).
Self-critique (if necessary):
Self-critique rating:
Question: `qGive your solution to 4 x ^ -.5
= 7
Your solution:
Confidence Assessment:
Given Solution:
`a** to solve the equation we first multiply both sides by
1/4 to get
x ^ -.5 = 7 / 4. Then we raise both sides to the -2 power:
(x^-.5)^-2 = (7/4)^-2 so
x = .327 approx **
STUDENT QUESTION
Why do we multiply by ¼?
INSTRUCTOR RESPONSE
4 is not part of the expression which is raised to the -.5
power.
^ precedes *. Since 4 x ^ -.5 means 4 * x ^ -.5, we do x^-.5 before we multiply
by 4.
So it's easier to multiply both sides by 1/4, so that the
left-hand side is just x^-.5, without the confusion of the 4.
Your solution illustrates why this is the best idea:
Quoting from your solution
4x^-.5 = 7 raise both sides by -1/.5
(4x^-.5)^-1/.5 = 7^-(1/.5) (this step is valid, even if it's not the
best way to proceed)
4x = 0.020 (this step is in error)
The error on that last step is that
(4 x ^-.5) ^ (-1 / .5) is not equal to 4x.
(4 x ^-.5) ^ (-1 / .5) =
4 ^ (-1 / .5) * (x^-.5) ^ (1 / (-.5) ) =
4^(-1/.5) * x
Self-critique (if necessary):
Self-critique rating:
Question: `qGive your solution to 14 x ^
(2/3) = 39
Your solution:
Confidence Assessment:
Given Solution:
`a** We first multiply both sides by 1/14 to get
x^(2/3) = 39/14. The we raise both
sides to the 3/2 power to get
x = (39/14)^(3/2) = 4.65. **
Self-critique (if necessary):
Self-critique rating:
Question: `qGive your solution to 5 ( 3 x / 8) ^ (-3/2) = 9
Your solution:
Confidence Assessment:
Given Solution:
`a** multiplying both sides by 1/5 we get
(3x/8)^(-3/2) = 9/5. Raising both
sides to the -2/3 power we have
3x / 8 = (9/5)^(-2/3). Multiplying
both sides by 8/3 we obtain
x = 8/3 * (9/5)^(-2/3) = 1.80 **
STUDENT QUESTION
Still confused on the steps after we get down to 9/5.
INSTRUCTOR COMMENT
We take the -2/3 power of the -3/2 power:
(3x/8)^(-3/2) = 9/5. Raising both sides to the -3/2 power we have
((3x / 8)^(-3/2) ) ^ (-2/3) = (9/5)^(-2/3), so that (by the laws of exponents)
(3x / 8)^(-3/2 * (-2/3) ) = (9/5)^(-2/3). Since -3/2 * -2/3 = 1 so we have
(3x / 8)^1 = (9/5)^(-2/3), that is
3 x / 8 = (9/5)^(-2/3). Multiplying both sides by 8/3 we complete the solution.
Self-critique (if necessary):
Self-critique rating:
Question: `qQuery problem 2. a(n+1) = a(n) + .5 n, a(0) = 2
What are a(1), a(2), a(3), a(4) and
a(5)?
Your solution:
Confidence Assessment:
Given Solution:
`a** Substituting n = 0 we get
a(0+1) = a(0) + .5 * 0 which we
simplify to get
a(1) = a(0). Substituting a(0) = 2 from the given information we get
a(1) = 2.
Substituting n = 1 we get
a(1+1) = a(1) + .5 * 1 which we
simplify to get
a(2) = a(1) + .5. Substituting a(1) = 2 from the previous step we get
a(2) = 2.5.
Substituting n = 2 we get
a(2+1) = a(2) + .5 * 2 which we
simplify to get
a(3) = a(2) + 1. Substituting a(2) = 2.5 from the previous step we get
a(3) = 2.5 + 1 = 3.5.
Substituting n = 3 we get
a(3+1) = a(3) + .5 * 3 which we
simplify to get
a(4) = a(3) + 1.5. Substituting a(3) = 3.5 from the previous step we get
a(4) = 3.5 + 1.5 = 5.
Substituting n = 4 we get
a(4+1) = a(4) + .5 * 4 which we
simplify to get
a(5) = a(4) + 2. Substituting a(4) = 5 from the previous step we get
a(5) = 5 + 2 = 7. **
Self-critique (if necessary):
Self-critique rating:
Question: `qWhat is your quadratic function
and what is its value for n = 4? Does it fit the sequence exactly?
Your solution:
Confidence Assessment:
Given Solution:
`a** Using points (1,2), (3,3.5)
and (7,5) we substitute into the form y = a x^2 + b x + c to obtain the three
equations
2 = a * 1^2 + b * 1 + c
3.5 = a * 3^2 + b * 3 + c
7 = a * 5^2 + b * 5 + c.
Solving the resulting system for a, b and c we obtain a = .25,
b = -.25 and c = 2, giving us the equation
0.25•x^2 - 0.25•x + 2. **
Self-critique (if necessary):
Self-critique rating:
Question: `qQuery problem 3. f(x) = .3 x^2
- 4x + 7, evaluate at x = 0, .4, .8, 1.2, 1.6 and 2.0.
Your solution:
Confidence Assessment:
Given Solution:
`a** We obtain the points
(0, 7)
(4, 5.448)
(.8, 3.992)
(1.2, 2.632)
(1.6, 1.368)
(2, .2)
y values are
7, 5.448, 3.992, 2.632, 1.368, 0.2.
Differences are 7-5.448 = -1.552, 3.992 - 2.632 = -1.456, etc.
The sequence of differences is
-1.552, -1.456, -1.36, -1.264, -1.168.
The rate of change of the original sequence is proportional
to this sequence of differences.
The differences of the sequence of differences (i.e., the
second differences) are .096, .096, .096, .096, .096..
These differences are constant, meaning that the sequence of
differences is linear..
This constant sequence is proportional to the rate of change
of the sequence of differences.
The differences are associated with the midpoints of the
intervals over which they occur. Therefore the difference -1.552,
which occurs between x = 0 and x = .4, is associated with x
= .2; the difference -1.456 occuring between x = .4
and x =
.8 is associated with x = .6, etc..
The table of differences vs. midpoints is
}
0.2, -1.552
.6, -1.456
1, -1.36
1.4, -1.264
1.8, -1.168
This table yields a graph whose slope is easily found to be
constant at .24, with y intercept -1.6. The function that
models these differences is therefore
y = 2.4 x - 1.6. **
STUDENT QUESTION:
I’m not quite sure I understand how you derived at your
model y = 2.4 x - 1.6
INSTRUCTOR RESPONSE:
For example the rise between the first and second point is
.096 and the run is .4, making the slope .096 / .4 = .24. You get the same
slope for any pair of points (you can easily see that the x values change by the
same amount each time, and if you calculate the changes in the y values you get
.096 for each change, leading fairly quickly to the conclusion that the slopes
are all the same).
Between 0 and 0.2 the 'rise' would therefore be 0.2 * .24 =
.048. This would put the x = 0 point .48 units lower than the x = .2
point, so that at the x = 0 point the y value is -1.552 - .048 = -1.6.
The line is therefore y = 2.4 x - 1.6.
STUDENT QUESTION:
I think I may have misunderstood the directions for
this one. I believe the problem statement
asked to graph the avg. rate of change vs the midpoints, which I assumed to be
the slopes vs the midpoints, not the
difference between x values (which would just be the run between each point?).
So my midpoint coordinates and the
resulting model do not match up to the given solution.
INSTRUCTOR RESPONSE:
Your solution was fine.
There is however a fine distinction in the wording of the question. The
statement referred to a 'quadratic sequence', which isn't the same thing as a
'quadratic function'.
A quadratic sequence is a sequence of numbers which can be obtained by
evaluating a quadratic function. Once the numbers are obtained, the function is
effectively discarded, and the sequence stands on its own.
The function yields the sequence in the manner you outline:
f(x) = 0.6x - 4
f(0.2) = 0.6(0.2) - 4 = -3.88
f(0.6) = 0.6(0.6) - 4 = -3.64
f(1.0) = 0.6(1.0) - 4 = -3.4
f(1.4) = 0.6(1.4) - 4 = -3.16
f(1.8) = 0.6(1.8) - 4 = -2.92
The original function f(x) is at this point forgotten. The sequence is -3.88,
-3.64, -3.4, -3.16, -2.92, ...
A sequence is numbered by integers, so that these values could now be referred
to as a(1), a(2), a(3), a(4) and a(5).
Since a sequence is numbered by consecutive integers, the rates at which a
sequence changes are just the differences in consecutive members of the
sequence.
You might question why we want to forget the origins of the sequence. The reason
is that the sequence is just simpler, and has certain properties and behaviors
that are better understood if we treat it as such.
Having done so we can relate our results for the sequence back to the original
function and its rate-of-change behavior.
Self-critique (if necessary):
Self-critique rating:
Question: `qQuery problem 4. f(x) = a x^2 +
b x + c
What symbolic expression stands for the average slope
between x = h and x = k?
Your solution:
Confidence Assessment:
Given Solution:
`a** The average slope is rise / run = [ f(k) - f(h) ] / (k
- h) = ( a k^2 + b k + c - ( a h^2 + b h + c) ) / ( k - h).
We simplify this to get
ave slope = ( a
( k^2 - h^2) + b ( k - h) ) / ( k - h), which we write as
ave slope = ( a
( k-h) ( k+h) ) + b ( k - h) ) / (k - h).
k - h is a factor of the numerator so we have the final form
ave slope = a (
k + h) + b. **
Question: `qQuery last asst
before test 1, problem 1.Give your solution to x ^ 3 / 17 = 58
Your solution:
Confidence Assessment:
Given Solution:
`a** The solution to x^3 / 17 = 58 is found by first
multiplying both sides by 17 to get
x^3 = 58 * 17
then taking the 1/3 power of both sides, obtaining
(x^3)^(1/3) = (58 * 17)^(1/3) or
x = 9.95, approx..
COMMON ERROR:
If you interpret the equation as x^(3/17)
= 58 you will get solution x = 58^(17/3) = 9834643694. However this is not
the solution to the given equation
To interpret x ^ 3 / 17 you have to follow the order of
operations. This means that x is first cubed (exponentiation
precedes multiplication or division) then divided by 17. If
you introduce the grouping x^(3/17) you are changing
the
meaning of the expression, causing 3 to be
divided by 17 before exponentiation. **
Self-critique (if necessary):
Question: `qGive your
solution to (3 x) ^ -2 = 19
Your solution:
Confidence Assessment:
Given Solution:
`a** (3x)^-2 = 19 is solved by
taking the -1/2 power of both sides, or the negative of the result:
((3x)^-2)^(-1/2))
= 19^(-1/2) gives us
3x = 19^(-1/2) so that
x = [ 19^(-1/2) ] / 3 = .0765 or
-.0765. **
Self-critique (if necessary):
Question: `qGive your
solution to 4 x ^ -.5 = 7
Your solution:
Confidence Assessment:
Given Solution:
`a** to solve the equation we first multiply both sides by
1/4 to get
x ^ -.5 = 7 / 4. Then we raise both sides to the -2 power:
(x^-.5)^-2 = (7/4)^-2 so
x = .327 approx **
Self-critique (if necessary):
Question: `qGive your
solution to 14 x ^ (2/3) = 39
Your solution:
Confidence Assessment:
Given Solution:
`a** We first multiply both sides by 1/14 to get
x^(2/3) = 39/14. The we raise both
sides to the 3/2 power to get
x = (39/14)^(3/2) = 4.65. **
Self-critique (if necessary):
If you understand the assignment and were
able to solve the previously given problems from your worksheets, you should be
able to complete most of the following problems quickly and easily. If you
experience difficulty with some of these problems, you will be given notes and
we will work to resolve difficulties.
Question: `qGive your
solution to 5 ( 3 x / 8) ^ (-3/2) = 9
Your solution:
Confidence Assessment:
Given Solution:
`a** multiplying both sides by 1/5 we get
(3x/8)^(-3/2) = 9/5. Raising both
sides to the -2/3 power we have
3x / 8 = (9/5)^(-2/3). Multiplying
both sides by 8/3 we obtain
x = 8/3 * (9/5)^(-2/3) = 1.80 **
Self-critique (if necessary):
Self-critique rating: