asst 1

course Mth 271

I got tripped up on this one

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assignment #001

001. `Query 1

Applied Calculus I

07-19-2007

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11:50:55

Section 0.1.26 solve x/2-x/3>5

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RESPONSE -->

mulitply by the least common denomenator

3x - 2x = 5\6

x = 5\6

confidence assessment: 1 I hate fractions

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11:51:54

It's easiest to avoid denominators where possible. So the preferred first step is to multiply both sides of the original equation by the common denominator 6:

6(x/2) - 6(x/3) = 6 * 5, which gives you3x - 2x = 6 * 5 which gives you

x > 6 * 5 which simplifies to

x > 30.

The interval associated with this solution is 30 < x < infinity, or (30, infinity).

To graph you would make an arrow starting at x = 30 and pointing to the right, indicating by an open circle that x = 30 isn't included.**

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RESPONSE -->

I knew I should have multiplied, but I'm never shure what to do, I knew this one.

self critique assessment: 3

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11:59:40

Section 0.1.28 solve 2x^2+1<9x-3

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RESPONSE -->

I started by subtracting one on both sides, then square rooting both sides

2x< 3x - 2

then add two to both sides, then subtract 2x from both sides

2 < x

confidence assessment: 3

The square root of 9x - 4 is not 3x - 2; among other things if you square 3x - 2 you get 9 x^2 - 6x + 4, not 9x - 4.

However this is a quadratic inequality and must be solved by factoring, as indicated below.

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12:01:45

The given inequality rearranges to give the quadratic 2x^2 - 9 x + 4 < 0.

The left-hand side has zeros at x = .5 and x = 4, as we see by factoring [ we get (2x-1)(x-4) = 0 which is true if 2x-1 = 0 or x - 4 = 0, i.e., x = 1/2 or x = 4. ]

The left-hand side is a continuous function of x (in fact a quadratic function with a parabola for a graph), and can change sign only by passing thru 0. So on each interval x < 1/2, 1/2 < x < 4, 4 < x the function must have the same sign.

Testing an arbitrary point in each interval tells us that only on the middle interval is the function negative, so only on this interval is the inequality true.

Note that we can also reason this out from the fact that large negative or positive x the left-hand side is greater than the right because of the higher power. Both intervals contain large positive and large negative x, so the inequality isn't true on either of these intervals.

In any case the correct interval is 1/2 < x < 4.

ALTERNATE BUT EQUIVALENT EXPLANATION:

The way to solve this is to rearrange the equation to get

2 x^2 - 9 x + 4< 0.

The expression 2 x^2 - 9 x + 4 is equal to 0 when x = 1/2 or x = 4. These zeros can be found either by factoring the expression to get ( 2x - 1) ( x - 4), which is zero when x = 1/2 or 4, or by substituting into the quadratic formula. You should be able to factor basic quadratics or use the quadratic formula when factoring fails.

The function can only be zero at x = 1/2 or x = 4, so the function can only change from positive to negative or negative to positive at these x values. This fact partitions the x axis into the intervals (-infinity, 1/2), (1/2, 4) and (4, infinity). Over each of these intervals the quadratic expression can't change its sign.

If x = 0 the quadratic expression 2 x^2 - 9 x + 4 is equal to 4. Therefore the expression is positive on the interval (-infinity, 1/2).

The expression changes sign at x = 1/2 and is therefore negative on the interval (1/2, 4).

It changes sign again at 4 so is positive on the interval (4, infinity).

The solution to the equation is therefore the interval (1/2, 4), or in inequality form 1/2 < x < 4. **

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RESPONSE -->

I don't understand that one... I hate inequalities

self critique assessment: 3

&#

In your self-critique you need a phrase-by-phrase analysis of the given solution, detailing everything you do and do not understand.

Deconstruct the given solution and explain in detail what you do and do not understand about every part.

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This is a pretty challenging question, and it's a good one for illustrating this process.

In analyzing the solution you should have asked yourself questions much like the following, and indicated for each one what you do and do not understand.

Of course few students would be able to construct this entire sequence of questions, so what follows is an idealized example. However, this example illustrates what you should strive for in the self-critique process:

The given inequality rearranges to give the quadratic 2x^2 - 9 x + 4 < 0.

Do you understand how this is equivalent to the given inequality?

The left-hand side has zeros at x = .5 and x = 4, as we see by factoring [ we get (2x-1)(x-4) = 0 which is true if 2x-1 = 0 or x - 4 = 0, i.e., x = 1/2 or x = 4. ]

Do you understand the factoring of the left-hand side 2x^2 - 9 x + 4?

Do you understand that in the factored form x = 1/2 or x = 4 makes the expression zero?

The left-hand side is a continuous function of x (in fact a quadratic function with a parabola for a graph), and can change sign only by passing thru 0. So on each interval x < 1/2, 1/2 < x < 4, 4 < x the function must have the same sign.

Do you understand that the left-hand side could be graphed to see where it is zero, where it is greater than zero and where it is less than zero?

Do you understand that the graph of a quadratic function is a parabola?

Do you understand that an expression changes sign when it goes from positive to negative or from negative to positive?

Do you understand that the only way an expression can change sign is therefore to pass through zero?

Do you understand that only points at which the left-hand side can be zero are x = 1/2 and x = 4?

Do you understand that these two points break the number line into three intervals?

Do you understand that x < 1/2 is an interval, as is 1/2 < x < 4 and as is 4 < x? Could you graph these intervals on the number line?

Do you understand that the function cannot therefore change sign except at these x values, so it must be either positive or negative on each of the intervals?

Testing an arbitrary point in each interval tells us that only on the middle interval is the function negative, so only on this interval is the inequality true.

What does it mean to 'test an arbitrary point' in an interval?

Select a point (a value of x) in each of the three intervals and see whether the expression 2x^2 - 9 x + 4 is less than or greater than zero.

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assignment #001

001. `Query 1

Applied Calculus I

07-19-2007"

See my notes and let me know if you have any questions.