precal1 orien

#$&*

course Mth163

5/30/12 9:13a

003. PC1 questions

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Question: `q001 A straight line connects the points (3, 5) and (7, 17), while another straight line continues on from (7, 17) to the point (10, 29). Which line is steeper and on what basis to you claim your result?

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Your solution:

With the coordinates (3,5) and (7,17) we can figure

m= (17-5)/ (7-3)= 10/4

m=3

with the coordinates of (7,17) and (10, 29) we can figure

m= (29-17)/ (10-7)= 12/3

m=4

So the second line with the slope of 4 is steeper because it is has the same run but more rise.

confidence rating #$&*:3

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Given Solution:

`aThe point (3,5) has x coordinate 3 and y coordinate 5. The point (7, 17) has x coordinate 7 and y coordinate 17. To move from (3,5) to (7, 17) we must therefore move 4 units in the x direction and 12 units in the y direction.

Thus between (3,5) and (7,17) the rise is 12 and the run is 4, so the rise/run ratio is 12/4 = 3.

Between (7,10) and (10,29) the rise is also 12 but the run is only 3--same rise for less run, therefore more slope. The rise/run ratio here is 12/3 = 4.

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Self-critique (if necessary):ok

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Self-critique Rating:

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Question: `q002. The expression (x-2) * (2x+5) is zero when x = 2 and when x = -2.5. Without using a calculator verify this, and explain why these two values of x, and only these two values of x, can make the expression zero.

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Your solution:

If x= 2 then (2-2) * (2(2)+5)

If we work inside the parentheses we get 0 * (4+5)= 0*9=0

If x=-2.5 then (-2.5-2) * (2(-2.5)+5)

If we work inside the parentheses we get -4.5 * (-5+5)

This gives us -4.5* 0=0

Theses are the only two numbers that can make one side a zero, ultimatley making a times 0 equation.

confidence rating #$&*:3

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Given Solution:

`aIf x = 2 then x-2 = 2 - 2 = 0, which makes the product (x -2) * (2x + 5) zero.

If x = -2.5 then 2x + 5 = 2 (-2.5) + 5 = -5 + 5 = 0.which makes the product (x -2) * (2x + 5) zero.

The only way to product (x-2)(2x+5) can be zero is if either (x -2) or (2x + 5) is zero.

Note that (x-2)(2x+5) can be expanded using the Distributive Law to get

x(2x+5) - 2(2x+5). Then again using the distributive law we get

2x^2 + 5x - 4x - 10 which simplifies to

2x^2 + x - 10.

However this doesn't help us find the x values which make the expression zero. We are better off to look at the factored form.

STUDENT QUESTION

I think I have the basic understanding of how x=2 and x=-2.5 makes this equation 0

I was looking at the distributive law and I understand the basic distributive property as stated in algebra

a (b + c) = ab + ac and a (b-c) = ab - ac

but I don’t understand the way it is used here

(x-2)(2x+5)

x(2x+5) - 2(2x+5)

2x^2 + 5x - 4x - 10

2x^2 + x - 10.

Would you mind explaining the steps to me?

INSTRUCTOR RESPONSE

The distributive law of multiplication over addition states that

a (b + c) = ab + ac

and also that

(a + b) * c = a c + b c.

So the distributive law has two forms.

In terms of the second form it should be clear that, for example

(x - 2) * c = x * c - 2 * c.

Now if c = 2 x + 5 this reads

(x-2)(2x+5) = x * ( 2 x + 5) - 2 * (2 x + 5).

The rest should be obvious.

We could also have used the first form.

a ( b + c) = ab + ac so, letting a stand for (x - 2), we have

(x-2)(2x+5) = ( x - 2 ) * 2x + (x - 2) * 5.

This will ultimately give the same result as the previous. Either way we end up with 2 x^2 + x - 10.

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Self-critique (if necessary):

??? I am seeing the answer correctly but I am rusty on the distributive law.I don't see how you decided which portion of the equation was a,b, and c.

It looks like you just randomly pulled them out of the equation. Is the first parentheses always a and b? Is the second parentheses always c

So for this question why is a=x?

Why is b=-2?

Why is c=2x+5?

And why is a= x-2 in the first form?

How and why does this differ?

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Self-critique Rating:2

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Good questions.

The distributive law can be written

a ( b + c) = a b + a c

or

(a + b) c = ac + bc.

If we use the first form, then a = x - 2, which means that b = 2x and c = 5. In this case

a ( b + c) gives us (x - 2) * 2x + (x - 2) * 5.

If we use the second form, then a = x and b = -2. In this case

(a + b) c = ac + bc gives us x (2x + 5) + (-2) ( 2 x + 5).

Either way, when we complete the last step of the expansion and collect terms we get the same final result

2x^2 + x - 10.

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Question: `q003. For what x values will the expression (3x - 6) * (x + 4) * (x^2 - 4) be zero?

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Your solution:For this expression to equal 0 1 part of the equation must equal 0. That way you are multiplying

by 0. So if 3x-6=0, then x=2

If x+4=0, then x=-4

If x^2-4=0, then x= +/- 2

So x can equal 2, -4, or +/-2

confidence rating #$&*:2

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Given Solution:

`aIn order for the expression to be zero we must have 3x-6 = 0 or x+4=0 or x^2-4=0.

3x-6 = 0 is rearranged to 3x = 6 then to x = 6 / 3 = 2. So when x=2, 3x-6 = 0 and the entire product (3x - 6) * (x + 4) * (x^2 - 4) must be zero.

x+4 = 0 gives us x = -4. So when x=-4, x+4 = 0 and the entire product (3x - 6) * (x + 4) * (x^2 - 4) must be zero.

x^2-4 = 0 is rearranged to x^2 = 4 which has solutions x = + - `sqrt(4) or + - 2. So when x=2 or when x = -2, x^2 - 4 = 0 and the entire product (3x - 6) * (x + 4) * (x^2 - 4) must be zero.

We therefore see that (3x - 6) * (x + 4) * (x^2 - 4) = 0 when x = 2, or -4, or -2. These are the only values of x which can yield zero.**

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Self-critique (if necessary):

???Once again I can arrive at the answer, but I am confused why when you take the square root of 4 in x^2-4=0

does the result become +/-. I remember it does from prior mathematics. But could you give me a reason why

so that I may fully understand why it's +/-?

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Self-critique Rating:

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x can be 2, or it can be -2, or it can be 4. In each case the resulting expression takes the values 0.

Since x can be 2 or -2, we abbreviate by saying the possible x values are +- 2.

You will also be familiar with the +- in the context of the quadratic formula:

x = (-b +- sqrt(b^2 - 4 a c) / (2 a).

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Usually the +- is written with the + above the -. That's not practical in a text editor so it's understood that this is the meaning of +-

You wrote +/-, which shows that the + is above the -. The / sign might be confusing so I don't use it, but that is a valid way to write it and if you prefer to write it that way I'll understand.

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Question: `q004. One straight line segment connects the points (3,5) and (7,9) while another connects the points (10,2) and (50,4). From each of the four points a line segment is drawn directly down to the x axis, forming two trapezoids. Which trapezoid has the greater area? Try to justify your answer with something more precise than, for example, 'from a sketch I can see that this one is much bigger so it must have the greater area'.

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Your solution:

By sketching the graph we can see the the trapezoid on the right has a greater area because it's width is so

much larger than the one on the left. The width (10,2) and (50,4) is 40 units while (3,5) and (7,9) is only 4 units.

confidence rating #$&*:3

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Given Solution:

`aYour sketch should show that while the first trapezoid averages a little more than double the altitude of the second, the second is clearly much more than twice as wide and hence has the greater area.

To justify this a little more precisely, the first trapezoid, which runs from x = 3 to x = 7, is 4 units wide while the second runs from x = 10 and to x = 50 and hence has a width of 40 units. The altitudes of the first trapezoid are 5 and 9,so the average altitude of the first is 7. The average altitude of the second is the average of the altitudes 2 and 4, or 3. So the first trapezoid is over twice as high, on the average, as the first. However the second is 10 times as wide, so the second trapezoid must have the greater area.

This is all the reasoning we need to answer the question. We could of course multiply average altitude by width for each trapezoid, obtaining area 7 * 4 = 28 for the first and 3 * 40 = 120 for the second. However if all we need to know is which trapezoid has a greater area, we need not bother with this step.

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Question: `q005. Sketch graphs of y = x^2, y = 1/x and y = `sqrt(x) [note: `sqrt(x) means 'the square root of x'] for x > 0. We say that a graph increases if it gets higher as we move toward the right, and if a graph is increasing it has a positive slope. Explain which of the following descriptions is correct for each graph:

As we move from left to right the graph increases as its slope increases.

As we move from left to right the graph decreases as its slope increases.

As we move from left to right the graph increases as its slope decreases.

As we move from left to right the graph decreases as its slope decreases.

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Your solution:

For y=x^2, we get the coordinates (1,1) (2,4) (3,9) (4,16). The graph is increasing and the the slope is increasing each time the x increases 1 unit.

For y=1/x, we get the coordinates (1,1) (2,.5) (3.,1/3) (4,1/4). The graph is decreasing and the slope is decreasing each time x increases 1 unit.

For y='sqrt(x) we get the coor. (1,1) (2,1.4) (3,1.7) (4,2). The graph is increasing and the slope is positive but slope is gradually decreasing

confidence rating #$&*:2

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Given Solution:

`aFor x = 1, 2, 3, 4:

The function y = x^2 takes values 1, 4, 9 and 16, increasing more and more for each unit increase in x. This graph therefore increases, as you say, but at an increasing rate.

The function y = 1/x takes values 1, 1/2, 1/3 and 1/4, with decimal equivalents 1, .5, .33..., and .25. These values are decreasing, but less and less each time. The decreasing values ensure that the slopes are negative. However, the more gradual the decrease the closer the slope is to zero. The slopes are therefore negative numbers which approach zero.

Negative numbers which approach zero are increasing. So the slopes are increasing, and we say that the graph decreases as the slope increases.

We could also say that the graph decreases but by less and less each time. So the graph is decreasing at a decreasing rate.

For y = `sqrt(x) we get approximate values 1, 1.414, 1.732 and 2. This graph increases but at a decreasing rate.

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Self-critique (if necessary):

???I am confused about the function y=1/x. Would you say that my answer is correct in saying the graph is decreasing and the slope is decreasing each time x increases 1 unit.

or would I need to elaborate further in saying the slope is negative, but the negative numbers are approaching 0. Therefore slope is increasing

as the graph decreases.???

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Self-critique Rating:

@&

You're on a roll. That's three really good questions in a row.

The graph of 1/x for x > 0 is decreasing toward the x axis, and it is getting less and less steep as you move from left to right.

So it might seem that the slope is decreasing. However that's not the case, since the slopes are negative. A negative slope that decreases would get more and more negative, which would cause the graph to curve downward.

In this case the slope gets less and less negative. In fact the slope approaches zero as the graph appropaches the x axis.

A negative number that's approaching zero is increasing.

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Question: `q006. If the population of the frogs in your frog pond increased by 10% each month, starting with an initial population of 20 frogs, then how many frogs would you have at the end of each of the first three months (you can count fractional frogs, even if it doesn't appear to you to make sense)? Can you think of a strategy that would allow you to calculate the number of frogs after 300 months (according to this model, which probably wouldn't be valid for that long) without having to do at least 300 calculations?

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Your solution:

If we figure out the first month 20*.1=2 and we add 20+2=22. We get 22 frogs for the first month.

We can then use 20x=22. 22/20=1.1, so x=1.1

We can then use F=20*(1.1^3)= 26.62 frogs for three months.

Continuing on with the expression F=20*(1.1^300)

confidence rating #$&*:

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Given Solution:

`aAt the end of the first month, the number of frogs in the pond would be (20 * .1) + 20 = 22 frogs. At the end of the second month there would be (22 * .1) + 22 = 24.2 frogs while at the end of the third month there would be (24.2 * .1) + 24.2 = 26.62 frogs.

The key to extending the strategy is to notice that multiplying a number by .1 and adding it to the number is really the same as simply multiplying the number by 1.1. We therefore get

20 * 1.1 = 22 frogs after the first month

22 * 1.1 = 24.2 after the second month

etc., multiplying by for 1.1 each month.

So after 300 months we will have multiplied by 1.1 a total of 300 times. This would give us 20 * 1.1^300, whatever that equals (a calculator, which is appropriate in this situation, will easily do the arithmetic).

A common error is to say that 300 months at 10% per month gives 3,000 percent, so there would be 30 * 20 = 600 frogs after 30 months. That doesn't work because the 10% increase is applied to a greater number of frogs each time. 3000% would just be applied to the initial number, so it doesn't give a big enough answer.

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Self-critique (if necessary):

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Question: `q007. Calculate 1/x for x = 1, .1, .01 and .001. Describe the pattern you obtain. Why do we say that the values of x are approaching zero? What numbers might we use for x to continue approaching zero? What happens to the values of 1/x as we continue to approach zero? What do you think the graph of y = 1/x vs. x looks for x values between 0 and 1?

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Your solution:

We get the pattern of 1,10,100,1000, so it's increasing by multiples of ten. X is approaching 0 because the vaules are decreasing from 1 getting closer and

closer to 0. The next number in the pattern would be .0001, then .00001, etc. to approach 0.

The values of 1/x will become larger and larger as we approach 0.

As for y=1/x for x values between 0 and 1. I would expect the y value to decrease as the x value increases. So the graph would be decreasing and the slope would be gradually increaing as x increased 1 unit.

confidence rating #$&*:2

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Given Solution:

`aIf x = .1, for example, 1 / x = 1 / .1 = 10 (note that .1 goes into 1 ten times, since we can count to 1 by .1, getting.1, .2, .3, .4, ... .9, 10. This makes it clear that it takes ten .1's to make 1.

So if x = .01, 1/x = 100 Ithink again of counting to 1, this time by .01). If x = .001 then 1/x = 1000, etc..

Note also that we cannot find a number which is equal to 1 / 0. Deceive why this is true, try counting to 1 by 0's. You can count as long as you want and you'll ever get anywhere.

The values of 1/x don't just increase, they increase without bound. If we think of x approaching 0 through the values .1, .01, .001, .0001, ..., there is no limit to how big the reciprocals 10, 100, 1000, 10000 etc. can become.

The graph becomes steeper and steeper as it approaches the y axis, continuing to do so without bound but never touching the y axis.

This is what it means to say that the y axis is a vertical asymptote for the graph .

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Self-critique (if necessary):

??? I thought you were asking for a graph as 0 approaches 1, when you asked What do you think the graph of y = 1/x vs. x looks for x values between 0 and 1?

That is why I explained it the way I did. I understand the increase in the numbers and y would never equal 0. Is this something that I would be counted

against???

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Self-critique Rating:

@&

Your answer is correct as far as it goes.

To fully answer the question you would want to address what the graph looks like near x = 0.

For a full description you would want to address this.

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Question: `q008. At clock time t the velocity of a certain automobile is v = 3 t + 9. At velocity v its energy of motion is E = 800 v^2. What is the energy of the automobile at clock time t = 5?

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Your solution:

If v=3t+9 and t=5, v=24

E=800(24^2)

E=460,800

confidence rating #$&*:3

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Given Solution:

`aFor t=5, v = 3 t + 9 = (3*5) + 9 = 24. Therefore E = 800 * 24^2 = 460800.

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Question: `q009. Continuing the preceding problem, can you give an expression for E in terms of t?

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Your solution:

All you need to do is plug in the value of v which is 3t+9 giving us

E=800*[(3t+9)^2]

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Given Solution:

`aSince v = 3 t + 9 the expression would be E = 800 v^2 = 800 ( 3t + 9) ^2. This is the only answer really required here.

For further reference, though, note that this expression could also be expanded by applying the Distributive Law:.

Since (3t + 9 ) ^ 2 = (3 t + 9 ) * ( 3 t + 9 ) = 3t ( 3t + 9 ) + 9 * (3 t + 9) = 9 t^2 + 27 t + 27 t + 81 = 9 t^2 + 54 t + 81, we get

E = 800 ( 9 t^2 + 54 t + 81) = 7200 t^2 + 43320 t + 64800 (check my multiplication because I did that in my head, which isn't always reliable).

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Self-critique (if necessary):

???Is it wrong to bracket the equation like I did; ex: E=800*[(3t+9)^2]. You arrive at the same answer don't you???

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Self-critique Rating:3

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&#Good work. See my notes and let me know if you have questions. &#