#$&* course 1/19 8:30pm 003. PC1 questions
.............................................
Given Solution: `aThe point (3,5) has x coordinate 3 and y coordinate 5. The point (7, 17) has x coordinate 7 and y coordinate 17. To move from (3,5) to (7, 17) we must therefore move 4 units in the x direction and 12 units in the y direction. Thus between (3,5) and (7,17) the rise is 12 and the run is 4, so the rise/run ratio is 12/4 = 3. Between (7,10) and (10,29) the rise is also 12 but the run is only 3--same rise for less run, therefore more slope. The rise/run ratio here is 12/3 = 4. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q002. The expression (x-2) * (2x+5) is zero when x = 2 and when x = -2.5. Without using a calculator verify this, and explain why these two values of x, and only these two values of x, can make the expression zero. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Anything multiplied by 0 results in 0. For the expression (x-2) * (2x+5), any value used for x that results in the term inside either of the parenthesis to become 0 will make the expression equal 0. With this in mind, we set the factors of the expression equal to 0 and solve for x as follows: First factor: x- 2 = 0 x = 2 (add 2 to both sides) If x = 2, then (2 - 2) * (2 * 2 + 5) = 0 * 9 = 0 Second factor: 2 x + 5 = 0 2 x = -5 (subtract 5 from both sides) x = -5 / 2 (divide 2 from both sides) If x = - 5 / 2, then (-5 / 2 - 2) * (-5 / 2 * 2 + 5) = - 9 / 2 * 0 = 0 Only when x is 2 or -5 / 2 will one of the terms inside the parenthesis become 0. All other values substituted for x will result in an expression other than 0. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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 dont 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q003. For what x values will the expression (3x - 6) * (x + 4) * (x^2 - 4) be zero? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Setting factors of the expression (3 x - 6) * (x + 4) * (x ^ 2 - 4) equal to 0 determines what x values will make the expression zero as follows: 3 x - 6 = 0 3 x = 6 (add 6 to both sides of the equation) x = 2 (divide both sides by 3) x + 4 = 0 x = - 4 (subtract 4 from both sides of the equation) x ^ 2 - 4 = (x + 2) * (x - 2) x + 2 = 0 x = - 2 (subtract 2 from both sides of the equation) x - 2 = 0 x = 2 (add 2 to both sides of the equation) Only when the x value is - 4, - 2, or 2 is the expression 0. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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.** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* 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'. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The area of a trapezoid is figured by the formula (1 / 2)h(a + b). A and b represent the length of the parallel sides and h is the distance between parallel sides. Trapezoid 1: We use the distance formula to get the distance between any two given points as follows: a = sqrt((3 - 3) ^ 2 + (0 - 5) ^ 2) a = sqrt(0 + 5^ 2) a = sqrt(0 + 25) a = sqrt(25) a = 5 b = sqrt((7 - 7) ^ 2 + (0 - 9) ^ 2) b = sqrt(0 + 9 ^ 2) b = sqrt(0 + 81) b = sqrt(81) b = 9 h = sqrt((7 - 3) ^ 2 + (0 - 0) ^ 2) h = sqrt(4 ^ 2 + 0) h = sqrt(16 + 0) h = sqrt(16) h = 4 We can now use the formula to find the area: area = (1 / 2) 4 (5 + 9) area = (1 / 2) 4 * 14 area = 56 / 2 area = 28 Trapezoid 2: We use the same steps above to find the terms needed for the area formula for trapezoid 2. a = 2 b = 4 h = 40 We can now use the formula to find the area: area = (1 / 2) 40 (2 + 4) area = (1 / 2) 40 * 6 area = 240 / 2 area = 120 The area of trapezoid 2 is greater than the area of trapezoid 1. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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. ********************************************* 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We have the function y = x ^ 2. We create a table of x and y values by substituting 1, 2, 3, 4 for x and solving for y. If x = 1, then y = 1 ^ 2 = 1 If x = 2, then y = 2 ^ 2 = 4 If x = 3, then y = 3 ^ 2 = 9 If x = 4, then y = 4 ^ 2 = 16 x y 1 1 2 4 3 9 4 16 The graph of y = x ^ 2 increases as its slope increases as we move from left to right. We have the function y = 1 / x. We create a table of x and y values by substituting 1, 2, 3, 4 for x and solving for y. If x = 1, then y = 1 / 1 = 1 If x = 2, then y = 1 / 2 If x = 3, then y = 1 / 3 If x = 4, then y = 1 / 4 x y 1 1 2 1 / 2 3 1 / 3 4 1 / 4 The graph of y = 1 / x decreases as its slope decreases as we move from left to right. We have the function y = `sqrt(x). We create a table of x and y values by substituting 1, 4, 9, 16 for x and solving for y. If x = 1, then y = `sqrt(1) = 1. If x = 4, then y = `sqrt(4) = 2. If x = 9, then y = `sqrt(9) = 3. If x = 16, then y = `sqrt(16) = 4 x y 1 1 4 2 9 3 16 4 The graph of y = `sqrt(x) increases as its slope decreases as we move from left to right. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To find the population of the frogs in a frog pond increased by 10% each month, starting with an initial population of 20 frogs we first create at table showing the population and amount of time. The frog population is figured in the following manner: 20 * .1 + 20 = 22 = population after month 1 22 * .1 + 22 = 24.2 = population after month 2 24.2 * .1 + 24.2 = 26.62 = population after month 3 26.62 * .1 + 26.62 = 29.282 = population after month 4 Months Population 1 22 2 24.2 3 26.62 4 29.282 From the data in the table we can use exponential regression to model the data in the form y = a * b ^ x. We obtain the function f(x) = 20 * 1.1 ^ x. f(300) = 20 * 1.1 ^ 300 = 5.234 E13 = the number of frogs after 300 months. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We have f(x) = 1 / x. If x = 1, then f(1) = 1 / 1 = 1 If x = .1, then f(.1) = 1 / .1 = 10 If x = .01, then f(.01) = 1 / .01 = 100 If x = .001, then f(.001) = 1 / .001 = 1000 The values used for x are said to be approaching zero by a factor of 10 because the fractional portion of 1 becomes smaller and smaller. The value of 1 / x increases at an increasing rate as the x values used approach zero. We can continue approaching zero by using .0001, .00001, and .000001 for x. The graph of y = 1 / x vs. x is decreasing at a decreasing rate for x values from 0 to 1. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `aIf x = .1, for example, 1 / x = 1 / .1 = 10 (note that .1 goes into 1 ten times, since we can count to 1 by .1, getting.1, .2, .3, .4, ... .9, 10. This makes it clear that it takes ten .1's to make 1. So if x = .01, 1/x = 100 Ithink again of counting to 1, this time by .01). If x = .001 then 1/x = 1000, etc.. Note also that we cannot find a number which is equal to 1 / 0. Deceive why this is true, try counting to 1 by 0's. You can count as long as you want and you'll ever get anywhere. The values of 1/x don't just increase, they increase without bound. If we think of x approaching 0 through the values .1, .01, .001, .0001, ..., there is no limit to how big the reciprocals 10, 100, 1000, 10000 etc. can become. The graph becomes steeper and steeper as it approaches the y axis, continuing to do so without bound but never touching the y axis. This is what it means to say that the y axis is a vertical asymptote for the graph . &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q008. At clock time t the velocity of a certain automobile is v = 3 t + 9. At velocity v its energy of motion is E = 800 v^2. What is the energy of the automobile at clock time t = 5? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If the velocity of an automobile is v = 3 t + 9 and the clock time t = 5, then v = 3 * 5 + 9 = 24. If at velocity v, its energy of motion is E = 800 v^2 and v = 24, then E = 800 * 24 ^ 2 = 460800. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `aFor t=5, v = 3 t + 9 = (3*5) + 9 = 24. Therefore E = 800 * 24^2 = 460800. ********************************************* Question: `q009. Continuing the preceding problem, can you give an expression for E in terms of t? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If v = 3 t + 9 and E = 800 v ^ 2, then E = 800 (3 t + 9) ^ 2. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `aSince v = 3 t + 9 the expression would be E = 800 v^2 = 800 ( 3t + 9) ^2. This is the only answer really required here. For further reference, though, note that this expression could also be expanded by applying the Distributive Law:. Since (3t + 9 ) ^ 2 = (3 t + 9 ) * ( 3 t + 9 ) = 3t ( 3t + 9 ) + 9 * (3 t + 9) = 9 t^2 + 27 t + 27 t + 81 = 9 t^2 + 54 t + 81, we get E = 800 ( 9 t^2 + 54 t + 81) = 7200 t^2 + 43320 t + 64800 (check my multiplication because I did that in my head, which isn't always reliable). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: For what x values is the value of the expression (2^x - 1) ( x^2 - 25 ) ( 2x + 6) zero? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Setting factors of the expression (2^x - 1) ( x^2 - 25 ) ( 2x + 6) equal to 0 determines what x values will make the expression zero as follows: 2 ^ x - 1 = 0 2 ^ x = 1 (add 1 to both sides of equation) 2 ^ x = 2 ^ 0 (express with same base) x = 0 (equate exponents) x^2 - 25 = (x + 5)(x - 5) (factor binomial) x + 5 = 0 x = -5 (subtract 5 from both sides of equation) x - 5 = 0 x = -5 (add 5 to both sides of equation) 2x + 6 = 2(x + 3) (factor) x + 3 = 0 x = - 3 (subtract 3 from both sides of the equation) Only when the x value is 0, -5, 5, - 3 is the expression 0. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ********************************************* Question: One straight line segment connects the points (3,5) and (7,9) while another connects the points (3, 10) and (7, 6). 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? Any solution is good, but a solution that follows from a good argument that doesn't actually calculate the areas of the two trapezoids is better. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The first trapezoid runs from x = 3 to x = 7 and the second runs from x = 3 to x = 7 so both have a width of 4 units. The altitudes of the first trapezoid are 5 and 9, so the average altitude is 7. The altitudes of the second trapezoid are 10 and 6, so the average altitude is 8, therefore the second trapezoid must have the greater area. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ********************************************* Question: Suppose you invest $1000 and, at the end of any given year, 10% is added to the amount. How much would you have after 1, 2 and 3 years? What is an expression for the amount you would have after 40 years (give an expression that could easily be evaluated using a calculator, but don't bother to actually evaluate it)? What is an expression for the amount you would have after t years? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The expression for y = the amount after t years is $1000 * 1.1 ^ t so for t = 1 year, t = 2 years and t = 3 years we have: t y 1 $1100 2 $1210 3 $1331 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ------------------------------------------------ Self-critique Rating: OK " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: