course Mth 271 007. `query 7*********************************************
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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. ** Self-critique Rating: I had to look ahead to understand how to approach the problem. ********************************************* 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: dT/dt = k(T Troom) dT/dt = k(T 20) This would result in an exponential equation when graphed
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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. ** ------------------------------------------------ Self-critique Rating: I feel fairly confident about this problem. ********************************************* 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 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: This is the function for the upper half of a circle. I know this because I put the function into my calculator. After reading a head I was able to deduce the ways to tell a circle solution.
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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. ** ------------------------------------------------ Self-critique Rating: I feel fairly confident about this solution. ********************************************* Question: 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I approached this problem like a growth rate problem. P(t) = P0(1 + r)^t The rate in this case is .88 (1 + -.12) So P(t) = 30(.88)^t
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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. ** ------------------------------------------------ Self-critique Rating: I feel very confident about this problem. ********************************************* Question: What is the formula for exponential function whose graph passes through the points (1,6) and (2,18)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: y = Ab^x (1, 6) and (2, 18) 6 = Ab^1 18 = Ab^2 18/6 = Ab^2/Ab^1 3 = b 6 = A(3)^1 A = 2 y = 2(3^x)
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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. ** ------------------------------------------------ Self-critique Rating: I feel confident about this problem. ********************************************* Question: 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The y intercept is (-1,0) and the x intercept is (0,1). The graph is increasing as x increases, but at a decreasing rate.
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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. ** ------------------------------------------------ Self-critique Rating: To answer this question, I entered the function into my calculator and used the table function to find the coordinates along the line.
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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 Rating: I feel very confident about this problem. "