#$&* course MTH 271 Question: `q **** Query class notes #07 **** Explain how we obtain the tangent line to a y = k x^3 function at a point on its graph, and explain why this tangent line gives a good approximation to the function near that point.
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Given Solution: `a If we know that y=kx^3, as in the sandpile model, we can find the derivative as y = 3kx^2. This derivative will tell us the rate at which the volume changes with respect to the diameter of the pile. On a graph of the y = k x^3 curve the slope of the tangent line is equal to the derivative. Through the given point we can sketch a line with the calculated slope; this will be the tangent line. Knowing the slope and the change in x we easily find the corresponding rise of the tangent line, which is the approximate change in the y = k x^3 function. In short you use y' = 3 k x^2 to calculate the slope, which you combine with the change `dx in x to get a good estimate of the change `dy in y. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `qQuery class notes #08 **** What equation do we get from the statement 'the rate of temperature change is proportional to the difference between the temperature and the 20 degree room temperature'? What sort of graph do we get from this equation and why? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: y proportional to x means that for some k we have y = k x. The rate of change of the temperature is the derivative dT/dt. The difference between temp and room temp is T - 20. dT/dt = k (T - 20). Whenever the rate dT/dt is proportional to a quantity like T - Troom, which is a linear function of T, the result is that T-Troom is an exponential function of clock time confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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 (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* 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: 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 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a It turns out that this graph is in fact the upper half of a circle of radius 3 centered at the origin. We can show that this graph is part of a circle: If y = `sqrt(9 - x^2) then y^2 = 9 - x^2 so x^2 + y^2 = 9. This is the Pythagorean Theorem for a right triangle defined by center (0,0) and legs x and y; we see that the square of the hypotenuse is 9 so the hypotenuse is 3. The hypotenuse represents the radius of the circle. ** PRETTY GOOD BUT ERRONEOUS STUDENT ANSWER If you take the square root of 9-x^2 you get y = -x + 9 which is linear. INSTRUCTOR COMMENTS Good try, and well expressed. However sqrt(9 - x^2) is not 9 - x. It's not 3 - x either. Be careful of the fallacy that sqrt(a^2 + b^2) = a + b. Seems like it ought to be so, but it's not. For example, sqrt(3^2 + 4^2) = sqrt( 9 + 16) = sqrt(25) = 5. We see that sqrt(3^2 + 4^2) is 5. So sqrt(3^2 + 4^2) is not 3 + 4 = 7. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `qExtra problem: (was problem 1.2.10) If air freshener initially contains 30 grams, what is the formula for the number of grams present if 12% of the amount present evaporate per day? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If 12% evaporates per day then 88% remains at the end of each day. That is, the growth rate is -.12 Growth factor is 1 + (-.12) = .88 Function is Q = 30 gram * .88^t confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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 (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q1.2.32 (was 1.2.28 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: 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 So the graph will be increasing at a decreasing rate, which means it is concave downward. 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 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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 (if necessary): ------------------------------------------------ Self-critique Rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!