course Mth 271 Sº©ÊĈþ”É›ÊÔ£ò¢Ò㼚«ºŽóassignment #015
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10:54:33 `q001. Using the differential and the value of the function at x = 3, estimate the value of f(x) = x^5 and x = 3.1. Compare with the actual value.
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RESPONSE --> I would estimate the answer to be close to 247. This is because 3^5 is 243. I was off because the actual answer is 286.29151 confidence assessment: 2
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10:57:04 The differential of a function y = f(x) is `dy = f ' (x) * `dx. Since for f(x) = x ^ 5 we have f ' (x) = 5 x^4, the differential is `dy = 5 x^4 `dx. At x = 3 the differential is `dy = 5 * 3^4 * `dx, or `dy = 405 `dx. Between x = 3 and x = 3.1 we have `dx = .1 so `dy = 405 * .1 = 40.5. Since at x = 3 we have y = f(3) = 3^5 = 243, at x = 3.1 we should have y = 243 + 40.5 = 283.5, approx.. Note that the actual value of 3.1 ^ 5 is a bit greater than 286; the inaccuracy in the differential is due to the changing value of the derivative between x = 3 and x = 3.1. Our approximation was based on the rate of change, i.e. the value of the derivative, at x = 3.
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RESPONSE --> I must have missed the differential part of the question. Sorry about that. I also do have a hard time with the differential, but I am getting better at it. self critique assessment: 2
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10:59:07 `q002. Using the differential and the value of the function at x = e, estimate the value of ln(2.8). Compare with the actual value.
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RESPONSE --> The differential of ln(2.8) is (1/2.8) * 'dx But from here, I don't know where to proceed. confidence assessment: 0
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11:00:46 The differential of the function y = f(x) = ln(x) is `dy = f ' (x) `dx or `dy = 1/x `dx. If x = e we have `dy = 1/e * `dx. Between e and 2.8, `dx = 2.8 - e = 2.8 - 2.718 = .082, approx.. Thus `dy = 1/e * .082 = .030, approx.. Since ln(e) = 1, we see that ln(2.8) = 1 + .030 = 1.030, approx..
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RESPONSE --> Oh ok, I approached it wrong. I should have had x as the denominator instead of 2.8. Then evaluate for x= e, then taken the difference and multiplying by 1/e. self critique assessment: 2
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11:03:12 `q003. Using the differential verify that the square root of a number close to 1 is twice as close to 1 as the number. Hint: Find the differential approximation for the function f(x) = `sqrt(x) at an appropriate point.
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RESPONSE --> The differential for the function 'sqrt(x) is 1/2 x ^(-1/2) If using 4 we get f'(4) = 1/2 (4^-1/2) f'(4) = 1/2 * 1/2 f'(4) = 1/4 confidence assessment: 1
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11:03:48 The differential for this function is easily seen to be `dy = 1 / (2 `sqrt(x) ) * `dx. At x = 1 the differential is therefore `dy = 1 / 2 * `dx. This shows that as we move away from x = 1, the change in y is half the change in x. Since y = f(x) = 1 when x = 1, as x 'moves away' from 1 we see that y also 'moves away' from 1, but only half as much.
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RESPONSE --> Ok I should have used 1, but I think I got the right answer. self critique assessment: 2
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11:05:39 `q004. Using the differential approximation verify that the square of a number close to 1 is twice as far from 1 as the number.
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RESPONSE --> The differential of this function is f'(x) = 2x If we plug 1 into both equations, we have f(x) = (1)^2 f(x) = 1 f'(x) = 2(1) f'(x) = 2 The square is twice as far from 1. confidence assessment: 3
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11:05:49 The differential for this function is easily seen to be `dy = 2 x * `dx. At x = 1 the differential is therefore `dy = 2 * `dx. This shows that as we move away from x = 1, the change in y is double the change in x. Since y = f(x) = 1 when x = 1, as x 'moves away' from 1 we see that y also 'moves away' from 1, but by twice as much.
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RESPONSE --> self critique assessment: 3
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11:07:19 `q005. The lifting strength of an athlete in training changes according to the function L(t) = 400 - 250 e^(-.02 t), where t is the time in weeks since training began. What is the differential of this function? At t = 50, what approximate change in strength would be predicted by the differential for the next two weeks?
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RESPONSE --> The differential of an ""e"" function still confuses me. I always tend to say that the differential is the same as the equation. confidence assessment: 0
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11:09:42 The differential is `dL = L ' (t) * `dt = -.02 ( -250 e^(-.02 t) ) `dt, so `dL = 5 e^(-.02 t) `dt. At t = 50 we thus have `dL = 5 e^(-.02 * 50) `dt, or `dL = 1.84 `dt. The change over the next `dt = 2 weeks would therefore be approximately `dL = 1.84 * 2 = 3.68.
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RESPONSE --> I do understand these steps past the differential. I still don't understand how or why we multiply by 5? self critique assessment: 1
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11:12:01 `q006. As you move away from a fairly typical source of light, the illumination you experience from that light is given by I(r) = k / r^2, where k is a constant number and r is your distance in meters from the light. Using the differential estimate the change in illumination as you move from r = 10 meters to r = 10.3 meters.
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RESPONSE --> The differential to this function is 'dr = k / 2r At r = 10, we have 'dr = 2(10) 'dr From 10 to 10.3, the change in r is .3, so 'dr = 20 * .3 = 6 confidence assessment: 1
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11:12:56 The differential is `dI = I ' (r) * `dr, where I ' is the derivative of I with respect to r. Since I ' (r) = - 2 k / r^3, we therefore have `dI = -2 k / r^3 * `dr. For the present example we have r = 10 m and `dr = .3 m, so `dI = -2 k / (10^3) * .3 = -.0006 k. This is the approximate change in illumination.
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RESPONSE --> I don't understand where I am messing up. On this problem, where did the -2 come from? self critique assessment: 1
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11:14:54 `q007. A certain crystal grows between two glass plates by adding layers at its edges. The crystal maintains a rectangular shape with its length double its width. Its width changes by .1 cm every hour. At a certain instant its width is 5 cm. Use a differential approximation to determine its approximate area 1 hour later.
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RESPONSE --> f(x) = 2w * w f'(x) = 2w f'(5) = 10 * dw f'(60) = 120 confidence assessment: 0
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11:16:45 If the width of the crystal is x then its length is 2x and its area is 2x * x = 2x^2. So we wish to approximate f(x) = 2x^2 near x = 5. f ' (x) = 4 x, so when x = 5 we have y = 2 * 5^2 = 50 and y ' = 4 * 5 = 20. The rate at which area changes with respect to width is therefore close to y ' = 20 units of area per unit of width. A change of .1 cm in width therefore implies a change of approximately 20 * .1 = 2 in area. Thus the approximate area should be 50 + 2 = 52. This can easily be compared with the accurate value of the area which is 52.02.
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RESPONSE --> I know I was way off. I can see the appropriate way to solve this problem now. I started off right, but sometimes I have a problem stepping out, and looking at the steps to take. self critique assessment: 2
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11:19:08 `q008. The radius of a sphere is increasing at the rate of .3 cm per day. Use the differential to determine the approximate rate at which its volume is increasing on a day when the radius is in the neighborhood of 20 cm.
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RESPONSE --> f(x) = pi * r^2 f'(x) = 2*pi*r f(20) = pi * (20^2) f(20) = 1256.64 f'(20) = 2 * pi * 20 f'(20) = 125.66 approx. confidence assessment: 1
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11:22:33 The volume of a sphere is V(r) = 4/3 * `pi * r^3. The rate at which the volume changes with respect to the radius is dV / dr = 4 * `pi * r^2. Thus when r = 20 the volume is changing at a rate of 4 * `pi * 20^2 = 1600 `pi cm^3 volume per cm of radius. It follows that if the radius is changing by .3 cm / day, the volume must be changing at 1600 `pi cm^3 / (cm of radius) * .3 cm of radius / day = 480 `pi cm^3 / day. Note that this is the instantaneous rate at the instant r = 20. This rate will increase as r increases.
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RESPONSE --> I didn't remember the volume of a sphere formula, this is where I messed up on this one. Other than that I would have taken the right steps. self critique assessment: 2
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