course Mth 173 ý}žV¥±àˆÝ„ñnÐÑû÷Íž‹§assignment #015
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11:57:27 `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 --> First I need to find the derivative of x^5: f' = 5 x^4 Now I need to find the value of the rate at x = 3: f'(3) = 405 Next I need to find the value of f(3): f(3) = 243 Finally I can multiply the rate by the change in position (0.1) and add it to the value of the function to estimate the change: Estimated value at x=3.1: 243 + (405 x 0.1) = 283.5 Actual value of function at t=3.1: f(3.1) = 286.29 confidence assessment: 2
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11:57:59 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 --> Ok. self critique assessment: 3
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12:16:32 `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(x) is 1/x. The rate at x=e: f'(e) = 0.367879 Value of function at x=e: ln(e) = 1 Now multiply the rate by the change in x and add to beginning value: 1 + (0.367879 * .081718) = 1.03 Actual value: ln(2.8) = 1.0296 confidence assessment: 2
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12:16:41 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 --> Ok. self critique assessment: 3
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14:00:53 `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 --> I know I need to find the differential of the 'sqrt function: f' = 1 / (2 'sqrt(x) ) But I am not sure what to do now. confidence assessment: 1
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14:02:37 07-13-2007 14:02:37 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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NOTES -------> Ok. Interesting.
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14:16:27 `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 an x^2 function is 2x. Therefore: 'dy = 2x * 'dx At x=1: 'dy = 2 As x grows, 'dy grows at twice the amount. confidence assessment: 2
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14:16:37 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 --> Ok. self critique assessment: 3
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14:23:56 `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 --> To find the differential, I need to use the chain rule f( g(x) ) = g'(x) * f'( g(x) ). L'(t) = -250 * -.02 * e^(-.02 t) ) L'(t) = 5 e^(-.02 t) At t=50, the approximate rate per week is: L'(50) = 1.83939 So for the next two weeks at this rate, the approximate change in strength would be: 1.83939 * 2 = 3.67878 confidence assessment: 2
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14:24:08 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 --> Ok. self critique assessment: 3
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14:28:44 `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 should be: l' = k / 2r Therefore, the change should be: 'dy = ( k / 2r ) * 'dt 'dy = ( k / 2 * 10 ) * 0.3 'dy = 0.3 k / 20 confidence assessment: 2
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14:32:20 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 missed the derivative of 1 / r^2, since I forgot that a number like that is the same as r^-2. self critique assessment: 3
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14:35:16 `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 --> If the length is twice the width, then L = 2W. At time t, W=5, so L=10. A = 2W * W = 2 W^2 The differential of this is: A' = 4 W Therefore, the rate at W=4 is: A'(5) = 20 Now to find the new area we take the old area and add the rate multiplied by the change: Area = 50 + (20 * .1) = 52. confidence assessment: 2
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14:44:05 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 --> Ok. self critique assessment: 3
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14:48:28 `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 --> The equation for the volume of the sphere would be: v(r) = (4/3) 'pi r^3 The differential would be: v' = 4 'pi r^2 Now to find the volume when radius=20cm: v(20) = 33510.32 cm^3 Next rate: v'(20) = 5026.55 cm^2 Now add together the original volume with the rate multiplied by the daily change: volume = 33510.32 cm^3 + (5026.55 cm^2 * .3) = 35018.29 Actual: v(20.3) = 35041.02 confidence assessment: 2
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14:52:10 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 --> Ok. self critique assessment: 3
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