course Mth 174 The relative accuracy of the trapezoidal and midpoint rules depends on the nature of the function. As shown in my previous notes for a positive function which is concave down the midpoint rule is more accurate. A brief numerical example is y = x^2 on the interval (-1, 0), where trap gives 1/2, actual is 1/3 and mid is 1/4. 1/3 is closer to 1/4 than to 1/2. The same is true if the function is positive and concave up, for the same reasons. In fact I think that the argument can be extended to show that if concavity doesn't change on an interval, mid has to beat trap. I'd have to draw a picture or two to be sure, but it seems to be fairly obvious so I'll leave that to you. Use my previous notes as a guide.
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********************************************* Question: 7.6.3 (previously 7.6.6) approx using n=10 is 2.346; exact is 4.0. What is n = 30 approximation if original approx used LEFT, TRAP, SIMP?
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21:44:07 ********************************************* Your solution: Error = actual value - approximation so 2.346-4.0 = 1.654 Left(30) = n1/n2 = 10/30 * 1.654= .55 4.0 - .55= 3.45 Trap(30)=(n1/n2)^2 = (10/30)^2 * 1.654 = .1837 4.0 - .1837= 3.8163 Simp(30)=(n1/n2)^4=(10/30)^4 * 1.654 = .0204 4.0- .0204= 3.9796 Confidence Assessment: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Confidence Assessment: 3
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Given Solution: LEFT and RIGHT approach the exact value in proportion to the number of steps used. MID and TRAP approach the exact value in proportion to the square of the number of steps used. SIMP approachs the exact value in proportion to the fourth power of the number of steps used. Using these principles we can work out this problem as follows: ** The original 10-step estimate is 2.346, which differs from the actual value 4.000 by -1.654. If the original estimate was done by LEFT then the error is inversely proportional to the number of steps and the n = 30 error is (10/30) * -1.654 = -.551, approximately. So the estimate for n = 30 would be -.551 + 4.000 = 3.449. If the original estimate was done by TRAP then the error is inversely proportional to the square of the number of steps and the n = 30 error is (10/30)^2 * -1.654 = -.184, approximately. So the estimate for n = 30 would be -.184 + 4.000 = 3.816. If the original estimate was done by SIMP then the error is inversely proportional to the fourth power of the number of steps and the n = 30 error is (10/30)^4 * -1.654 = -.020, approximately. So the estimate for n = 30 would be -.02 + 4.000 = 3.98. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: ********************************************* Question: problem 7.6.5 (previously problem 7.6.10) If TRAP(10) = 12.676 and TRAP(30) = 10.420, estimate the actual value of the integral. **** What is your estimate of the actual value and how did you get it? ********************************************* Your solution: I’m not sure how to go about doing this problem. Confidence Assessment: 0 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Confidence Assessment: 0
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Given Solution: ** You need to use the inverse square proportionality of the error with the number of steps. Trap(30) is approximately (10/30)^2 = 1/9 of TRAP(10). So the difference 10.420 - 12.676 = -2.256 between TRAP(10) and TRAP(30) is approximately 8/9 of the error of TRAP(10). It follows that the error of TRAP(10) is 9/8 * -2.256 = -2.538. Our best estimate of the integral is therefore -2.538 + 12.676 = 10.138. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I know that Trap(30) is = 1/9 of Trap(10) because (10/30)^2 = (1/3)^2 = 1/9. The difference is 10.420 - 12.676 = -2.256 How do you figure out that the difference between TRAP(10) and TRAP(30) is approximately 8/9 of the error of TRAP(10)? Self-critique Rating:3 ********************************************* Question: problem 7.7.2 (previously 7.7.12) integrate 1 / (u^2-16) from 0 to 4 if convergent ********************************************* Your solution: The limit I undefined at 0 so I substituted b for 0. integral 1 / (u^2-16) from 0 to 4= limit as b approaches 0 = integral of b to 4 of 1 / (u^2-16) = limit as b approaches 0 of -1/2 (u^2 – 16)^-2 = limit as b approaches 0 of -1/2(b^2 – 16) + ½ (4^2 – 16) Since the limit as b approaches 4 of -1/2(b^2 – 16) does not exist, the integral diverges. Confidence Assessment: 1-2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Confidence Assessment: 1-2
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Given Solution: 1 / (u^2-16) = 1 / [(u+4)(u-4)] . Since for 0 < x < 4 we have 1/8 < 1 / (u+4) < 1/4, the integrand is at most 1/4 times 1/(u-4) and at least 1/8 of this quantity, so the original integral is at most 1/4 as great as the integral of 1 / (u-4) and at least 1/8 as great. That is, 1/8 int(1 / (u-4), u, 0, 4) < int(1 / (u^2-4), u, 0, 4) < 1/4 int(1 / (u-4), u, 0, 4). Thus if the integral of 1 / (u-4) converges or diverges, the original integral does the same. An antiderivative of 1 / (u-4) is ln | u-4 |, which is just ln(4) at the limit u=0 of the integral but which is undefined at the limit u = 4. We must therefore take the limit of the integral of 1/(u-4) from u=0 to u=x, as x -> 4. The integral of 1 / (u-4) from 0 to x is equal to ln (4) - ln(x-4) = ln( 4 / (x-4) ). As x approaches 4 the denominator approaches 0 so the fraction approaches infinity and the natural log approaches infinity. Thus the integral diverges. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Did you get 1/8 and 1/4 by substituting 0 and 4 in for u in the original function? If this is the case then I understand the first part of the solution.
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Given Solution: ** First graph the function using standard graphing techniques from first-semester calculus: This graph increases at first as you move to the right from t = 0. However e^(-.5 t) eventually approaches zero much faster than t increases so the graph has an asymptote at the positive t axis. So it increases for small positive t but eventually returns almost to the t axis, and it can't be strictly increasing. Its concavity changes from downward (negative) for small positive t to upward for larger t; the point at which the concavity changes is important. We use the standard technique from first-semester calculus to find the point at which this function maximizes. The first derivative is dr/dt = 1000 e^(-.5 t) - 500 t e^(-.5 t). Setting this derivative equal to 0 we get • 1000 e^(-.5 t) - 500 t e^(-.5 t) = 0; dividing through by e^-.5 t we get the equation 1000 - 500 t = 0, which is easily solved to obtain t = 2. A first-or second-derivative test confirms that the t = 2 graph point is a relative maximum. Concavity is determined by the second derivative r'' = e^(-.5 t) [ -1000 + 250 t ], which is 0 when t = 4. This is a point of inflection because the second derivative changes from negative to positive at this point. So the function is concave downward on the interval (-infinity, 4) and concave upward on (4, infinity). The first derivative has a critical point where the second derivative is zero. This occurs at x = 4, which was identified in the preceding paragraph as the point of inflection for the original function. Since the second derivative goes from negative to positive, this point is a minimum of the first derivative. The first derivative is a decreasing function from t = 0 to t = 4 (2d derivative is negative) and is then an increasing function with asymptote y = 0, the x axis, which it approaches through negative values. Its maximum value for t >= 0 is therefore at t = 0. ** People are getting sick the fastest when the rate of infection is highest. This occurs at the relative maximum of the rate function, which was found above to occur at t = 2. Thus people are getting sick the fastest 2 days after the epidemic begins. To find how many people get sick during a time interval, you integrate the rate function over that interval. In this case the interval doesn't end; so you need to integrate the rate function r = 1000 t e^(-.5t) from t = 0 until forever, i.e., from t = 0 to t = infinity. An antiderivative of the function is F(t) = 1000 int ( t e^(-.5 t)) = 1000 [ -2 t e^(-.5t) - int ( e^(-.5 t) ) ] = 1000 [ -2 t e^(-.5 t) - 4 e^(-.5 t) ]. Integrating from 0 to x gives F(x) - F(0) = 1000 [ -2 t e^(-.5 x) - 4 e^(-.5 x) ] - 1000 [ -2 * 0 e^(-.5 *0 ) - 4 e^(-.5 * 0 ) ] = 1000 e^-(.5 x) [ -2 t - 4 ] - (-4000). As x -> infinity, e^-(.5 x) [ -2 t - 4 ] -> 0 since the exponential will go to 0 very much faster than (-2 x - 4) will approach -infinity. This leaves only the -(-4000) = 4000. ** The calculator is fine for checking yourself, but you need to use the techniques of calculus to determine inflection points, maxima, minima etc.. The careful use the calculator to enhance rather than replace mathematical understanding. I get a lot of students in these courses who are now at 4-year institutions and who have taken courses based on the graphing calculator, or even TI-92, and many of them tend to have a very difficult time in courses that don't permit them, and in courses were mathematical understanding is required. ** ** You have to use the techniques of calculus to determine these behaviors. Plugging values in won't show you the exact location of intercepts, maxima, minima, etc.. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I didn’t realize you had to take the derivative of the function to find how many people are sick at what time. I thought you could just substitute the amount of days into t of the original function to get this. After reading the solution I now understand this problem.