Webwork Test 1.pdf - Zachary Dorff Assignment Test 1 Fall B 2019 due at 03:36pm MST Problem 2 8(1 point Consider the four functions shown below On the

Webwork Test 1.pdf - Zachary Dorff Assignment Test 1 Fall B...

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Zachary Dorff Zhu MAT 266 ONLINE B Fall 2019 Assignment Test 1 Fall B 2019 due 10/29/2019 at 03:36pm MST Problem 1. 4. (1 point) Estimate the area under the graph in the figure by using (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule, each with n = 4. T 4 M 4 S 4 Solution: SOLUTION We have a = 0 , b = 4 and n = 4. So Δ x = b - a n = 4 4 = 1, and x k = a + k Δ x so x 0 = 0, x 1 = 1 , x 2 = 2, x 3 = 3, and x 4 = 4. We use the graph of the function to estimate the values of the function at the required values of x : 1. T 4 = Δ x 2 h f ( x 0 )+ 2 f ( x 1 )+ 2 f ( x 2 )+ 2 f ( x 3 )+ f ( x 4 ) i = 1 2 h f ( 0 )+ 2 f ( 1 )+ 2 f ( 2 )+ 2 f ( 3 )+ f ( 4 ) i 1 2 h 0 + 2 · 3 + 2 · 5 + 2 · 3 + 1 i = 11 . 5 2. The midpoints are ¯ x 1 = 0 . 5, ¯ x 2 = 1 . 5, ¯ x 3 = 2 . 5 and ¯ x 4 = 3 . 5 M 4 = Δ x h f ( ¯ x 1 )+ f ( ¯ x 2 )+ f ( ¯ x 3 )+ f ( ¯ x 4 ) i = 1 · h f ( 0 . 5 )+ f ( 1 . 5 )+ f ( 2 . 5 )+ f ( 3 . 5 ) i 1 · h 1 + 4 . 5 + 4 . 5 + 2 i = 12 3. S 4 = Δ x 3 h f ( x 0 )+ 4 f ( x 1 )+ 3 f ( x 2 )+ 4 f ( x 3 )+ f ( x 4 ) i = 1 3 h f ( 0 )+ 4 f ( 1 )+ 2 f ( 2 )+ 4 f ( 3 )+ f ( 4 ) i 1 3 h 0 + 4 · 3 + 2 · 5 + 4 · 3 + 1 i = 35 3 Correct Answers: 1/2*(0+2*3+2*5+2*3+1) 1*(1+4.5+4.5+2) 1/3*(0+4*3+2*5+4*3+1) Problem 2. 8. (1 point) Consider the four functions shown below. On the first two, an approximation for R b a f ( x ) dx is shown. 1. 2. 3. 4. (Click on any graph to get a larger version.) 1. For graph number 1, Which integration method is shown? A. right rule B. midpoint rule C. left rule D. trapezoid rule Is this method an over- or underestimate? A. over B. under 2. For graph number 2, Which integration method is shown? A. trapezoid rule B. left rule C. right rule D. midpoint rule Is this method an over- or underestimate? A. over B. under 3. On a copy of graph number 3, sketch an estimate with n = 2 subdivisions using the midpoint rule. Is this method an over- or underestimate? A. under B. over 4. On a copy of graph number 4, sketch an estimate with n = 2 subdivisions using the trapezoid rule. Is this method an over- or underestimate? A. under B. over 1
Solution: SOLUTION We can see by inspection or by looking at how the functions increase/decrease or their concavity that graph 1 shows the right rule. Similarly, graph 2 shows the right rule. The graphs showing the numerical estimates for graphs 3 and 4 are shown below. 3. 4. (Click on any graph to get a larger version.) Then graph 3, showing the midpoint rule. Then graph 4, showing the trapezoid rule.

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