Donnie Smith Assignment Section 6.5 due 04/13/2017 at 11:59pm MST 1. (1 point) Given the following integral and value of n, approximate the following integral using the methods indicated (round your answers to six decimal places): Z 1 Sharma MAT 266 ONLINE B Spring 2017 \u0011 2 e5x dx, n = 4 = Midpoint = \u0010 0 (a) Trapezoidal Rule (b) Midpoint Rule \u0011 = Trapezoid = Simpson's = (c) Simpson's Rule Answer(s) submitted: Answer(s) submitted: 0.3954 0.3959 0.3955 (correct) 2. (1 point) A radar gun was used to record the speed of a runner during the first 5 seconds of a race (see table). Use Simpson's rule to estimate the distance the runner covered during those 5 seconds. t(s) v(m/s) 0 0 0.5 2.4 1 3.3 1.5 6.2 2 6.75 2.5 8.55 3 9.45 3.5 10.15 4 10.45 Answer(s) submitted: 37.141 (correct) 3. (1 point) Estimate \u0010 Left = R 4.75 1.5 |2 x| dx using 5 divisions and 4.5 10.75 1.25 1 2.25 3.5 4.75 6 21.875 1.25 2.25 3.5 4.75 6 7.25 29.6875 5 1.25 10.75 1.125 1.75 (score 0.217391304347826) 4. (1 point) R x Consider the integral approximation T20 of 05 2e 4 dx. Does T20 overestimate or underestimate the exact value? \u0011 A. overestimates B. underestimates = Right = Find the error bound for T20 without calculating TN using the result that M(b a)3 Error(TN ) , 12N 2 \u0010 1 where M is the least upper bound for all absolute values of the x second derivatives of the function 2e 4 on the interval [a, b]. Error(T20 ) Answer(s) submitted: A 0.003255 (correct) 1. Take the sample points from the left-endpoints. Answer: L4 = 2. Is your estimate L4 an underestimate or overestimate of the true area? Choose one Underestimate Overestimate 3. Take the sample points from the right-endpoints. Answer: R4 = 4. Is your estimate R4 an underestimate or overestimate of the true area? Choose one Underestimate Overestimate 5. Use the Trapezoid Rule with n = 4. Answer: T4 = 6. Is your estimate T4 an underestimate or overestimate of the true area? Choose one Underestimate Overestimate 5. (1 point) Use six rectangles to find an estimate of each type for the area under the given graph of f from x = 0 to x = 12. 1. Take the sample points from the left-endpoints. Answer: L6 = 2. Is your estimate L6 an underestimate or overestimate of the true area? Choose one Underestimate Overestimate 3. Take the sample points from the right-endpoints. Answer: R6 = 4. Is your estimate R6 an underestimate or overestimate of the true area? Choose one Underestimate Overestimate 5. Take the sample points from the midpoints. Answer: M6 = Note: You can click on the graph to enlarge the image. Answer(s) submitted: 20.8 Underestimate 25.2 Overestimate 23 Overestimate (correct) Note: You can click on the graph to enlarge the image. 7. (1 point) R The graph of a function f is given below. Estimate 08 f (x) dx using four subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints. R (a) 08 f (x) dx R (b) 08 f (x) dx R (c) 08 f (x) dx Answer(s) submitted: 47.78 Overestimate 39.78 Underestimate 44.12 (correct) 6. (1 point) Use four rectangles to find an estimate of each type for the area under the given graph of f from x = 1 to x = 9. 2 0.8675, 0.8632, and 0.9540, and the same number of subintervals were used in each case. Answer(s) submitted: 4.2 6.2 10 (a) Which rule produced which estimate? (correct) ? ? ? ? 8. (1 point) Given the following graph of the function y = f (x) and n = 6, answer the following questions about the area under the curve from x = 0 to x = 6. of 1. 2. 3. 4. (b) Between which two approximations does the true value 0 f (x) dx lie? R2 1. Use the Trapezoidal Rule to estimate the area. Answer: T6 = 2. Use Simpson's Rule to estimate the area. Answer: S6 = Right-hand estimate Left-hand estimate Midpoint Rule estimate Trapezoidal Rule estimate A. 0.8632 < 02 f (x) dx < 0.8675 B. No conclusions can be drawn. R C. 0.8675 < 02 f (x) dx < 0.9540 R D. 0.7811 < 02 f (x) dx < 0.8632 R Answer(s) submitted: Note: You can click on the graph to enlarge the image. 7811 9540 8632 8675 A (correct) Answer(s) submitted: 11. (1 point) Consider the four functions shown below. On R the first two, an approximation for ab f (x) dx is shown. 30.5 29.667 (incorrect) 9. (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. T4 M4 S4 1. 2. 3. 4. Answer(s) submitted: (Click on any graph to get a larger version.) 1. For graph number 1, 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 11.5 12 11.66 (correct) 10. (1 point) The left, right, Trapezoidal, and Midpoint Rule approximaR tions were used to estimate 02 f (x) dx, where f is the function whose graph is shown below. The estimates were 0.7811, 3 B. under 2. For graph number 2, Which integration method is shown? A. left rule B. trapezoid rule C. midpoint rule D. right rule Is this method an over- or underestimate? A. under B. over 3. On a copy of graph number 3, sketch an estimate with n = 2 subdivisions using the right rule. Is this method an over- or underestimate? A. over B. under 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 U B (correct) 13. (1 point) Use six rectangles to find an estimate of each type for the area under the given graph of f from x = 0 to x = 12. 1. Take the sample points from the left-endpoints. Answer: L6 = 2. Is your estimate L6 an underestimate or overestimate of the true area? Choose one Underestimate Overestimate 3. Take the sample points from the right-endpoints. Answer: R6 = 4. Is your estimate R6 an underestimate or overestimate of the true area? Choose one Underestimate Overestimate 5. Take the sample points from the midpoints. Answer: M6 = Answer(s) submitted: D A A A B B (correct) 12. (1 point) R Estimate 01 cos(x2 ) dx using (a) the Trapezoidal Rule and (b) the Midpoint Rule, each with n = 4. Give each answer correct to five decimal places. (a) T4 = (b) M4 = Note: You can click on the graph to enlarge the image. Answer(s) submitted: (c) By looking at a sketch of the graph of the integrand, determine for each estimate whether it overestimates, underestimates, or is the exact area. ? 1. M4 ? 2. T4 (score 0.4000000059604645) (d) What can you conclude about the true value of the integral? 75.6 Overestimate 83.6 Underestimate 80.2 14. (1 point) Use six rectangles to find an estimate of each type for the area under the given graph of f from x = 0 to x = 12. A. M4 < 01 cos(x2 ) dx < T4 R B. T4 < 01 cos(x2 ) dx < M4 C. No conclusions can be drawn.R R D. T4 < 01 cos(x2 ) dx and M4 < 01 cos(x2 ) dx R R E. T4 > 01 cos(x2 ) dx and M4 > 01 cos(x2 ) dx R Answer(s) submitted: 0.895759 0.908907 O 4 1. Take the sample points from the left-endpoints. Answer: L6 = 2. Is your estimate L6 an underestimate or overestimate of the true area? Choose one Underestimate Overestimate 3. Take the sample points from the right-endpoints. Answer: R6 = 4. Is your estimate R6 an underestimate or overestimate of the true area? Choose one Underestimate Overestimate 5. Take the sample points from the midpoints. Answer: M6 = Answer(s) submitted: (incorrect) 15. (1 point) Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as divergent . Z 0 Answer(s) submitted: 7 Note: You can click on the graph to enlarge the image. (correct) c Generated by WeBWorK, http://webwork.maa.org, Mathematical Association of America 5 7ex dx