HSC Mathematics integration 5.3 numerical methods

Transcription

1 MD53a HSC Mathematics integration 5.3 numerical methods student name Monday, 3 May 6 number о number о a 8 b 9 9 a b a 3 b 4 5 result out of 6 applied to the system

2 MD53a

3 5.3 numerical methods 3 Question Calculate the area between the curve the trapezoidal rule with one strip. y x, the x-axis and the bounds x and x using Question Calculate the area between the curve trapezoidal rule with two strips. y x, the x-axis, and the bounds x and x using Question 3 A geographer takes depth readings across a river at 5 metres intervals, as shown in the table. Use the trapezoidal rule to estimate the area of the cross-section. distance across depth in metres Question 4 Use the trapezoidal rule once to estimate the area enclosed by the curve lines x, x and the x-axis. y x x, the Question 5 Use the trapezoidal rule to estimate the area enclosed by a curve which passes through the points given in the table and the x-axis. x 3 4 y Question 6 Use the trapezoidal rule with two strips to estimate the area bounded by the curve the lines x, x, and the x-axis. y x,

4 4 5.3 numerical methods Question 7 Use the trapezoidal rule with three function values to estimate in fraction. dx leaving the answer x Question 8 Use the trapezoidal rule with seven intervals to estimate to three significant figures. 6 xdx, giving your answer correct 9 Question 9 Use the trapezoidal rule with four intervals to estimate to three significant figures. sinxdx, giving your answer correct Question Calculate x dx using Simpson s rule with three functions values. Question Calculate x dx using Simpson s rule with five function values. Question A surveyor measures offsets from a straight road to a river. The offsets are spaced at metres intervals. The results are given in the table. distance 3 4 offset Estimate the area of this shape using Simpson s rule correcting to decimal place.

5 5.3 numerical methods 5 Question 3 Calculate dx using Simpson s rule with five function values correcting to 3 decimal places. x Question 4 The velocity of a particle is measured at second intervals and the results tabulated below. time (seconds) velocity (m/s) Use Simpson s rule to estimate the distance travelled correcting to decimal place. Question 5 The curve y x x between x and x is rotated about the x-axis. Estimate the volume of the solid formed using Simpson s rule with four sub-intervals correcting to decimal places. Question 6 Use Simpson s rule with three function values to estimate the area between the curve the lines x and x, and the x-axis. y x, Question 7 Use Simpson s rule with five function values to estimate the area bounded by the curve y x, the lines x and x, and the x-axis. Question 8 For an approximation to the area under f x Simpson s rule with four equal sub-intervals; 4 x above the x-axis, x and x.6, using (a) setup the expression for the approximation (b) calculate the approximation correcting to four decimal places.

6 6 5.3 numerical methods Question 9 A researcher takes depth readings at metre intervals across a canal. His results are shown in the table. distance across depth in metres (a) Use Simpson s rule to estimate the cross-sectional area. (b) If the water in the canal was flowing at km/h, estimate the volume of the flowing past this point each hour. Question Use Simpson's rule to find an approximation for 5 dx using 6 subintervals. x Question For an approximate value for the area bounded by y, the x-axis and the ordinates at x and x x, using the trapezoidal rule with ten subintervals; (a) setup the expression for the approximation (b) calculate the approximation, correcting to seven decimal places. Question For an approximation to the value of calculate the approximation. x 3 dx, using the trapezoidal rule with five trapezia, Question 3 For an approximation to 5 log 3 xdx using Simpson rule with three function values, calculate the approximation correcting to nine decimal places.