31.5. Let be the circle |+1| = 2 traversed twice in the clockwise direction. Evaluate dz (22 + 2)²

Using FDF, BDF, and CDF, find the first derivative; 1. The distance x of a runner from a fixed point is measured (in meters) at an interval of half a second. The data obtained is: t 0 x 0 0.5 3.65 1.0 1.5 2.0 6.80 9.90 12.15 Use CDF to approximate the runner's velocity at times t = 0.5s and t = 1.5s 2. Using FDF, BDF, and CDF, find the first derivative of f(x)=x Inx for an input of 2 assuming a step size of 1. Calculate using Analytical Solution and Absolute Relative Error: = True Value - Approximate Value| x100 True Value 3. Given the data below where f(x) sin (3x), estimate f(1.5) using Langrage Interpolation. x 1 1.3 1.6 1.9 2.2 f(x) 0.14 -0.69 -0.99 -0.55 0.31 4. The vertical distance covered by a rocket from t=8 to t=30 seconds is given by: 30 x = Loo (2000ln 140000 140000 - 2100 9.8t) dt Using the Trapezoidal Rule, n=2, find the distance covered. 5. Use Simpson's 1/3 and 3/8 Rule to approximate for sin x dx. Compare the results for n=4 and n=8

1. A Blue Whale's resting heart rate has period that happens to be approximately equal to 2π. A typical ECG of a whale's heartbeat over one period may be approximated by the function, f(x) = 0.005x4 2 0.005x³-0.364x² + 1.27x on the interval [0, 27]. Find an nth-order Fourier approximation to the Blue Whale's heartbeat, where n ≥ 3 is different from that used in any other posts on this topic, to generate a periodic function that can be used to model its heartbeat, and graph your result. Be sure to include your chosen value of n in your Subject Heading.

7. The demand for a product, in dollars, is p = D(x) = 1000 -0.5 -0.0002x² 1 Find the consumer surplus when the sales level is 200. [Hints: Let pm be the market price when xm units of product are sold. Then the consumer surplus can be calculated by foam (D(x) — pm) dx]

4. Find the general solution and the definite solution for the following differential equations: (a) +10y=15, y(0) = 0; (b) 2 + 4y = 6, y(0) =

5. Find the solution to each of the following by using an appropriate formula developed in the lecture slides: (a) + 3y = 2, y(0) = 4; (b) dy - 7y = 7, y(0) = 7; (c) 3d+6y= 5, y(0) = 0

1. Evaluate the following improper integrals: (a) fe-rt dt; (b) fert dt; (c) fi da dx

8. Given the rate of net investment I(t) = 9t¹/2, find the level of capital formation in (i) 16 years and (ii) between the 4th and the 8th years.

9. If the marginal revenue function of a firm in the production of output is MR = 40 - 10q² where q is the level of output, and total revenue is 120 at 3 units of output, find the total revenue function. [Hints: TR = √ MRdq]