Reference no: EM132347926

Math Questions -

Q1. Evaluate the improper integrals:

(a) ₀∫^∞ e^-2x sinx dx

(b) ₁∫¹⁰ 1/√(x-1) dx (the integrand is discontinuous at x = 1)

Q2. (a) A radar gun was used to record the speed of a car during the first 4 seconds after starting (see the table). Find the Midpoint Rule, the Trapezoidal Rule, and the Simpson's Rule approximation with n = 4 subintervals to estimate the distance the car covered during these 4 seconds (recall that distance x = _a∫^bv(t)dt).

t(s)	v (in sec)
0	0
0.5	17
1.0	26.5
1.5	35
2.0	42
2.5	57

 

x(s)	v (in sec)
3.0	72
3.5	77.5
4.0	82

(b) Use Simpson's Rule with n = 8, to approximate ₀∫^2x√(1+cos2(x))dx.

Q3. (a) Compute the length of the curve y = ¼x² - ½lnx, 1 ≤ x ≤ 2.

(b) Compute the area of the surface obtained by rotating y = 1/6 x³ + 1/2x, ½ ≤ x ≤ 1, about the x-axis.

(c) Compute the area of the surface obtained by rotating y = -2x + 8, 0 ≤ x ≤ 4, about the y-axis. (You will need to solve for x, to obtain x = g(y), 0 ≤ y ≤ 8, and apply the surface area formula). Verify that your result is S = πrl, with r = radius of the circular base and l = slant height of the resulting cone.

Q4. Sketch the graph of f(x) = 1/6(x²+4)^½, 0 ≤ x ≤ 6, and find the length of this curve.

Q5. (a) A system of four masses, of 20kg, 16kg, 10 kg and 5 kg are located at (-3, 4), (2, 6), (4, -6), and (5, 2), respectively. Find the center of mass.

(b) Find the centroid of the region bounded by the lines x = 5, y = 6/5 x and y = 1/5 x.

(c) Find the centroid of the region bounded by the x-axis, the line x = 9, and the curve y = 3√x.

Q6. Solve the initial value problem: dy/dx = (x - 2)(y + 1), y(2) = 5.

Q7. A bottle of soda at room temperature of 72^oF is placed in a refrigerator where the temperature is 42^oF. After 15 minutes, the soda has cooled to 60^oF.

(a) Determine an equation that gives the temperature of the soda after t minutes, where t represents the time in minutes since the bottle of soda was placed inside the refrigerator.

(b) What will be the temperature of the soda after another 15 minutes (that is, ¼ hour from the time when it was placed in the refrigerator)?

(c) How long will it take for the bottle of soda to cool down to 50^oF?

Q8. (a) Find T₁(x), the Taylor polynomial of degree 3 centered at a = 5, for the function f(x) = √(x-4).

(b) Draw a sketch of the graphs of y = √(x-4) and y = T₁(x).

(c) Evaluate f(5.22) and T₁(5.22).