Assignment:
Q1. Determine the following antiderivatives (don’t worry about simplifying, just show the rules)
a. ∫(x200 + √x- e2πx) dx
b. ∫x(4x2 - 3)47 dx
c. ∫(3/x + 5/x4 - 8) dx
Q2. Calculate the value of each definite integral (Show work!):
a. ∫10 (x2 / 2x3 + 9) dx
b. ∫42 x2 (6x3 + 3) dx
c. ∫∞2 3/x4 dx
Q3. a. Approximate the area under the curve f(x) = 1/√x and above the x-axis by splitting the region from x=1 to x = 9 into 4 equal subintervals (rectangles) and using the midpoints of the subintervals as the heights.
b. Use the Simpson’s Rule with n = 4 to estimate area under the curve f(x) = 1/√x.
c. Find the exact value of ∫91 1/√x dx , and compare with the answer obtained from part (a), and (b). Which method is more accurate, part A or part B?
Q4. Determine the area between the curves f(x) = x2-2 and g(x) = x.
Q5. Use geometry to determine the value of the following definite integral: ∫3-3 √9-x2 dx
Q6. A stock analyst plots the price per share of a certain stock as a function of time and finds that it can be modeled by the function s(t) = 25-2√10t where t is the time (in years) since the stock was purchased. Find the average price of the stock over the first 2 years of its purchase.
Q7. Use the consumer’s surplus formula ∫qo0 [D(q) - po] dq to determine the consumer’s surplus for the demand equation D(q) = 1400- 5q, if we assume supply and demand are in equilibrium when q = 15.
Q8. Sketch the region and then calculate the volume of the solid of revolution formed by rotating the region bounded by f(x) = x2 + 3 y = 0 , x= 1 and x = 5 around the x-axis.
Q9. The function represents the rate of flow of money in dollars per year. Assume a 8-year period for t and a rate r of 10% compounded continuously and determine the following:
a. The present value (P = ∫t0 f(x)e-rx dx)
b. The accumulated amount (A = ert ∫t0 f(x)e-rx dx)
Provide complete and step by step solution for the question and show calculations and use formulas.