Determine the domain and range of the variables - determine

Question

Question 1) Consider quadratic function y = ax² + bx + c:

a. Determine the equation of the above function if the parabola passes through a point with x-intercept of 10 and another point with y-intercept of 0.

b. Determine the domain and range of the variables.

c. How should your solutions to parts (a) and (b) be modified it y = ax² + bx + c is a revenue function?

Question 2) Prove the following summation expressions:

∑_t=0^∞ x_i = 1/(1-x)    where |x| < 1

Question 3) Consider a firm's quadratic revenue function, R =p.q = aq² + bq, linear cost function. C = vq + f, where a and b are constant terms, v is unit variable cost and f is total fixed cost. Variables R, C, and p are measured in dollars and q equals number of units sold. The firm experiences 2 break-even points at (500, $10,000) and (1,800, $12,600).

a. Determine the equation of total revenue R = f(q)= aq² + bq .
b. Determine the quantity q* and price p* that maximizes total revenue R .
c. Calculate the maximum total revenue R*.
d. Determine the restricted domain of p and restricted ranges of q and R
e. Determine the equation of total cost C and calculate it at the quantity q*.
f. Calculate variable and fixed costs at the quantity q*.
g. Calculate average cost, AC at the quantity q*.
h. Determine the restricted range of C.
i. Write the equation of profit function Π.
j. Determine the level of quantity q₁ and price p₁ that maximizes profit.
k. Calculate total revenue R₁, total cost C₁ and profit Π₁, at the quantity q₁.
l. Determine the restricted range of Π.
m. On one graph depict total revenue, total cost, and profit functions. Label the axes, the functions, the intercepts, the optimum point(s) and the origin.