Reference no: EM13347108

1. Graph each function, then specify its domain, range, x and y intercepts, vertical and horizontal asymptotes, relative maxima and minima, inflection points, the regions where the function is increasing and decreasing, and the regions where the function is concave up and concave down.

(A) f(x) = -2x³ - (x²/2) + x -3  (B) g(x) = 2x²/(3 - x)

2. The cost function to produce q electric cat brushes is given by C(q) = -10q² + 250q. The demand function is given by p = -q²-3q+299, where p is the price in dollars.

(A) State the profit function. (B) Find the number of brushes that will produce the maximum profit. (C) Find the price that produces the maximum profit. (D) Find the maximum profit.

3. A fence must be built in a large field to enclose a rectangular area of 25,000 square meters. One side of the area is bounded by an existing fence. And no fence is needed there. Material for the fence costs $3 per meter for the two ends and $1.50 per meter for the side opposite the existing fence. Find the cost of the least expensive fence and provide the dimensions that minimize the cost of the fencing.

4. Suppose the price p (in dollars) of a product is given by the demand function p = (1000-10x)/(400-x), where x represents the quantity demanded. If the daily demand is decreasing at a rate of 20 units per day, at what rate is the price changing when the demand is 20 units?

5. A cable TV company has 1,000 customers paying $80 each month. For each $4 reduction in price, it attracts 100 new customers. Find the price that yields maximum revenue, the quantity that maximizes revenue, and the maximum revenue.

6. Given f(x) = (1.3x⁴-.02x³+.25x²-x+5). Evaluate the first and second derivatives when x = -1, x = 0, x = 1, x=2, and x=3.

7. Profit is maximized when marginal revenue (MR) equals marginal cost (MC).  Use this fact to maximize profit when C(q) = 5000 + 250q -  .01q² and R(q) = 400q - .02q². Then obtain the profit function and maximize it. Then compare your answers.

8. Differentiate the following functions implicitly. (A) x³ + y³ = 3      (B)  √x + √y = 2  (C ) xy  + y² = x

9. Find the equation of the tangent line to the curve passing through the given point.                    

(A) f(x) = 4x(x⁵ +1), (1,8).       (B) f(x) = x² / (1 + √x , (4, (16/3)). Use calculators to find m.

10.    Differentiate the following functions. (A) f(x) = (1 - 5x)⁴     (B)  f(x) =  1 / (√x + 3)⁴.