When finding relative maxima and minima don’t forget to use the first or second derivative test to show that the critical value is a relative maxima or minima. Also, when sketching a graph use EXCEL or graph paper with the scale of the axes labeled. Any other sketch will be marked incorrect.
problem1. Graph each function, then specify its domain, range, x and y intercepts, vertical and horizontal asymptotes, relative minima and maxima, inflection points, the regions where the function is decreasing and increasing, 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)
problem2. The cost function to create 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 out the number of brushes that will create the maximum profit. (C) Find out the price which produces the maximum profit. (D) Find out the maximum profit.
problem3. A fence should 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 required 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 out the cost of the least expensive fence and give the dimensions which minimize the cost of the fencing.
problem4. 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 declining at a rate of 20 units per day, at what rate is the price changing when the demand is 20 units?
problem5. A cable TV company has 1,000 customers paying $80 every month. For each $4 decline in price, it attracts 100 new customers. Find out the price which yields maximum revenue, the quantity that maximizes revenue, and the maximum revenue.
problem6. Given f(x) = (1.3x4-.02x3+.25x2-x+5). find out the first and second derivatives when x = -1, x = 0, x = 1, x=2, and x=3.
problem7. Profit is maximized when marginal revenue (MR) equals marginal cost (MC). Use this information 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.
problem8. Distinguish the following functions implicitly. (A) x3 + y3 = 3 (B) √x+√y=2 (C ) xy + y2 = x
problem9. Find out 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)).
problem10. Differentiate the following functions. (A) f(x) = (1 – 5x)4 (B) f(x) = 1 / (√x + 3)4.