Rates of Change Velocity Recall that the difference between speed and velocity is that
velocity has direction and speed does not. In other words, the speed is
the absolute value of velocity. We have seen that the secant line can be
used to approximate the velocity. The formula for average
velocity is
Example The height of a pine cone t seconds after falling from a tree is given by s = -16t2 + 100 A. Find the average velocity in the interval [0,2] B. Find the instantaneous velocity for t = 0 and t = 1 C. How long will it take to hit the ground D. Find the velocity when it hits the ground Solution: A. We compute
s(b) - s(a)
100 - 36
B. s'(t) = -32t so s'(0) = 0 and s'(1) = -32
C. We solve -16t2 + 100 = 0 or t = 2.5 D. s'(2.5) = -80
Marginals In economics the key terms are revenue, cost, and profit. We use the word marginal to indicate the additional cost of producing one more. In calculus terms marginal means the derivative. Example Suppose that the cost of producing x burgers per hour is C(x) = 1000/x + x for x > 35 The burgers cost 2.50 each. Find the marginal cost, marginal revenue, and marginal profit of producing 40 burgers
Solution: The marginal cost is C'(x) = (1000x -1 + x)' = -1000x -2 + 1 so that C'(40) = -1000/1600 + 1 = .375 so that the additional cost of producing 40 burgers is about 38 cents
Revenue is price times number sold R = 2.5x so R' = 2.5 so that the marginal revenue is $2.50. The profit is Profit = Revenue - Cost so the marginal profit is marginal revenue - marginal cost = 2.5 - .38 = 2.12
Example You are the owner of the Tahoe View Inn and have experimented with different pricing strategies. When you charged $80 per night, you were able to average 50 occupied rooms and when you charged $90 per night, you were able to average 45 occupied rooms. Find the marginal revenue at an average of 50 occupied rooms assuming the demand function is linear. Then interpret its meaning.
Solution We need to first find the demand function. Since it is linear, we need to find its slope. We calculate
90 - 80 Using the point slope formula for the equation of a line gives p - 80 = -2(x - 50) = -2x + 100 p = -2x + 180 Since Revenue = (Price)(Quantity) we have R(x) = (-2x + 180)(x) = -2x2 + 180x To find the marginal revenue we calculate R'(x) = -4x + 180 Now plug in x = 50 to get R'(50) = -4(50) + 180 = -20 Since the marginal revenue is negative, this says that increasing the occupancy rate will decrease revenue at a rate of $20 per unit. We should strive to decrease the occupancy rate by raising the price. Later on, we will learn how to find the price at which the maximal revenue will be achieved.
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