Chapter 2.3 Rates of Change: Velocity and Marginals

two primary applications of derivatives*

1. slope: the derivative of f is a function that gives the slope of the graph of f at point (x, f(x)) 2. rate of change: the derivative of f is a function that gives the rate of change of f(x) with respect to x at the point (x,f(x))

an equation that relates marginal profit, marginal revenue, and marginal cost

P = R - C where P = total profit, R = total revenue, and C = total cost

rates of change in economics

another important use of rates of change is in the field of economics. Economists refer to marginal profit, marginal revenue, and marginal cost as the rates of change of the profit, revenue, and cost with respect to x, the number of units produced or sold.

when determining the rate of change of one variable with respect to another, you must be careful to distinguish between what

average and instantaneous rates of change. (the distinction between these two rates of change is comparable to the distinction between the slope of the secant line through two points on a graph and the slope of the tangent line at one point on the graph)

how to approximate instantaneous rate of change

calculate the average rate of change over smaller and smaller intervals of the form [x, x+(delta(x))]

derivatives of profit, revenue, and cost*

dP/dx: marginal profit, dR/dx: marginal revenue, and dC/dx: marginal cost

definition of average rate of change**

if y = f(x), then the average rate of change of y with respect to x on the interval [a,b] is = [ f(b) - f(a) ] / (b-a) = delta(y)/delta(x) (note that f(a) is the value of the function at the left endpoint of the interval, f(b) is the value of the function at the right endpoint of the interval, and b-a is the width of the interval.

discrete variable*

in many business and economics problems, the number of units produced or sold is restricted to nonnegative integer values

average velocity*

of an object that is moving in a straight line: average velocity = change in distance / change in time

study tip for real-life rate of change problems

remember to include units for x and y! For example, if y is measured in miles and x is measured in hours, then delta(y) / delta(x) is measured in miles per hour

derivative of the position function is what?*

the derivative of the position function is the velocity function. h' = -32t + v(sub0)

position function*

the general position function for a free-falling object, neglecting air resistance, is h = -16t^2 + v(sub0)t + h(sub0) where h is height (in feet), t is time (in seconds), v(sub0) is the initial velocity (in feet per second) and h(sub0) is the initial height (in feet). remember that the model assumes that positive velocities indicate upward motion and negative velocities indicate downward motion.

definition of instantaneous rate of change**

the instantaneous rate of change (or simply rate of change) of y = f(x) at x is the limit of the average rate of change on the interval [x, x + delta(x)] as delta(x) approaches 0. lim(delta(x)->0) delta(y) / delta(x) = lim(delta(x)->0) [ f(x+delta(x) - f(x) ] / delta(x) if y is a distance and x is time, then the rate of change is a velocity

study tip: relationship between instantaneous rate of change and derivative of f at x *

the limit in the definition of instantaneous rate of change is the same as the limit in the definition of the derivative of f at x. this is the second major interpretation of the derivative -- as an instantaneous rate of change in one variable with respect to another. Recall that the first interpretation of the derivative is as the slope of the graph of f at x.

demand function *

the number of units x that consumers are willing to purchase at a given price per unit p is given by p = f(x)

relationship between profit, sales, and price

the profit function is unusual in that the profit continues to increase as long as the number of units sold increases. In practice, it is more common to encounter situations in which sales can be increased only be lowering the price per item. Such reductions in price will ultimately cause the profit to decline.

the absolute value of velocity is what*

the speed of the object

revenue function *

the total revenue R is related to price per unit and the quantity demanded (or sold) by the equation R = xp

when is average velocity negative

when the object is moving downward

demand function economics vs. consumer point of view

writing a demand function in the form p = f(x) is a convention used in economics. From a consumer's point of view, it might seem more reasonable to think that the quantity demanded is a function of the price. Mathematically, however, the two points of view are equivalent because a typical demand function is one-to-one and so has an inverse function.

how to analyze a function of a discrete variable x

you can temporarily assume that x is a continuous variable and is able to take on any real value in a given interval. then, you can use the methods of calculus to find the x-value that corresponds to the marginal revenue, maximum profit, minimum cost, or whatever is called for. Finally, you should round the solution to the nearest sensible x-value -- cents, dollars, units, or days, depending on the context of the problem.