Derivatives V(T) – Kinematics And Calculus

Determine a new value of a quantity from the old value and the amount of change. Calculate the average rate of change and explain how it differs from the instantaneous rate of change. Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line. Predict the future population from the present value and the population growth rate. Use derivatives to calculate marginal cost and revenue in a business situation.

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In this section we look at some applications of the derivative by focusing on the interpretation of the derivative as the rate of change of a function. These applications include acceleration and velocity in physics, population growth rates in biology, and marginal functions in economics.

Amount of Change Formula

One application for derivatives is to estimate an unknown value of a function at a point by using a known value of a function at some given point together with its rate of change at the given point. If (f(x)) is a function defined on an interval (), then the amount of change of (f(x)) over the interval is the change in the (y) values of the function over that interval and is given by

The average rate of change of the function (f) over that same interval is the ratio of the amount of change over that interval to the corresponding change in the (x) values. It is given by

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As we already know, the instantaneous rate of change of (f(x)) at (a) is its derivative

For small enough values of (h), (f′(a)≈frac{f(a+h)−f(a)}{h}). We can then solve for (f(a+h)) to get the amount of change formula:

We can use this formula if we know only (f(a))and (f′(a)) and wish to estimate the value of (f(a+h)). For example, we may use the current population of a city and the rate at which it is growing to estimate its population in the near future. As we can see in Figure (PageIndex{1}), we are approximating (f(a+h)) by the (y) coordinate at a+h on the line tangent to (f(x)) at (x=a). Observe that the accuracy of this estimate depends on the value of (h) as well as the value of (f′(a)).

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Exercise (PageIndex{3})

The current population of a mosquito colony is known to be 3,000; that is, (P(0)=3,000). If (P′(0)=100), estimate the size of the population in 3 days, where (t) is measured in days.

Hint

Use (P(3)≈P(0)+3P′(0))

Answer

3,300

Changes in Cost and Revenue

In addition to analyzing motion along a line and population growth, derivatives are useful in analyzing changes in cost, revenue, and profit. The concept of a marginal function is common in the fields of business and economics and implies the use of derivatives. The marginal cost is the derivative of the cost function. The marginal revenue is the derivative of the revenue function. The marginal profit is the derivative of the profit function, which is based on the cost function and the revenue function.

Example (PageIndex{6}): Applying Marginal Revenue

Assume that the number of barbeque dinners that can be sold, (x), can be related to the price charged, (p), by the equation (p(x)=9−0.03x,0≤x≤300).

In this case, the revenue in dollars obtained by selling (x) barbeque dinners is given by

(R(x)=xp(x)=x(9−0.03x)=−0.03x^2+9x; ext{ for }0≤x≤300).

Use the marginal revenue function to estimate the revenue obtained from selling the (101^{ ext{st}}) barbeque dinner. Compare this to the actual revenue obtained from the sale of this dinner.

Solution

First, find the marginal revenue function: (MR(x)=R′(x)=−0.06x+9.)

Next, use (R′(100)) to approximate (R(101)−R(100)), the revenue obtained from the sale of the (101^{ ext{st}}) dinner. Since (R′(100)=3), the revenue obtained from the sale of the (101^{ ext{st}}) dinner is approximately $3.

The actual revenue obtained from the sale of the (101^{ ext{st}}) dinner is

(R(101)−R(100)=602.97−600=2.97,) or ($2.97.)

The marginal revenue is a fairly good estimate in this case and has the advantage of being easy to compute.

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Exercise (PageIndex{4})

Suppose that the profit obtained from the sale of (x) fish-fry dinners is given by (P(x)=−0.03x^2+8x−50). Use the marginal profit function to estimate the profit from the sale of the (101^{ ext{st}}) fish-fry dinner.

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