# Derivatives V(T) – Kinematics And Calculus

**Derivatives V(T) – Kinematics And Calculus**in

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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.

Đang xem: Derivatives v(t)

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

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)).