## Calculating rate of change using derivatives

30 Mar 2016 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 Calculate the average rate of change and explain how it differs from the for derivatives is to estimate an unknown value of a function at a point by using a A summary of Rates of Change and Applications to Motion in 's Calculus AB: Applications of the Derivative. Learn exactly what happened in this chapter, scene, Sal finds the average rate of change of a curve over several intervals, and uses This problem requests the equation in a different form called "point-slope form. (1.9) and the coordinates of a point through which the line passes (7, 109.45).

## Lecture 6 : Derivatives and Rates of Change Some limits are easy to calculate when we recognize them as derivatives: Example The following limits represent the derivative of a function fat a number a. In each case, The derivative, f0(a) is the instantaneous rate of change of y= f(x) with respect to xwhen x= a. When

Section 4-1 : Rates of Change The purpose of this section is to remind us of one of the more important applications of derivatives. That is the fact that \(f'\left( x \right)\) represents the rate of change of \(f\left( x \right)\). The Derivative as an Instantaneous Rate of Change. The derivative tells us the rate of change of one quantity compared to another at a particular instant or point (so we call it "instantaneous rate of change"). This concept has many applications in electricity, dynamics, economics, fluid flow, population modelling, To find the derivative of a function y = f(x) we use the slope formula: Slope = Change in Y Change in X = ΔyΔx. And (from the diagram) we see that: Now follow these steps: Fill in this slope formula: ΔyΔx = f(x+Δx) − f(x)Δx. Simplify it as best we can. Then make Δx shrink towards zero. Differentiation is a method to calculate the rate of change (or the slope at a point on the graph); we will not Recall that these derivatives represent the rate of change of \(f\) as we vary \(x\) (holding \(y\) fixed) and as we vary \(y\) (holding \(x\) fixed) respectively. We now need to discuss how to find the rate of change of \(f\) if we allow both \(x\) and \(y\) to change simultaneously. Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line. 3.4.4. Predict the future population from the present value and the population growth rate. 3.4.5. Use derivatives to calculate marginal cost and revenue in a business situation. So, in this section we covered three “standard” problems using the idea that the derivative of a function gives the rate of change of the function. As mentioned earlier, this chapter will be focusing more on other applications than the idea of rate of change, however, we can’t forget this application as it is a very important one.

### The Derivative as an Instantaneous Rate of Change. The derivative tells us the rate of change of one quantity compared to another at a particular instant or point (so we call it "instantaneous rate of change"). This concept has many applications in electricity, dynamics, economics, fluid flow, population modelling,

calculate the average rate of change between P(x, f(x)) and a nearby point Q(x Again using the preceding “limit definition” of a derivative, it can be proved that Use the definition of the derivative to calculate derivatives. Average rates of change: We are all familiar with the concept of velocity (speed): If Note: We can discuss the instantaneous rate of change of any function using the method above .

### Lecture 6 : Derivatives and Rates of Change Some limits are easy to calculate when we recognize them as derivatives: Example The following limits represent the derivative of a function fat a number a. In each case, The derivative, f0(a) is the instantaneous rate of change of y= f(x) with respect to xwhen x= a. When

Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line. 3.4.4. Predict the future population from the present value and the population growth rate. 3.4.5. Use derivatives to calculate marginal cost and revenue in a business situation.

## The table at the left shows the results of similar calculations for the slopes of other secant lines. SECTION 2.1 DERIVATIVES AND RATES OF CHANGE. □. 1.

19 Mar 2013 Description: This calculation computes the approximate rate of change at each point of a function f(x), using finite differences. Example

22 Jan 2011 In general a rate of change may be the change in anything divided by units for this display You may select the lower value of t using the cursor. have defined the derivative of a function at a point, we could calculate it for It is commonly interpreted as instantaneous rate of change. Let's think about how we can calculate the derivative at a point for a function y=f(x). but we can approach a tangent line by starting with a secant line (passes through two points) . Chapter 2, Section 2.6: Derivatives and Rates of Change. Recall: Slope of the secant line through two points on a curve. If y = f(x) and the points P(a, f(a)) and calculate the average rate of change between P(x, f(x)) and a nearby point Q(x Again using the preceding “limit definition” of a derivative, it can be proved that