If you’re looking at a graph and trying to find the instantaneous rate of change there are a few things you need to keep in mind. First the rate of change is going to be different at different points on the graph. Second you need to be able to identify the points on the graph where the rate of change is the greatest.
Here are a few tips on how to find the instantaneous rate of change from a graph:
1. Look for the points on the graph where the slope is the steepest. The steepness of the slope is directly proportional to the rate of change.
2. Identify the points on the graph where the line is horizontal. These points represent points of inflection and the rate of change is zero at these points.
3. Use the points you’ve identified in steps one and two to draw a tangent line to the graph. The slope of this line is the instantaneous rate of change.
4. Use calculus to find the precise value of the instantaneous rate of change. This can be done by taking the derivative of the function that represents the graph.
Hopefully these tips will help you next time you’re trying to find the instantaneous rate of change from a graph!
What is the average rate of change of f(x) on the interval [ax]?
The average rate of change of f(x) on the interval [ax] is the slope of the line segment joining the points (af(a)) and (xf(x)) on the graph of f.
What is the instantaneous rate of change of f(x) at the point x=a?
The instantaneous rate of change of f(x) at the point x=a is the slope of the tangent line to the graph of f at the point (af(a)).
What is the instantaneous rate of change of f(x) at the point x=b?
The instantaneous rate of change of f(x) at the point x=b is the slope of the tangent line to the graph of f at the point (bf(b)).
What is the average rate of change of f(x) on the interval [ab]?
The average rate of change of f(x) on the interval [ab] is the slope of the line segment joining the points (af(a)) and (bf(b)) on the graph of f.
What is the instantaneous rate of change of f(x) at the point x=c?
The instantaneous rate of change of f(x) at the point x=c is the slope of the tangent line to the graph of f at the point (cf(c)).
What is the instantaneous rate of change of f(x) at the point x=d?
The instantaneous rate of change of f(x) at the point x=d is the slope of the tangent line to the graph of f at the point (df(d)).
What is the average rate of change of f(x) on the interval [cd]?
The average rate of change of f(x) on the interval [cd] is the slope of the line segment joining the points (cf(c)) and (df(d)) on the graph of f.
What is the instantaneous rate of change of f(x) at the point x=e?
The instantaneous rate of change of f(x) at the point x=e is the slope of the tangent line to the graph of f at the point (ef(e)).
What is the instantaneous rate of change of f(x) at the point x=f?
The instantaneous rate of change of f(x) at the point x=f is the slope of the tangent line to the graph of f at the point (ff(f)).
What is the average rate of change of f(x) on the interval [ef]?
The average rate of change of f(x) on the interval [ef] is the slope of the line segment joining the points (ef(e)) and (ff(f)) on the graph of f.
What is the instantaneous rate of change of f(x) at the point x=g?
The instantaneous rate of change of f(x) at the point x=g is the slope of the tangent line to the graph of f at the point (gf(g)).
What is the instantaneous rate of change of f(x) at the point x=h?
The instantaneous rate of change of f(x) at the point x=h is the slope of the tangent line to the graph of f at the point (hf(h)).
What is the average rate of change of f(x) on the interval [gh]?
The average rate of change of f(x) on the interval [gh] is the slope of the line segment joining the points (gf(g)) and (hf(h)) on the graph of f.
What is the instantaneous rate of change of f(x) at the point x=i?
The instantaneous rate of change of f(x) at the point x=i is the slope of the tangent line to the graph of f at the point (if(i)).
What is the instantaneous rate of change of f(x) at the point x=j?
The instantaneous rate of change of f(x) at the point x=j is the slope of the tangent line to the graph of f at the point (jf(j)).
