Average Rate of Change Calculator

Compute (f(b) − f(a)) ÷ (b − a) from a function you type or from two measured points.

Use + − * / ^ and ( ). Functions: sqrt, sin, cos, tan, ln, log, exp, abs. Constants: pi, e

How to use the Average Rate of Change Calculator

  1. Choose the input type.
  2. Enter the f(x) = – Use + − * / ^ and ( ). Functions: sqrt, sin, cos, tan, ln, log, exp, abs. Constants: pi, e.
  3. Enter the a (start of interval).
  4. Enter the b (end of interval).
  5. Enter the f(a).
  6. Enter the f(b).
  7. Press Calculate. The result updates automatically as you type.

About the Average Rate of Change Calculator

The average rate of change tells you how fast a quantity changed, on average, over an interval. If a car's position is f(t) miles after t hours, the average rate of change from t = 1 to t = 4 is its average speed over those three hours. For a straight line it is just the slope; for curves it depends on the interval you choose.

In precalculus this is the stepping stone to the derivative: as b gets closer and closer to a, the secant line approaches the tangent line and the average rate becomes the instantaneous rate. You can type a formula such as x^2 + 3x, 2^x or sqrt(x+1), or switch to points mode and enter measured data such as population counts in two different years.

Formula

Average rate of change = (f(b) − f(a)) ÷ (b − a)

It equals the slope of the secant line through (a, f(a)) and (b, f(b)). Trig functions use radians; log means base 10 and ln means base e.

Example calculation

Using the default values:

  • Input type = Function f(x)
  • f(x) = = x^2 + 3x
  • a (start of interval) = 1
  • b (end of interval) = 4

the calculator returns:

  • Average rate of change: 8
  • f(a): 4
  • f(b): 28
  • Secant line: y = 8x − 4

Frequently asked questions

How do you find the average rate of change?

Evaluate the function at both ends of the interval, subtract, and divide by the length of the interval. For f(x) = x² + 3x on [1, 4]: (28 − 4) ÷ 3 = 8.

Is the average rate of change the same as slope?

Yes, it is the slope of the secant line connecting the two points on the graph.

What units does the rate of change have?

Output units per input unit, such as miles per hour, dollars per year or degrees per minute.

Last updated: · Category: Math calculators

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