Average and instantaneous rates of change
Ratio, proportion and rates • Ratio and proportion
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Definition of average rate of change
Average rate of change equals the change in the dependent variable divided by the change in the independent variable over a specific interval. For a function f, the average rate between x = a and x = b equals (f(b) - f(a)) / (b - a). This value equals the gradient of the chord that connects the points (a, f(a)) and (b, f(b)) on the graph. Average rates provide information about overall behaviour across an interval but do not reveal variation within the interval. Larger intervals may hide local increases or decreases because the result summarises the net change.
Definition of instantaneous rate of change
Instantaneous rate of change represents how rapidly the dependent variable changes at a single point on the curve. Graphically, the instantaneous rate appears as the gradient of the tangent line at that point. The tangent line gives the best linear approximation to the curve at the point, and its gradient represents the local change per unit of the independent variable. Instantaneous rates require a smooth curve at the point. Points with corners, cusps or discontinuities do not have well-defined instantaneous rates because no single tangent line exists at those locations.
Gradient of a chord (average rate) - graphical and numerical
The gradient of a chord between two points (x1, y1) and (x2, y2) equals (y2 - y1) / (x2 - x1). Graphical estimation uses coordinates read from the graph, then applies the formula. Numerical problems supply coordinates directly and require arithmetic or algebraic substitution. Careful reading of coordinates and use of consistent units prevents errors. If the chord interval is small, the chord gradient approximates the instantaneous gradient more closely; if the interval is large, the chord gradient may differ substantially from the local rate.
Gradient of a tangent (instantaneous rate) - estimation from chords
Estimation of a tangent gradient uses gradients of chords with one endpoint fixed at the point of interest and the other endpoint approaching that point. As the second point moves closer, the chord gradient approaches the tangent gradient. Graphically, drawing secant lines with progressively smaller intervals gives increasingly accurate estimates of the instantaneous rate. Algebraic estimation uses pairs of x-values close to the target x-value to compute difference quotients. Repeated estimates with narrower intervals reveal convergence toward the instantaneous rate, subject to rounding and measurement limits.
Algebraic methods without formal limits
Algebraic tasks supply two nearby x-values or an expression for f(x) and require computation of (f(x+h) - f(x)) / h for small h values, but without invoking limit notation. Substitution of small numeric h values produces approximate instantaneous rates. Simplification of difference quotients for polynomial functions often produces exact expressions that reveal the behaviour as h becomes small. Exact algebraic simplification provides precise average rates for specified intervals. When simplification removes the dependency on h in the numerator, substitution of h = 0 can give the tangent gradient as an algebraic result; however, such substitution is valid only after algebraic cancellation removes any zero denominators.
Limitations and conditions
Well-defined rates require a single-valued function and consistent units. Instantaneous gradients require local smoothness and no vertical tangent. Discontinuities or corners cause undefined or ambiguous instantaneous rates. Measurement errors, coarse graph scales and wide intervals cause poor approximations when estimating instantaneous rates from chords. Numerical approximations involve rounding error and depend on machine or pencil-and-paper precision. Algebraic cancellation must precede substitution where denominators become zero; failure to do so leads to incorrect conclusions.
Key notes
Important points to keep in mind