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NSWMaths AdvancedQuick questions
Year 11: Introduction to Differentiation
Quick questions on Average and instantaneous rates of change for HSC Maths Advanced: the average rate of change as the gradient of a chord (secant) between two points on a curve or in a table, the instantaneous rate of change as the gradient of the tangent at a point, estimating an instantaneous rate with a short chord, reading and interpreting rates from real graphs such as distance-time and cooling curves, and the idea of the derivative as a new function giving the gradient at each point
2short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.
What is average rate of change is the gradient of a chord?Show answer
A chord (also called a secant) is the straight line joining two points on a curve. If a quantity has value at and value at , the average rate of change of with respect to over that interval is the gradient of the chord:
What is instantaneous rate of change is the gradient of a tangent?Show answer
The instantaneous rate of change is the rate at one precise moment, not averaged over a stretch. On a graph it is the steepness of the curve at a single point, and the line that captures that steepness is the tangent: the straight line that touches the curve at that point and heads off in the exact direction the curve is going. Think of the beam from a car's headlights as it rounds a bend, always pointing along the curve at the instant.
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