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Rate of Change

11.3 Rate of Change, Total Change

Integration and Total Changes

• In differentiation, we know that f^{\prime}(x) shows the rate of change of f(x). Therefore, by utilising integration, we can recover the original function from its rate of change function.
• Essentially, from the concept of antiderivatives we have \int f^{\prime}(x) d x=f(x)+c.
• Since f^{\prime}(x) represents rates of changes, thus integrating rates of changes basically gives the total change of a variable.
• We have already seen this in 11.2 Average Value, Straight Line Motion, where we integrate the velocity (which is represents the rate of change of displacement) to find the total change in displacement.
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1.1 Derivatives Basic Concepts – Rate of Changes

Average Rate of Change

• The average change of y with respect to x over the interval [a, b] is the gradient of the line connecting the points A(a,f(a)) and B(b,f(b)).

Note: [a,b] represents a range between a and b, inclusive. In other words, if x \in[a, b] , then \mid a \leq x \leq b . Also, we will always have a<b .

• It is given by:
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Applications of Derivatives

This tutorial covers material encountered in chapter 10 of the VCE Mathematical Methods Textbook, namely: Tangents and normals Finding and classifying stationary points Maximum and… Read More »Applications of Derivatives