14H - Functions and derivatives
Strictly increasing and strictly decreasing functions:
14H - Content video: Strictly increasing and strictly decreasing functions
This video covers the theory and several examples relating to strictly increasing and strictly decreasing functions. VCAA also released a supplement in April 2011 relating to strictly increasing and strictly decreasing functions.
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■ A function is said to be strictly increasing if each proceeding \(y\)-value is greater than the previous. Mathematically, a function is strictly increasing over an interval if \(x_2>x_1\) gives \(f(x_2)>f(x_1)\).
■ A function is said to be strictly decreasing if each proceeding \(y\)-value is less than the previous. Mathematically, a function is strictly increasing over an interval if \(x_2>x_1\) gives \(f(x_2)<f(x_1)\).
■ A function is said to be strictly decreasing if each proceeding \(y\)-value is less than the previous. Mathematically, a function is strictly increasing over an interval if \(x_2>x_1\) gives \(f(x_2)<f(x_1)\).
Graphs of functions and their derivatives:
Calculating the derivative of a function results in another function which can also be graphed. In this section we will examine several important features of a graph of a function and the graph of its derivative function. Consider a function \(f(x)\) and its derivative function \(f'(x)\):
Graph of \(f(x)\) |
Graph of \(f'(x)\) |
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The graphs of \(f(x)=\frac{1}{3}x^3-4x\) and its derivative \(f'(x)=x^2-4\) are shown to the right. Consider the key features of \(f(x)\) and the associated features on \(f'(x)\):
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14H - Example 1: Graphs of functions and their derivatives
Sketch the graphs of both \(f'(x)=x^3+6x^2+9x\) and an approximation of \(f(x)\) on the same set of axes.
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14H - Example 1: Video solution
14H - Example 1: Practice
Question 1: ABC Question 2: ABC 14H - Example 1: Solutions
Question 1: ABC Question 2: ABC |