7B - Rates of change
Average rate of change:
When the relationship between two variables is variable (non-linear), we can find the average rate of change. The average rate of change is found by calculating the gradient of a line segment drawn between two points on a graph: \[AvROC=m=\frac{y_2-y_1}{x_2-x_1}\]
■ For any function, \(y=f(x)\), the average rate of change over an interval \([a,b]\) is given by \[\frac{f(b)-f(a)}{b-a}\]
7B - Example 1: Finding the AVERAGE rate of change (VCAA, 2007)
[VCAA, 2007 Exam 2 MC Question 4]
The average rate of change of the function with rule \(f(x)=x^3-\sqrt{x+1}\) between \(x=0\) and \(x=3\) is A. \(0\) B. \(12\) C. \(\frac{26}{3}\) D. \(\frac{25}{3}\) E. \(8\) |
7B - Example 1: Video solution
7B - Example 1: Practice
Question 1: ABC 7B - Example 1: Solutions
Question 1: ABC |
7B - Example 2: Finding the AVERAGE rate of change (VCAA, 2010)
[VCAA, 2010 Exam 2 MC Question 2]
For \(f(x)=x^3+2x\), the average rate of change with respect to \(x\) for the interval \([1,5]\) is A. \(18\) B. \(20.5\) C. \(24\) D. \(32.5\) E. \(33\) |
7B - Example 2: Video solution
7B - Example 2: Practice
Question 1: ABC 7B - Example 2: Solutions
Question 1: ABC |
Instantaneous rate of change:
7B - Example 3: Finding the instantaneous rate of change (VCAA, 2009)
[VCAA, 2009 Exam 2 MC Question 7]
For \(y=e^{2x}cos(3x)\) the rate of change of \(y\) with respect to \(x\) when \(x=0\) is A. \(0\) B. \(2\) C. \(3\) D. \(-6\) E. \(-1\) |
7B - Example 3: Video solution
7B - Example 3: Practice
Question 1: ABC 7B - Example 3: Solutions
Question 1: ABC |
7B - Example 4: Finding the instantaneous rate of change (VCAA, 2003)
[VCAA, 2003 Exam 1 MC Question 10]
If \(y=x log_e(x)\), then the rate of change of \(y\) with respect to \(x\) when \(x=2\) is equal to A. \(log_e(2)\) B. \(1\) C. \(1+log_e(2)\) D. \(2\) E. \(1+log_2(e)\) |
7B - Example 4: Video solution
7B - Example 4: Practice
Question 1: ABC 7B - Example 4: Solutions
Question 1: ABC |