The gradient of a graph indicates the rate of change. In the example below, the rate of change of y with respect to x (in other words, how y is changing when x is changing) increases as the value of x increases.
The gradient of a graph can be obtained by two methods. An average rate of change can be obtained by drawing a chord over the point. The second method is to take a tangent at the point, and to determine the gradient of the tangent.
P259_z3;P259_z3.png;rate of change on a graph - tangent
In this instance, the average (chord) method gives the same answer as the instantaneous (tangent) method, but the average method will be unsuitable for graphs with turning points.
| 1 | What is the average gradient for the function `y = 3x^3 -x + 5` at `x` = 0? Use the points between `x=-3` and `x=+3`? |
| Answer: | -0.1 |
y coordinate for `x=-3` is `y = 5.3` and for `x=+3` is `y=4.7` The gradient = `frac(4.7 - 5.3)(3 - -3)` = `frac(-0.6)(6)` = -0.1 |
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| 2 | P259_z4;P259_z4.png;P259 question 2 What is the instantaneous rate of change for the function `y = 3x^3 -x + 5` when `x=0`? The tangent has been drawn for you, and the end points of the tangent are (-3, 8) and (3, 2).
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| Answer: | -1 |
The gradient is `frac(text(change in )y)(text(change in ) x)` = `frac(2 - 8)(3 - -3)` = `frac(-6)(6)` = -1 |