Functions and Graphs - Estimating Gradients from Tangents
Functions and Graphs - Estimating Gradients from Tangents
Secondary 3
The variables \(x\) and \(y\) are related by the equation \(y = \frac{1}{7}x(10 - x^2)\).
Some corresponding values are given in the table below.
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) |
|---|---|---|---|---|---|---|---|---|
| \(y\) | \(-1.71\) | \(-1.29\) | \(0\) | \(1.29\) | \(1.71\) | \(0.43\) | \(a\) | \(-10.71\) |
- Find the value of \(a\).
- Taking \(2\) cm to represent \(2\) units for the \(y\)-axis and \(2\) cm to represent \(1\) unit for the \(x\)-axis, draw the graph of \(y = \frac{1}{7}x(10 - x^2)\) for the range \(-2 \le x \le 5\).
- Use your graph to find the values of \(x\) in the range \(-2 \le x \le 5\) for which \(\frac{1}{7}x(10 - x^2) = -1\).
- By drawing a tangent, find the gradient of the curve at the point \((3, 0.43)\).
- Use your graph to solve the equation \(x(10 - x^2) = -14x\).
Solution:
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Answer:(a) \(-3.43\). (c) \(-0.74,3.47\). (d) About \(-2.43\). (e) \(0,4.90\).
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