Draw a graph from the following data. Draw tangents at x = 2, 6 and 8. Find the slopes of these tangents. Verify that the curve draw is y = 2x² and the slope of tangent is $\tan θ = \frac{d y}{d x} = 4 x$ .
$\begin{matrix} x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ y & 2 & 8 & 18 & 32 & 50 & 72 & 98 & 128 & 162 & 200 \end{matrix}$

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Solution

Note: Students should draw the graph y = 2x² on a graph paper for results.

To find a slope at any point, draw a tangent at the point and extend the line to meet the x-axis. Then find tan θ as shown in the figure.

The above can be checked as follows:
$Slope = \tan θ = \frac{d y}{d x} = \frac{d}{d x} (2 x^{2}) = 4 x$
Here, x = x-coordinate of the point where the slope is to be measured

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$\begin{matrix} x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 y & 2 & 8 & 18 & 32 & 50 & 72 & 98 & 128 & 162 & 200 \end{matrix}$

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Draw a graph from the following data. Draw tangents at x = 2, 6 and 8. Find the slopes of these tangents. Verify that the curve draw is y = 2x2 and the slope of tangent is tan θ=dydx=4x. x12345678910y281832507298128162200