Dynamic Calculus
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Tangents and normals
The graph of the parabola \(y = {x^2}\) is drawn below. What are the equations of the tangent and normal to the curve at the point \(P\)?
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]
- On grid paper, draw an accurate sketch of the parabola \(y = {x^2}\). Draw the tangent to the curve at the point \((-4, 16)\) and measure the gradient.
- Using calculus, find the gradient of the tangent to the curve at the point \((-4, 16)\). How does your answer from question (1) compare?
- Using the 'point-gradient' formula (or otherwise), write down the equation of the tangent at the point \((-4, 16)\). Check your solution by clicking the 'show equation and graph of tangent' checkbox.
- What is the geometric relationship between the tangent and the normal to a curve at a point? On your sketch, draw the normal to the curve at the point \((-4, 16)\).
- Using the 'point-gradient' formula (or otherwise), write down the equation of the normal at the point \((-4, 16)\). Check your solution by clicking the 'show equation and graph of normal' checkbox.
- Reposition the point \(P\) by moving the 'slider' left and right. What is the location of the point \(P\) if the normal has equation \(y = \frac{x}{2} + 1.5\)?