Question

At what point on the curve \( y=x^2\) does the tangent make an angle of \( 45^{\circ} \) with the axis.

Answer 1

Answer:

x = 1/2

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Step-by-step explanation:

The tangent makes an angle of 45° with the x-axis if its gradient is 1.

[ In general, the angle θ and the gradient m are related by m = tan θ.  Notice that here, tan 45° = 1. ]

So the question becomes:

At what point on the curve y = x² does the tangent have gradient 1?

The gradient of the tangent is the value of the derivative dy/dx.  So the question becomes:

At what point on the curve y = x² do we have dy/dx = 1?

Given y = x², differentiating gives dy/dx = 2x.  So...

dy/dx = 1  <=>  2x = 1  <=>  x = 1/2.