Question

At what point of the curve $ y={{x}^{2}} $ does the tangent make an angle of $ 45{}^\circ $ with the x-axis?

Answer 1

Hint: We will look at the graph of the given function. Using the given angle between the x-axis and the tangent, we can find the slope of the tangent. Given a function, the equation of the tangent is the derivative of the function. So, we will calculate the derivative of the function. The derivative of a function at a particular point gives us the slope of the tangent at that point. So, we will equate the slope of the tangent line and the derivative of the function at a point to find the coordinates of the point.

Complete step by step answer:
The given function is $ y={{x}^{2}} $ . The graph of the given function looks like the following,

Now, we know that the tangent line forms an angle of $ 45{}^\circ $ with the x-axis. Therefore, the slope of the tangent line is $ \tan 45{}^\circ =1 $ . We know that the equation of tangent to a curve is the differentiation of the function representing the curve. We will differentiate the equation of the curve in the following manner,
 $ \dfrac{dy}{dx}=\dfrac{d}{dx}\left( {{x}^{2}} \right) $
The formula for differentiating polynomial function is, $ \dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}} $ . Using this formula, we have the following,
 $ \dfrac{dy}{dx}=2x $
We know that the derivative of a function at a particular point gives us the slope of tangent at that point. We will now equate the equation of the tangent at point $ \left( x,y \right) $ with the slope of the tangent line. So, we get the following,
 $ \begin{align}
  & {{\left. \dfrac{dy}{dx} \right|}_{\left( x,y \right)}}=1 \\
 & \Rightarrow 2x=1 \\
 & \therefore x=\dfrac{1}{2} \\
\end{align} $
Substituting the value of $ x $ in the given function, we get
 $ \begin{align}
  & y={{\left( \dfrac{1}{2} \right)}^{2}} \\
 & \therefore y=\dfrac{1}{4} \\
\end{align} $
Hence, the point at which the tangent to the curve $ y={{x}^{2}} $ makes an angle of $ 45{}^\circ $ with the x-axis is $ \left( \dfrac{1}{2},\dfrac{1}{4} \right) $.
Note:
The derivative of a function is the ratio of change in the value of the function and the change in the value of the inputs. Therefore, the derivative represents the tangent of the function and the derivative at a particular point gives us the slope of the tangent. The derivative is a very important tool for studying different types of functions. It is useful to know the formulae for calculating the derivatives of standard functions.