Finding the Normal Equations
A normal curve is a straight line passing through the point where the tangent touches the curve and is perpendicular (at right angles) to the tangent at that point.
The gradient of the tangent to a curve is $m$, then the gradient of the normal is $\displaystyle -\dfrac{1}{m}$, as the product of the gradients of $2$ perpendicular lines equals to $-1$.
The gradient of the tangent at $x=a$ is $f^{\prime}(a)$. Therefore the gradient of the normal is $\displaystyle -\dfrac{1}{f^{\prime}(a)}$. The equation of the normal is:
$$y-f(a) = -\dfrac{1}{f^{\prime}(a)}(x-a)$$
Example 1
Find the gradient of the normal to the curve $f(x)=2x^3-x^2+1$ at $x=1$.
\( \begin{align} \displaystyle \require{AMSsymbols} \require{color}
f^{\prime}(x) &= 6x^2-2x \\
f^{\prime}(1) &= 6 \times 1^2-2 \times 1 &\color{red} \text{gradient of tangent} \\
&= 4 \\
\end{align} \)
Therefore, the normal gradient is $\displaystyle -\dfrac{1}{4}$.
Example 2
Find the equation of the normal to $f(x)=x^3-2x+3$ at the point $(1,2)$.
\( \begin{align} \displaystyle \require{AMSsymbols} \require{color}
f^{\prime}(x) &= 3x^2-2 \\
f^{\prime}(1) &= 3 \times 1^2-2 &\color{red} \text{gradient of tangent} \\
&= 1
\end{align} \)
Therefore the gradient of the normal is $-1$.
The equation of the normal is:
\( \begin{align} \displaystyle \require{color}
y-2 &= -1(x-1) \\
\therefore y &= -x+3
\end{align} \)
Example 3
Find the equation of the normal to $f(x)=3x^2-1$ at $x=1$.
\( \begin{align} \displaystyle \require{AMSsymbols} \require{color}
f(1) &= 3 \times 1^2-1 \\
&= 2 &\color{red} y\text{-coordinate of the point} \\
f^{\prime}(x) &= 6x \\
f^{\prime}(1) &= 6 \times 1 &\color{red} \text{gradient of tangent} \\
&= 6
\end{align} \)
Therefore, the normal gradient is $\displaystyle -\dfrac{1}{6}$.
The equation of the normal is:
\( \begin{align} \displaystyle \require{AMSsymbols} \require{color}
y-2 &= -\dfrac{1}{6}(x-1) &\color{red} y-f(1) = -\dfrac{1}{f^{\prime}(1)}(x-1) \\
6y-12 &= -x+1 \\
\therefore x+6y-13 &= 0
\end{align} \)
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume
Responses