Tangents, Normals and Approximations: Quick Revision

A brief note on Tangents, Normals and Approximations. Read it before a test. Try each question at the end before you open its answer.

Tangents and normals

• Gradient (slope): the gradient of the tangent at $(x_0, y_0)$ is $f'(x_0)$.
• Tangent: $y - y_0 = f'(x_0)(x - x_0)$.
• Normal: gradient $-\dfrac{1}{f'(x_0)}$, through the same point. A horizontal tangent has the vertical normal $x = x_0$.
• Implicit and parametric curves: differentiate both sides, or use $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$.

Given directions and angles

• Parallel to a line of gradient $m$: $f'(x) = m$. Perpendicular: $f'(x) = -\dfrac{1}{m}$. Horizontal: $f'(x) = 0$.
• From an outside point: let the point of contact be $(t, f(t))$, write the tangent, and solve for $t$.
• Angle between curves: $\tan\varphi = \left|\dfrac{m_1 - m_2}{1 + m_1m_2}\right|$. Equal gradients mean they touch. $m_1m_2 = -1$ means they cut at a right angle.

Small changes and errors

• Differential: $dy = f'(x)\,dx$, and $f(x + dx) \approx f(x) + f'(x)\,dx$.
• Errors: absolute $dx$, relative $\dfrac{dx}{x}$, percentage $\dfrac{dx}{x} \times 100\%$.
• Powers: for $y = x^n$, the relative error in $y$ is $n$ times that in $x$.
• Radians: use radians when the derivative of a sine or cosine is used.

A worked example

Find the tangent to $y = x^2 - 4x + 7$ at $x = 3$, and use it to estimate $y$ at $x = 3.05$.

1. The point and gradient. $y_0 = 4$ and $\dfrac{dy}{dx} = 2x - 4 = 2$.
2. The tangent. $y - 4 = 2(x - 3)$, so $y = 2x - 2$.
3. The estimate. $y(3.05) \approx 4 + 2 \times 0.05 = 4.1$. The exact value is $4.1025$.

Practice with new numbers

A program to try

The program finds the tangent and the normal at $x = 3$ for the curve above, and compares the estimate of $f(3.02)$ with the exact value.

Mistakes to avoid

• Using $f(x_0)$ for the gradient. The gradient is $f'(x_0)$.
• Forgetting the minus sign and the reciprocal for the normal.
• Using degrees in a derivative of a trigonometric function.
• Treating the point of contact as known for a tangent from an outside point.

Check yourself

1. Find the tangent to $y = x^2$ at $x = 2$.

2. Find the gradient of the normal to $y = x^2$ at $x = 2$.

3. For which $x$ is the tangent to $y = x^2 - 6x$ horizontal?

4. What is the angle between two lines with gradients $3$ and $-\tfrac{1}{3}$?

5. The radius of a circle is $5\ \text{cm}$, with an error of $0.1\ \text{cm}$. Find the percentage error in the area.

Sources

• NCERT, Mathematics Textbook for Class XII, Part I, Chapter 6 (Application of Derivatives).
• OpenStax, Calculus Volume 1, Section 3.1 (Defining the Derivative) and Section 4.2 (Linear Approximations and Differentials).