Radford Mathematics IB Mathematics resources • AA HL

IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus

Tangents and Normals

Use derivatives to construct tangent and normal lines at a point on a curve.

AA HL · SL 5.4

Learning goal

Find tangent and normal gradients and write equations of the corresponding lines.

Syllabus link

AA HL: SL 5.4 · Topic 5 Calculus.

Big idea

At a point, the tangent shares the curve gradient; the normal is perpendicular to the tangent.

Key relationship

\(m_{\rm tangent}=f\'(a)\)

1

Core idea

At a point, the tangent shares the curve gradient; the normal is perpendicular to the tangent.

Find tangent and normal gradients and write equations of the corresponding lines.

2

Key results and method

\[m_{\rm tangent}=f\'(a)\]
\[m_{\rm normal}=-\frac{1}{f\'(a)}\]
\[y-y_1=m(x-x_1)\]
  1. Find the point on the curve.
  2. Evaluate the derivative to obtain the tangent gradient.
  3. Use the negative reciprocal for the normal when the tangent gradient is non-zero.
  4. Use point-gradient form.
3

Worked example

Question

For \(y=x^2+1\), find the tangent at \(x=2\).

Solution
  1. The point is \((2,5)\).
  2. \(y\'=2x\), so the gradient is \(4\).
  3. Use \(y-5=4(x-2)\).
Answer: \(y=4x-3\).
4

Practice

Try these questions before moving on.

  1. Find the normal to \(y=x^3\) at \(x=1\).
  2. Find a tangent parallel to \(y=6x-4\).
  3. Explain the special case of a horizontal tangent and vertical normal.

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5

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