Back to Topic 5.4 — Tangents and normals

5.4.2Math AI SL SL8 flashcards

Normal Lines

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Card 1 of 85.4.2

5.4.2

Question

State the relationship between the tangent gradient and the normal gradient.

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All 8 Flashcards — Normal Lines

Card 1formula

Question

State the relationship between the tangent gradient and the normal gradient.

Answer

m_tangent × m_normal = −1, so m_normal = −1/m_tangent. The normal is perpendicular to the tangent.

Card 2formula

Question

The tangent gradient at a point is 5. What is the normal gradient?

Answer

m_n = −1/5.

Card 3formula

Question

The tangent gradient at a point is −3. What is the normal gradient?

Answer

m_n = −1/(−3) = 1/3. Two negatives cancel.

💡 Hint

Watch the signs — two negatives make positive.

Card 4formula

Question

Find the gradient of the normal to y = x² − 2x at x = 3.

Answer

dy/dx = 2x − 2. m_t = 4. m_n = −1/4.

Card 5formula

Question

Find the equation of the normal to y = x² at (3, 9).

Answer

dy/dx = 2x → m_t = 6 → m_n = −1/6. Normal: y − 9 = −(1/6)(x − 3) → y = −(1/6)x + 19/2.

Card 6concept

Question

The tangent at a point is horizontal. What does the normal look like?

Answer

The normal is vertical: a line of the form x = x₁. You cannot divide −1 by zero.

Card 7concept

Question

Both the tangent and normal pass through the same point. True or false?

Answer

True. Both lines pass through the point of tangency (x₁, y₁). They differ only in their gradients.

Card 8concept

Question

What is the single most common error in normal-line questions?

Answer

Using the tangent gradient (from f′) directly as the normal gradient, without applying m_n = −1/m_t. Always take the negative reciprocal.

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Tangents and normals

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