Back to Topic 5.4 — Tangents and normals

5.4.1Math AI SL SL8 flashcards

Tangent Lines

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Card 1 of 85.4.1

5.4.1

Question

State the point-slope form used to write a tangent equation.

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All 8 Flashcards — Tangent Lines

Card 1formula

Question

State the point-slope form used to write a tangent equation.

Answer

y − y₁ = m(x − x₁), where m is the gradient and (x₁, y₁) is the point of tangency.

Card 2concept

Question

The three steps for finding a tangent equation — what are they?

Answer

1. Differentiate f(x) to get f′(x).\n2. Substitute x₁ into f′(x) to get the gradient m.\n3. Write y − y₁ = m(x − x₁) and simplify.

Card 3formula

Question

Find the gradient of the tangent to y = x² at x = 3.

Answer

dy/dx = 2x. At x = 3: m = 6.

Card 4formula

Question

Find the equation of the tangent to y = x² + 1 at x = 2.

Answer

dy/dx = 2x → m = 4. y₁ = 5. Tangent: y − 5 = 4(x − 2) → y = 4x − 3.

Card 5concept

Question

Why do you substitute x₁ into f(x) (not f′(x)) to find y₁?

Answer

Because f(x) gives y-values (heights). f′(x) gives gradients. You need the y-coordinate of the point of tangency — that comes from the original function.

Card 6concept

Question

How do you find x when you are given the tangent gradient instead of the x-value?

Answer

Set f′(x) = given gradient and solve for x. There may be one or two solutions. Find y at each solution using f(x).

Card 7formula

Question

Find the tangent to f(x) = x³ at x = −1.

Answer

f′(x) = 3x². m = 3. f(−1) = −1 → point (−1, −1). Tangent: y + 1 = 3(x + 1) → y = 3x + 2.

💡 Hint

Check signs carefully.

Card 8concept

Question

What does the tangent line tell you about the curve near the point of tangency?

Answer

The tangent is the best linear approximation to the curve at that point. It has exactly the same gradient as the curve at that point — but the curve will curve away from the tangent for x-values further away.

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Topic 5.4 hub

Tangents and normals

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