36 exam-style questions on Tangents and normals and their equations — topic 5.4 of Calculus in the IB Mathematics: Applications and Interpretation syllabus — with answers and explanations. Samples below; the full set is free with an account.
The curve y = x² + 1 passes through P(2, 5). Determine the gradient of the tangent at P.
Answer: 4
dy/dx = 2x; at x = 2 the gradient is 4.
Find the equation of the tangent to y = x² at the point (3, 9).
Answer: y = 6x − 9
dy/dx = 2x = 6 at x = 3. Tangent: y − 9 = 6(x − 3), so y = 6x − 9.
The tangent to a curve at a point has gradient 2. Determine the gradient of the normal at that point.
Answer: −1/2
The normal is perpendicular to the tangent, so its gradient is the negative reciprocal: −1/2.
For y = x² − 4x, find the x-coordinate of the point where the tangent is horizontal.
Answer: 2
A horizontal tangent has gradient 0: dy/dx = 2x − 4 = 0 gives x = 2.
The curve y = x³ − 2x passes through (1, −1). Find the equation of the normal at this point.
Answer: y = −x
dy/dx = 3x² − 2 = 1 at x = 1, so the normal gradient is −1. Normal: y − (−1) = −1(x − 1), giving y = −x.
The tangent to y = 2x² at x = 1 meets the y-axis at which point?
Answer: (0, −2)
dy/dx = 4x = 4 at x = 1; point (1, 2). Tangent: y − 2 = 4(x − 1), so y = 4x − 2. At x = 0, y = −2.
That was 6 of 36 Tangents and normals and their equations questions.
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36 Tangents and normals and their equations questions — free with an account. Spaced repetition, streaks and full exam simulations included.
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