Unit 5: Calculus
Topic 5.4: Optimisation Questions
Practice 20 exam-style questions for IB Math AI SL Topic 5.4. Review the question stems below, then unlock the full Question Bank to access markschemes, model answers, and AI grading.
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The tangent to y = x² at x = 3 has gradient:
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Find the equation of the tangent to y = 3x² − 2x at x = 2. Give your answer in the form y = mx + c.
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The tangent gradient at a point is −4. What is the normal gradient?
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The tangent to a curve at a point has gradient 5. What is the gradient of the normal?
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Find the equation of the normal to y = x² − 4 at x = 3.
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To write the equation of a tangent at x = 2, which substitution gives the y-coordinate of the point?
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Find the gradient of the normal to f(x) = 3x² − 2x at x = 1.
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Write down the gradient of the normal there.
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Which equation represents the normal to y = x³ at (1, 1)?
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A ball is thrown upward. Its height h (metres) at time t (seconds) is h(t) = −5t² + 20t.
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Explain the geometric relationship between the tangent and normal at a point. At x = 2, f(x) = x² − 3x.
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The curve y = x³ − 3x has a tangent and a normal at the point (2, 2).
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The tangent to y = 2x² at a point has gradient 12. Find the coordinates of the point of tangency.
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At x = 2, f(x) = x³ − 3x has f′(2) = 9. What is the equation of the normal at (2, 2)?
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Find the equation of the normal to f(x) = x³ − x at x = −1.
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(a) Find the equation of the tangent to the curve y = x² − 3x + 4 at the point where x = 2.
(b) State where this tangent crosses the y-axis.
(b) State where this tangent crosses the y-axis.
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The curve f(x) = x³ − 5x passes through the point (2, −2). Find the equation of the tangent at this point.
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Find the equation of the tangent to f(x) = x³ − 3x² + 2 at x = 1.
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The gradient of the normal to y = x² at x = 3 is:
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The tangent to f(x) = x² + 2x at x = 1 has equation:
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