Practice IB Mathematics Applications & Interpretation (AI) Topic SL 5.4 Tangents and Normals with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 5.4 Tangents and Normals and mirrors Paper 1, 2, 3 style where relevant.
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A section of the cross-section of a garden arch is modelled by . The coordinates and are measured in metres relative to a fixed origin.
(a) Find an expression for .
(b) the equation of the normal line to the curve at the point where .
An engineer models the cross-section of a reflector by , where and are measured in centimetres.
(b) Find the slope of the tangent line at .
(c) Find the equation of the normal line to the function at .
A curve has equation .
(a) Find an expression for in terms of and .
(b) Find the equations of the tangents to this curve at the points where the curve intersects the line .
The cross-section of a tunnel entrance is modeled by the polynomial .
(a) Obtain an expression for .
The volume of water in a storage tank is modeled by the function , where represents the number of hours since filling began and is the volume of water in litres.
(a) Find the derivative that represents the rate at which the volume of water changes over time.
(b) Find the gradient of the volume curve at .
(d) the meaning of the gradient of the tangent line in the context of the volume of water in the tank.
(a) the derivative .
(d) the value of for which the slope of the tangent is .
(e) the domain and range of the function .
(b) the equation of the normal line to the curve at the point where .
(c) the equation of the tangent line to the volume curve at the point where hours. Give your answer in the form .
