Maths · Calculus · chapter 5 of 13
Class 12 Maths Chapter 5: Continuity and Differentiability case study question with answers
One case-based question of the kind the board asks: a passage, then 3 parts worth 4 marks in all, with the answer and the marking scheme for each. Written from the NCERT chapter.
Case study · 4 marks
An engineer in Pune designs a ramp for a parking structure. Measured along the ground, x metres from the start, the ramp's height is modelled by h(x) = x24 for 0 ≤ x ≤ 2 (a curved lead-in) and h(x) = ax + b for 2 < x ≤ 6 (a straight section). The two sections must meet without a step, which means h must be continuous at x = 2. For a smooth drive the ramp must also have no corner at x = 2, which means h must be differentiable there.
Part 1 · 1 mark
The height of the ramp at x = 2 is
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Answer: (a) 1
h(2) = 22/4 = 1 m.
Part 2 · 1 mark
For the ramp to have no corner at x = 2, the value of a must be
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Answer: (a) 1
The left hand derivative of x2/4 at 2 is x/2 = 1, the right hand derivative of ax + b is a, so a = 1.
Part 3 · 2 marks
With the value of a found above, find b and the height of the ramp at x = 6.
Answer and marking scheme
Continuity at 2: 2a + b = 1, so b = 1 − 2 = −1. Then h(6) = 6 − 1 = 5 m.
- b = −1 from continuity at x = 2 (1)
- h(6) = 5 m (1)
Or
Suppose instead the straight section is laid with a = 12 and joined without a step. Find b, and show that the ramp then has a corner at x = 2.
Answer and marking scheme
Continuity: 2(1/2) + b = 1, so b = 0. Left hand derivative at 2 is 1; right hand derivative is 1/2. They differ, so h is continuous but not differentiable at 2: a corner.
- b = 0 (1)
- one-sided derivatives 1 and 1/2 differ: a corner (1)
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