Road-to-Bridge Transition: Matching Value and Slope
A civil engineer is joining a straight approach road to a curved bridge deck. For 0≤x≤1, measured in tens of metres from the junction, the approach profile is f(x)=x²+3x+2. For x>1 the deck profile is modelled by g(x)=a√x+b, with x in the same units and height in metres. The design brief requires the road height and slope to be continuous at x=1; otherwise vehicles would feel a jolt at the junction. The engineer also wants to know whether the curvature changes abruptly, because a sharp change in curvature can cause discomfort at speed. A junior engineer suggests choosing a and b so that the two pieces have the same value and derivative at x=1, and then claims the result is 'perfectly smooth'. The senior engineer warns that matching first derivatives is not the same as matching second derivatives, and asks for a quantitative check before approving the profile.
1. Find a and b so that the height and the first derivative match at x=1.
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For the approach, f(1)=1+3+2=6, and f′(x)=2x+3, so f′(1)=5. For the deck, g(1)=a+b and g′(x)=a/(2√x), so g′(1)=a/2. Height matching gives a+b=6. Slope matching gives a/2=5, hence a=10. Then 10+b=6, so b=−4. The deck model is therefore g(x)=10√x−4. With these values the two pieces agree in value and first derivative at x=1, so the profile is continuous and has no slope jump at the junction.
2. Compute the second derivatives at x=1 and decide whether the junior engineer's 'perfectly smooth' claim is justified.
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On the approach, f″(x)=2 for all x, so f″(1)=2. On the deck, g′(x)=5x^(−1/2), so g″(x)=−5/2·x^(−3/2). At x=1 this gives g″(1)=−5/2=−2.5. The second derivatives are not equal: there is a jump from 2 to −2.5. Thus the joined profile is continuous and differentiable at x=1, but it is not twice differentiable there. The junior engineer's phrase 'perfectly smooth' is unjustified if smoothness is meant in the C² sense. The curvature jump is real and should be assessed against the speed and comfort criteria rather than dismissed because the first derivatives match.
3. Can a deck profile of the form g(x)=a√x+b satisfy height, slope and second-derivative matching at x=1? Show the conflict and state what must change.
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Height and slope matching already force a=10 and b=−4. The second derivative of a√x+b at x=1 is −a/4. Matching the approach value 2 would require −a/4=2, so a=−8. This contradicts a=10. Therefore no function of the form a√x+b can match all three quantities at the junction while keeping the given approach curve. To remove the curvature jump, the engineer must use a richer family, such as a quadratic or cubic deck segment with enough parameters, or move the junction to a point where the required conditions are compatible. Otherwise the design must explicitly accept a C¹ but not C² transition.