Science:Math Exam Resources/Courses/MATH102/December 2016/Question A 06

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MATH102 December 2016

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Question A 06
The graphs below represent the position, velocity and acceleration of a child swinging on a swing. Identify the correct relationships between these functions. (i) $B''(t)=C'(t)=A(t)$ (ii) $B''(t)=A'(t)=C(t)$ (iii) $A''(t)=C'(t)=B(t)$ (iv) $A''(t)=B'(t)=C(t)$ (v) $C''(t)=A'(t)=B(t)$ (vi) $C''(t)=B'(t)=A(t)$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Start with the fact that where a differentiable function $x(t)$ has a (local) max/min (zero slope) its derivative must become zero (intersects x-axis). The same reasoning will help you determine which one is the second derivative as $x''(t)$ is the derivative of the derivative.

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Solution
At the points where $x(t)$ has a min/max (slope of tangent line =0), $v(t)=x'(t)$ must be equal to zero, i.e. intersects x-axis. This fact eliminates the option of $x(t)=B(t)$ , because if so, we then see that at its maximum point neither $A(t)$ nor $C(t)$ vanishes. Now we have two choices, for each of which we check whether the graphs match: If $x(t)=C(t)$ , we see that at $C$ 's max and min, $B(t)$ intersects x-axis, this means that $v(t)=x'(t)=B(t)$ so we must have $(x')'(t)=B'(t)=A(t)$ which implies that where $B(t)$ has a max or min $A(t)$ must become zero, however, we've already seen that at $B$ 's maximum $A(t)$ is NOT zero. $\Rightarrow {\text{NOT a choice}}$ . If $x(t)=A(t)$ , we see that at $A$ 's max and min, $C(t)$ intersects x-axis, this means that $v(t)=x'(t)=C(t)$ so we must have $(x')'(t)=C'(t)=B(t)$ which implies that where $C(t)$ has a max or min $B(t)$ must become zero, which we see that it is in fact true. Therefore, the correct choice is $x(t)=A(t)$ , $x'(t)=A'(t)=C(t)$ , and $x''(t)=\color {blue}A''(t)=C'(t)=B(t)$ . Answer: $\color {blue}(iii)$