Science:Math Exam Resources/Courses/MATH104/December 2016/Question 02 (b)

MATH104 December 2016

• Q1 (a) • Q1 (b) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q7 (a) • Q7 (b) • Q8 (a) • Q8 (b) • Q9 (a) • Q9 (b) • Q9 (c) • Q10 • Q11 (a) • Q11 (b) • Q11 (c) • Q11 (d) • Q12 • Q13 •

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Question 02 (b)
The graph of the position of a particle is shown below, where $t$ is measured in second and the dots are local extrema or points of inflection. Determine when the particle is speeding up. Hint: a particle is "speeding up" when its velocity and acceleration have the same sign.

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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
Followed the hint from question, the velocity is $f'$ , and the acceleration is the derivative of velocity $f''$ . Find the sign of $f',f''$

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Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The first derivative of m is straightforward, if the derivative is positive, then $f'>0$ . For $f''$ , we use the points of inflection (3 and 5 from question). 3 is one inflection point. From 0 to 3, $f'$ goes from positive to negative, thus during that area, $f''<0$ . For $t\in [3,5],f''>0$ . For $t>5,f''<0$ . The part where $f',f''$ are both positive: $\left(4,5\right)$ The part where $f',f''$ are both negative: $\left(2,3\right)$ answer: $\color {blue}\left(2,3\right),\left(4,5\right)$