Science:Math Exam Resources/Courses/MATH110/April 2011/Question 03 (b)

MATH110 April 2011

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Question 03 (b)
In this question you will state and then apply the Mean Value Theorem. What does the Mean Value Theorem tell us about the function $f(x)=\sin x$ on the interval [0,b], where b is any positive number?

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
Does $\sin x$ satisfy the conditions of the Mean Value Theorem on the interval $[0,b]$? What conclusion can you draw?

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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. Because the function $\sin x$ is continuous on the interval $[0,b]$ and is differentiable on the interval $(0,b)$, we can apply the Mean Value Theorem. Thus we know that there is some $c$ in the interval $(0,b)$ so that: ${\begin{aligned}\cos(c)&={\frac {\sin(b)-\sin(0)}{b-0}}\\&={\frac {\sin(b)}{b}}\end{aligned}}$

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