Science:Math Exam Resources/Courses/MATH110/April 2013/Question 03 (c)

MATH110 April 2013

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[hide] Question 03 (c)
Let $f(x)={\frac {\sin(x)}{e^{x}}}$ on the interval $[0,\pi ]$ . In this question you will sketch this function. (c) Determine where ƒ is increasing and where it is decreasing. Does ƒ have any local extrema?

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[show] Hint
To determine where $f(x)$ is increasing and decreasing, we should find the first derivative, its critical points, and do the first derivative test.

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[show]

Solution

Using the quotient rule, we find:

{\begin{aligned}f'(x)&={\frac {(\cos x)(e^{x})-(\sin x)(e^{x})}{(e^{x})^{2}}}\\&={\frac {e^{x}(\cos x-\sin x)}{(e^{x})^{2}}}\\&={\frac {\cos x-\sin x}{e^{x}}}\end{aligned}}

Setting $\displaystyle f'(x)=0$ , we can clear the denominator and get:

\displaystyle \cos x-\sin x=0

or, equivalently,

\displaystyle \cos x=\sin x

This equation has infinitely many solutions, but only one which is in the interval [0,π]. The trig ratios $\cos x$ and $\sin x$ are equal on a 45-45-90 degree triangle, so the angle in the interval [0,π] that solves the above equation is 45 degrees, or in radian measure, $x=\pi /4$ .

Next, we check to see if there are any critical points where $f'(x)$ is undefined. The derivative would be undefined where the denominator is equal to zero, but because $e^{x}$ is always positive, we don't need to worry about this here.