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How do you find the average value of #f(x)=cosx# as x varies between #[1,5]#?

Answer 1

Emily Ammerman

#1/4(sin(5)-sin(1))approx-0.45010#

The average value of the function #f# on the interval #[a,b]# is

#1/(b-a)int_a^bf(x)dx#

With the given information this translates into

#1/(5-1)int_1^5cos(x)dx#

The antiderivative of #cos(x)# is #sin(x)# so

#=1/4[sin(x)]_1^5=1/4(sin(5)-sin(1))#

This is as simplified as we can get without using a calculator.

#1/4(sin(5)-sin(1))approx-0.45010#

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Answer 2

Aurora Alvarado

To find the average value of ( f(x) = \cos(x) ) over the interval ([1, 5]), you use the formula:

[ \text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(x) , dx ]

Where ( a = 1 ) and ( b = 5 ) in this case. Integrating ( \cos(x) ) over the interval ([1, 5]) gives:

[ \int_{1}^{5} \cos(x) , dx = \sin(5) - \sin(1) ]

So, the average value is:

[ \frac{1}{5 - 1} \times (\sin(5) - \sin(1)) ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.