What is the average value of a function #sec^2x# on the interval #[pi/6, pi/4]#?
The average value then equals:
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To find the average value of the function ( \sec^2(x) ) on the interval ( \left[ \frac{\pi}{6}, \frac{\pi}{4} \right] ), follow these steps:
 Evaluate the integral of ( \sec^2(x) ) over the given interval.
 Divide the result from step 1 by the length of the interval.
Here are the steps in detail:
 Integrate ( \sec^2(x) ) over the interval ( \left[ \frac{\pi}{6}, \frac{\pi}{4} \right] ): [ \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \sec^2(x) , dx ]
Using the integral of ( \sec^2(x) ), which is ( \tan(x) + C ), evaluate the integral at the upper and lower limits of integration: [ \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \sec^2(x) , dx = \tan\left(\frac{\pi}{4}\right)  \tan\left(\frac{\pi}{6}\right) ]

Calculate the length of the interval: [ \text{Length} = \frac{\pi}{4}  \frac{\pi}{6} = \frac{\pi}{12} ]

Compute the average value of ( \sec^2(x) ) over the interval by dividing the result from step 1 by the length of the interval: [ \text{Average value} = \frac{\tan\left(\frac{\pi}{4}\right)  \tan\left(\frac{\pi}{6}\right)}{\frac{\pi}{12}} ]
[ \text{Average value} = \frac{1  \frac{\sqrt{3}}{3}}{\frac{\pi}{12}} ]
[ \text{Average value} = \frac{12}{\pi} \left(1  \frac{\sqrt{3}}{3}\right) ]
[ \text{Average value} \approx \frac{12}{\pi} \left(1  \frac{1.732}{3}\right) ]
[ \boxed{\text{Average value} \approx \frac{12}{\pi} \left(1  0.577\right) \approx \frac{12}{\pi} \times 0.423 \approx \frac{50.76}{\pi}} ]
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When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
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