What is the slope of the line normal to the tangent line of #f(x) = sec^2xxcos(xpi/4) # at # x= (7pi)/4 #?
The reqd. slope =
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To find the slope of the line normal to the tangent line of ( f(x) = \sec^2(x)  x \cos(x  \frac{\pi}{4}) ) at ( x = \frac{7\pi}{4} ), we need to first find the derivative of the function and evaluate it at ( x = \frac{7\pi}{4} ) to get the slope of the tangent line. Then, we'll find the negative reciprocal of this slope to get the slope of the normal line.

Find the derivative of ( f(x) ): [ f'(x) = 2\sec(x)\tan(x) + \cos(x  \frac{\pi}{4}) + x\sin(x  \frac{\pi}{4}) ]

Evaluate ( f'(x) ) at ( x = \frac{7\pi}{4} ): [ f'\left(\frac{7\pi}{4}\right) = 2\sec\left(\frac{7\pi}{4}\right)\tan\left(\frac{7\pi}{4}\right) + \cos\left(\frac{7\pi}{4}  \frac{\pi}{4}\right) + \frac{7\pi}{4}\sin\left(\frac{7\pi}{4}  \frac{\pi}{4}\right) ]

Simplify ( f'\left(\frac{7\pi}{4}\right) ): [ f'\left(\frac{7\pi}{4}\right) = 2\sec\left(\frac{7\pi}{4}\right)\tan\left(\frac{7\pi}{4}\right) + \cos\left(\frac{3\pi}{2}\right) + \frac{7\pi}{4}\sin\left(\frac{3\pi}{2}\right) ] [ f'\left(\frac{7\pi}{4}\right) = 2(1)(1) + 0 + \frac{7\pi}{4}(1) ] [ f'\left(\frac{7\pi}{4}\right) = 2  \frac{7\pi}{4} ]

This is the slope of the tangent line. Now, find the negative reciprocal to get the slope of the normal line: [ \text{Slope of normal line} = \frac{1}{f'\left(\frac{7\pi}{4}\right)} = \frac{1}{2  \frac{7\pi}{4}} = \frac{1}{2 + \frac{7\pi}{4}} = \frac{1}{2}  \frac{\pi}{8} ]
So, the slope of the line normal to the tangent line of ( f(x) ) at ( x = \frac{7\pi}{4} ) is ( \frac{1}{2}  \frac{\pi}{8} ).
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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.
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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