What is the equation of the normal line of #f(x)= xsinx-cos^2x# at #x = pi/8#?

Answer 1

#y =-0.69x-0.43# [To 2D]

#f(x) = xsinx-cos^2x#

Apply product rule, chain rule and standard derivatives.

#f'(x) = xcosx+sinx- 2cosx*(-sinx)#

#= xcosx+sinx+2sinxcosx#

#= xcosx+sinx+sin(2x)#

The slope of #f(x)# at #x=pi/8# is #f'(pi/8)#

#f'(pi/8) = pi/8*cos(pi/8) +sin(pi/8) + sin(pi/4)#

#approx 1.542597# (Calculator)

The slope of a tangent (#m_1#) #xx# slope of a normal (#m_2#) at any point on a curve is given by: #m_1*m_2 =-1#

#:.# Slope of the normal to #f(x)# at #x=pi/8 approx -1/1.542597#

#approx -0.688422#

The equation of a straight line of slope #m# passing through the point #(x_1,y_1)# is given by:

#(y-y_1) = m(x-x_1)#

In this example, our normal will have the equation:

#y-f(pi/8) approx -0.688422 (x-pi/8)#

#y- (-0.703274) approx -0.688422 (x- 0.392699)# (Calculator)

#y+0703274 approx -0.688422x+0.270343#

#y approx -0.688422x-0.432931#

#y =-0.69x-0.43# [To 2D]

We can see #f(x)# and the normal at #x=pi/8# on the graphic below.

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Answer 2

The equation of the normal line of f(x) = xsinx - cos^2x at x = pi/8 is y = -8x + 1.

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Answer 3

To find the equation of the normal line to ( f(x) = x \sin(x) - \cos^2(x) ) at ( x = \frac{\pi}{8} ), follow these steps:

  1. Calculate the derivative of ( f(x) ) to get the slope of the tangent line. [ f'(x) = \sin(x) + x \cos(x) + 2 \cos(x) \sin(x) ]

  2. Evaluate the derivative at ( x = \frac{\pi}{8} ): [ f'\left(\frac{\pi}{8}\right) = \sin\left(\frac{\pi}{8}\right) + \frac{\pi}{8} \cos\left(\frac{\pi}{8}\right) + 2 \cos\left(\frac{\pi}{8}\right) \sin\left(\frac{\pi}{8}\right) ]

  3. Calculate the slope of the tangent line at ( x = \frac{\pi}{8} ).

  4. The slope of the normal line is the negative reciprocal of the slope of the tangent line.

  5. Use the point-slope form of a line to write the equation of the normal line using the slope from step 4 and the point ( \left(\frac{\pi}{8}, f\left(\frac{\pi}{8}\right)\right) ).

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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.

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.

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.

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.

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