What is the equation of the line tangent to # f(x)=2x^2 + cos(x)# at # x=pi/3#?

Answer 1

#y-(4pi^2+9)/18=(8pi-3sqrt3)/6(x-pi/3)#

Find #f'(pi/3)#, the slope of the tangent line at the point on #f(x)# where #x=pi/3#.
The point is at #(x,f(pi/3))#.
#f(pi/3)=2(pi/3)^2+cos(pi/3)=(2pi^2)/9+1/2=(4pi^2+9)/18#
#(pi/3,(4pi^2+9)/18)#

Calculate the derivative.

#f'(x)=4x-sin(x)#
#f'(pi/3)=4(pi/3)-sin(pi/3)=(4pi)/3-sqrt3/2=(8pi-3sqrt3)/6#

In point-slope form, write:

#y-(4pi^2+9)/18=(8pi-3sqrt3)/6(x-pi/3)#
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Answer 2

The equation of the line tangent to f(x)=2x^2 + cos(x) at x=pi/3 is y = 3√3x - 2√3π + 2/3.

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Answer 3

To find the equation of the line tangent to the function ( f(x) = 2x^2 + \cos(x) ) at ( x = \frac{\pi}{3} ), we need to follow these steps:

  1. Find the derivative of the function ( f(x) ) using differentiation rules.
  2. Evaluate the derivative at ( x = \frac{\pi}{3} ) to find the slope of the tangent line.
  3. Use the point-slope form of the equation of a line, where the slope is the derivative evaluated at ( x = \frac{\pi}{3} ), and the point is ( (x, f(x)) ) where ( x = \frac{\pi}{3} ).

Step 1: Find the derivative of ( f(x) ): [ f'(x) = 4x - \sin(x) ]

Step 2: Evaluate the derivative at ( x = \frac{\pi}{3} ): [ f'\left(\frac{\pi}{3}\right) = 4\left(\frac{\pi}{3}\right) - \sin\left(\frac{\pi}{3}\right) ] [ = \frac{4\pi}{3} - \frac{\sqrt{3}}{2} ]

Step 3: Use the point-slope form to find the equation of the tangent line: [ y - f\left(\frac{\pi}{3}\right) = f'\left(\frac{\pi}{3}\right)(x - \frac{\pi}{3}) ] [ y - \left(2\left(\frac{\pi}{3}\right)^2 + \cos\left(\frac{\pi}{3}\right)\right) = \left(\frac{4\pi}{3} - \frac{\sqrt{3}}{2}\right)\left(x - \frac{\pi}{3}\right) ]

This is the equation of the tangent line to ( f(x) ) at ( x = \frac{\pi}{3} ).

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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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