How do you find the linearization of #f(x)=x^3# at the point x=2?
The linearization is the equation of the tangent line. (Often presented in a different form.)
The linearization is
The equation of the tangent line
In pointslope form the equation is
When thinking of it as a linear approximation, we write
More explanation
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To find the linearization of ( f(x) = x^3 ) at the point ( x = 2 ), follow these steps:

Find the value of the function ( f(x) ) at the given point ( x = 2 ): ( f(2) = 2^3 = 8 ).

Find the derivative of the function ( f'(x) = 3x^2 ).

Evaluate the derivative at the point ( x = 2 ): ( f'(2) = 3(2)^2 = 12 ).

Write the equation of the tangent line using the pointslope form: ( y  y_1 = m(x  x_1) ), where ( (x_1, y_1) ) is the point ( (2, 8) ) and ( m ) is the slope ( f'(2) ).

Substitute the values into the equation: ( y  8 = 12(x  2) ).

Simplify the equation: ( y  8 = 12x  24 ).

Rearrange the equation to isolate ( y ): ( y = 12x  24 + 8 ).

Simplify further: ( y = 12x  16 ).
So, the linearization of ( f(x) = x^3 ) at the point ( x = 2 ) is ( L(x) = 12x  16 ).
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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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