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How do you find the linearization at a=1 of #f(x)=x^(3/4)#?

Answer 1

Charles Arocha

It is the equation of the tangent line at #(1,f(1))#

#f(x) = x¾#

#f(1) - 1^(3/4) = 1#

#f'(x) = 3/4 x^(-1/4)#, so #f'(1) = 3/4#

The tangent line has point-slope equation #y-1 = 3/4(x-1)#,

so the linearization is

#L(x) = 1 + 3/4(x-1)#

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Answer 2

Isaiah Allen

To find the linearization at ( a = 1 ) of ( f(x) = x^{3/4} ), follow these steps:

Find the derivative of ( f(x) ): ( f'(x) = \frac{3}{4}x^{-1/4} ).
Evaluate ( f'(1) ) to find the slope of the tangent line at ( x = 1 ): ( f'(1) = \frac{3}{4} ).
Use the point-slope form of a line to find the equation of the tangent line at ( x = 1 ): [ y - f(1) = f'(1)(x - 1) ].
Substitute ( f(1) = 1^{3/4} = 1 ) and ( f'(1) = \frac{3}{4} ) into the equation: [ y - 1 = \frac{3}{4}(x - 1) ].
Simplify the equation to get the linearization at ( a = 1 ): [ y = \frac{3}{4}x - \frac{1}{4} ].

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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.