How do you find the linearization at a=1 of #f(x) = ln(x)#?

Answer 1

#L=x#

#L=f(a)+f'(a)(x-a)#
#f(x) = lnx#
#f(a) = f(1) =0#
#f'(x) = 1/x#
#f'(a) = f'(1) = 1/1=1#
#L=x#
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Answer 2

To find the linearization of ( f(x) = \ln(x) ) at ( a = 1 ), follow these steps:

  1. Calculate the derivative of ( f(x) = \ln(x) ) using the differentiation rule for natural logarithm. ( f'(x) = \frac{1}{x} )

  2. Evaluate the derivative at ( x = 1 ) to find the slope of the tangent line at ( x = 1 ). ( f'(1) = \frac{1}{1} = 1 )

  3. Use the point-slope form of a line to write the equation of the tangent line. ( y - y_1 = m(x - x_1) )

  4. Substitute the slope ( m ) and the point ( (x_1, y_1) = (1, \ln(1)) ) into the equation. ( y - \ln(1) = 1(x - 1) )

  5. Simplify the equation. ( y - 0 = x - 1 )

  6. The linearization of ( f(x) = \ln(x) ) at ( a = 1 ) is given by: ( L(x) = x - 1 )

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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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