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

I found:

Here we need to locally substitute the actual curve of our function with the line tangent to our curve at point of coordinate

Basically instead of considering the real curve we substitute it with a straight line (the tangent). This works locally, i.e., in a very small interval around our point so we must be careful not to overdo it!

We will need the equation of the line in the general form:

where:

In our case we have:

substituting into our function we see that:

Now we need the slope

evaluated at

Now we use:

or

Graphically:

You can see that around

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

- Find the first derivative of ( f(x) ).
- Evaluate the derivative at ( x = 1 ) to find the slope of the tangent line.
- Plug ( x = 1 ) into ( f(x) ) to find the corresponding y-coordinate.
- Use the point-slope form of a linear equation ( y - y_1 = m(x - x_1) ), where ( (x_1, y_1) ) is the point on the function and ( m ) is the slope.
- Substitute the slope and the point into the point-slope form to find the linearization.

Here are the calculations:

- ( f'(x) = \frac{3}{4}x^{-1/4} )
- ( f'(1) = \frac{3}{4}(1)^{-1/4} = \frac{3}{4} )
- ( f(1) = 1^{3/4} = 1 )
- Use the point ( (1,1) ) and slope ( \frac{3}{4} )
- ( y - 1 = \frac{3}{4}(x - 1) )

Therefore, the linearization of ( f(x) = x^{3/4} ) at ( x = 1 ) is ( y = \frac{3}{4}(x - 1) + 1 ).

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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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