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

Answer 1

I found: #y=3/4x+1/4#

Here we need to locally substitute the actual curve of our function with the line tangent to our curve at point of coordinate #x=1#.

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:
#y-y_0=m(x-x_0)#
where:
#m# is the slope;
#x_0,y_0# are the coordinates of the point of interest.
In our case we have:
#x_0=1# (given)
substituting into our function we see that: #f(1)=y_0=1#
Now we need the slope #m#; we can find it by deriving our function and evaluating it at #x=1#:
#f'(x)=3/4x^(3/4-1)=3/4x^(-1/4)#
evaluated at #x=1#:
#f'(1)=m=3/4#

Now we use: #y-y_0=m(x-x_0)# to get:
#y-1=3/4(x-1)#
or
#y=3/4x-3/4+1#
#y=3/4x+1/4#

Graphically:

You can see that around #x=1# the line and the curve are almost the same thing, so, if you need, you can study the line instead of the original curve (where the curve may represent a phenomenon or tendency)!

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Answer 2

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

  1. Find the first derivative of ( f(x) ).
  2. Evaluate the derivative at ( x = 1 ) to find the slope of the tangent line.
  3. Plug ( x = 1 ) into ( f(x) ) to find the corresponding y-coordinate.
  4. 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.
  5. Substitute the slope and the point into the point-slope form to find the linearization.

Here are the calculations:

  1. ( f'(x) = \frac{3}{4}x^{-1/4} )
  2. ( f'(1) = \frac{3}{4}(1)^{-1/4} = \frac{3}{4} )
  3. ( f(1) = 1^{3/4} = 1 )
  4. Use the point ( (1,1) ) and slope ( \frac{3}{4} )
  5. ( 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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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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