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How do you find the linearization at a=1 of # f(x) = x^4 + 4x^2#?

Answer 1

A linearization is a tangent line.

The linearization of # f(x) = x^4 + 4x^2# at #a=1# is one form of the equation of the line tangent to the graph of #f# at the point #(1,f(1))#.

For # f(x) = x^4 + 4x^2#, we have #f(1) = 5# and #f'(x) = 4x^3+8x#, so #f'(1) = 12#.

The tangent line has point-slope form:

#y-f(1) = f'(1)(x-1)# #" "# #" "# #y-5=12(x-1)#.

The linearization is:

#y=f(1) + f'(1)(x-1)# #" "# #" "# #y=5 + 12(x-1)#.

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Answer 2

Evelyn Agard

To find the linearization of ( f(x) = x^4 + 4x^2 ) at ( a = 1 ), follow these steps:

Find the first derivative of ( f(x) ), denoted as ( f'(x) ).
Evaluate ( f'(1) ) to find the slope of the tangent line at ( x = 1 ).
Use the point-slope form of a linear equation ( y - y_1 = m(x - x_1) ), where ( (x_1, y_1) ) is the point ( (1, f(1)) ) and ( m ) is the slope found in step 2.

The linearization at ( a = 1 ) of ( f(x) = x^4 + 4x^2 ) is given by the equation of the tangent line at ( 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.