Approximate the value of ^3 sqrt 27.3 using the tangent line of f(x)= ^3 sqrt x at point x= 27?

Answer 1
The point of tangency will be #f(27) = root(3)(27) = 3# so #(27, 3)#.

The slope of the tangent at this point is given by the derivative.

#f(x) = x^(1/3) -> f'(x) = 1/3x^(-2/3)#
#f'(27) = 1/3(27)^(-2/3) = 1/(3(27)^(2/3)) = 1/27#

Recall the equation of a line is given by

#y -y_1 = m(x- x_1)#
#y - 3 = 1/27(x - 27)#
#y = 1/27x - 1 +3#
#y = 1/27x + 2#
So at #27.3#, the tangent line approximation gives us
#y(27.3) = 1/27(27.3) + 2 = 3.011#
If you check using a calculator, your answer will be the same as the one obtained using the tangent line up to #4# decimal places (so our answer, which is to three decimal places, is the same as the actual value of #root(3)(27.3)#).

Hopefully this helps!

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Answer 2

The approximate value of ( \sqrt[3]{27.3} ) using the tangent line of ( f(x) = \sqrt[3]{x} ) at ( x = 27 ) is approximately ( 3.29 ).

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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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