Question

Find the linear approximation l(x) of the function g(x) = (1 x)^1/3 at a = 0. a)L(x)=1−1/3x b)L(x)=1+1/3x c)L(x)=1+x d)L(x)=1−x

Answer 1

The linear approximation l(x) of the function g(x) = (1 - x)^(1/3) at a = 0 is L(x) = 1 + (1/3)x.

Correct option is b.

Step-by-step explanation:

The linear approximation l(x) of the function g(x) = (1 - x)^(1/3) at a = 0 can be found using the formula:

L(x) = g(0) + g'(0)(x - 0)

First, let's find g(0):

g(0) = (1 - 0)^(1/3) = 1

Next, let's find g'(x) (the derivative of g(x)):

g'(x) = (1/3)(1 - x)^(-2/3)

Now, plug in x = 0 into g'(x) to find g'(0):

g'(0) = (1/3)(1 - 0)^(-2/3) = (1/3)

Finally, substitute g(0) = 1 and g'(0) = 1/3 into the linear approximation formula:

L(x) = 1 + (1/3)(x - 0) = 1 + (1/3)x

Therefore, the linear approximation of g(x) = (1 - x)^(1/3) at a = 0 is L(x) = 1 + (1/3)x.

Correct option is b.