a.) Given that f(3)=5 and f'(x)=x/((x^3)+3), find the linear approximation of f(x) at x=3.

b.)If the linear approximation is used to estimate the value of f(2.9), will it be an overestimation or underestimation? Show justification

answered below

a.) To find the linear approximation of f(x) at x=3, we can use the formula for linear approximation:

L(x) = f(a) + f'(a)(x - a)

Given that f(3) = 5, we know that a = 3 and f(a) = 5.

Now, we need to find the value of f'(a) at x = 3. Using the given function f'(x) = x/((x^3)+3), we can substitute a = 3:

f'(3) = 3/((3^3)+3) = 3/30 = 1/10

Now, we can plug in a = 3, f(a) = 5, and f'(a) = 1/10 into the linear approximation formula:

L(x) = 5 + (1/10)(x - 3)

Therefore, the linear approximation of f(x) at x=3 is: L(x) = 5 + (1/10)(x - 3).

b.) To estimate the value of f(2.9) using the linear approximation, we can substitute x = 2.9 into the linear approximation formula we found in part a:

L(2.9) = 5 + (1/10)(2.9 - 3)

Simplifying this expression, we get:

L(2.9) = 5 + (1/10)(-0.1) = 5 - 0.01 = 4.99

Comparing this with the actual value of f(2.9), we can determine if it is an overestimation or underestimation. However, we don't have the actual value of f(2.9), so we cannot directly determine if the linear approximation is an overestimation or underestimation.

To justify our estimation, we can consider the behavior of the function f(x) and its derivative f'(x) around x = 3.

Note that f'(x) = x/((x^3)+3) is always positive for x > 0. This implies that the function f(x) is increasing for x > 0.

Since 2.9 is slightly less than 3, and f(x) is increasing around x = 3, it is likely that f(2.9) is less than the linear approximation value of 4.99. Therefore, it is reasonable to conclude that the linear approximation will be an overestimation of f(2.9).