Final answer:
To find the tangent line approximation for f(1.1), find the derivative of f(x) and plug in the values. The equation of the tangent line can be found using the point-slope form. Determine if the actual value is an overestimate or underestimate by analyzing the graph's concavity, which requires the second derivative.
Step-by-step explanation:
To find the tangent line approximation for f(1.1), we start by finding the derivative of f(x) using the given expression f'(x) = cos((1/x^2)+x). Plugging in x = 1, we get f'(1) = cos((1/1^2)+1) = cos(2).
Next, we find the equation of the tangent line using the point-slope form y - y1 = m(x - x1) where (x1, y1) is the point on the graph. Plugging in the values x1 = 1 and y1 = f(1) = 2, and m = f'(1) = cos(2), we get y - 2 = cos(2)(x - 1).
To determine if f(1.1) = 2.12 is an overestimate or underestimate, we can look at the shape of the graph of f(x) near x = 1. If the graph is concave up (opening upwards) near x = 1, then the tangent line approximation would be an underestimate, and vice versa. We can determine the concavity by analyzing the second derivative of f(x). However, since the second derivative is not given in the question, we cannot determine the concavity with the given information.