AP Calculus AB Exam Question:

Given the function f(x) = 3x^2 - 2x + 4, find the linear approximation of f(x) near x = 2, and use it to estimate f(2.1). Show all work and round your final answer to three decimal places.

Solution:

To find the linear approximation of f(x) near x = 2, we need to find the equation of the tangent line at that point.

Step 1: Find the slope of the tangent line by calculating the derivative of f(x). f'(x) = d/dx (3x^2 - 2x + 4) = 6x - 2

Step 2: Evaluate the derivative at x = 2 to find the slope of the tangent line. f'(2) = 6(2) - 2 = 10

Step 3: Find the y-coordinate of the point (2, f(2)) on the function. f(2) = 3(2)^2 - 2(2) + 4 = 12

Step 4: Use the point-slope form of a line to write the equation of the tangent line. y - f(2) = f'(2) * (x - 2)

Substituting the values we found: y - 12 = 10(x - 2)

Step 5: Simplify the equation. y - 12 = 10x - 20 y = 10x - 8

Now, we can use this line to estimate f(2.1).

Step 6: Plug in x = 2.1 into the equation y = 10x - 8. f(2.1) ≈ 10(2.1) - 8 = 21 - 8 = 13

Therefore, the linear approximation of f(x) near x = 2 gives us an estimate of f(2.1) as approximately 13.

Answer: f(2.1) ≈ 13.