Question

Find the tangent line approximation T to the graph of f at the given point. Then complete the table. (Round your answers to four decimal places.) f(x)=x2, (8, 64)

T(x) = x 7.9 7.99 8 8.01 8.1
f(x)=?
T(x)=?

Answer 1

To find the tangent line approximation to the curve f(x) = x^2 at the point (8, 64), we first calculate the slope of the tangent (which is the derivative of the function at that point) and then use the point-slope form to write the equation for the tangent line. The specified values of x can then be substituted into this equation to complete the table.

Step-by-step explanation:

The subject of this question is finding the equation for the tangent line to the function f(x) = x2 at the point (8, 64). The steps to solve for the tangent line, T(x), include:

1. Calculate the derivative of f(x) to find the slope of the tangent line at x = 8.
2. Use the point-slope form of a straight line to find the equation of the tangent line.
3. Evaluate this tangent line equation at specified values of x to fill in the table with T(x).

The derivative of f(x) = x2 is f'(x) = 2x. So, at x = 8, the slope is f'(8) = 2(8) = 16. The equation of the tangent line using point-slope form is T(x) = f'(8)(x - 8) + f(8), which simplifies to T(x) = 16(x - 8) + 64. Substituting the values 7.9, 7.99, 8.01, and 8.1 into T(x), we can determine the corresponding approximate values for T(x).