Tangent Approximation

Question

Let f(x)=1/x². Determine the equation for a tangent line to f at a=2, and use this tangent equation to approximate f(2.25).

Answer 1

In case you have not learned the power rule for taking the derivative:

Then calculate the equation of the line as outlined by Mark M.

Answer 2

f(x) = 1/x² = x^-2

So, f'(x) = -2x^-3 = -2 / x³. Slope of tangent line = f'(2) = -1/4.

Point on tangent line = (2, f(2)) = (2, 1/4)

Equation of tangent line: y - 1/4 = -1/4(x - 2). So, y = -1/4x + 3/4.

f(2.25) ≈ -(1/4)(2.25) + 3/4 = -9/16 + 3/4 = 3/16 = 0.1875