Q. 2 TB - Calculus

Question

If a function \(f\) is differentiable at the point $x_o$, what is the equation of the line tangent to the graph of \(f\) at $x_o$? Why is this line a good approximation for \(f\) near $x_o$?

Step-by-Step Solution

Answer

The equation of the tangent line to the graph of \(f\) at $x_o$ is given as: \(y=f'(x_0)(x-x_0)+f(x_0)\).

Step 1: Equation of a tangent line to the graph of \(f\) at the given point.

We know that if a function f is differentiable at x=$x_o$, we can compute f′($x_o$) to get the slope of the tangent line to y=f(x) at ($x_o$,f($x_o$) ). The equation in point-slope form for the resulting tangent line through ($x_o$,f($x_o$) )  with slope \(m=f'(x_0)\) is given by

\(y=f'(x_0)(x-x_0)+f(x_0)\).                                           

Step 2: Linear Approximation

• In order to approximate the value of a function at a point using a line, one must use a linear approximation of the function.
• The value of the function at any point that is very close to the provided point can be generally determined using the equation of the tangent line at the given point.
• The linear approximation formula is:

  L(x) = f($x_o$) + f '($x_o$) (x - $x_o$)

  where,

  • L(x) is the linear approximation of f(x) at x = $x_o$.
  • f '($x_o$) is the derivative of f(x) at x = $x_o$.