How do you find the linear approximation of the function f(x) = 4x at a = 0, and how can you use it to approximate the values of 39 and 399?

by

To find the linear approximation of the function f(x) = 4x at a = 0, we start by using the formula for linear approximation, which is given by:

L(x) = f(a) + f'(a)(x - a)

Where:

  • L(x) is the linear approximation,
  • f(a) is the value of the function at a,
  • f'(a) is the derivative of the function at a,
  • x is the point at which we want to approximate the function,
  • a is the point at which we are approximating (in this case, 0).

Step 1: Calculate f(a):

f(0) = 4 * 0 = 0

Step 2: Calculate the derivative of the function f(x):

f'(x) = 4

Step 3: Calculate f'(a):

f'(0) = 4

Step 4: Plug everything into the linear approximation formula:

L(x) = f(0) + f'(0)(x - 0)
L(x) = 0 + 4x = 4x

The linear approximation of the function f(x) = 4x at the point a = 0 is L(x) = 4x.

Now, we can use this linear approximation to estimate the values of 39 and 399:

To approximate 39:

L(39) = 4 * 39 = 156

To approximate 399:

L(399) = 4 * 399 = 1596

So, using the linear approximation, the estimated values are:

  • For 39: approximately 156
  • For 399: approximately 1596

Leave a Comment