Calculus

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Linear Approximation

How to Calculate a Linear Approximation

A linear approximation is a way to approximate what a function looks like at a point along its curve. We find the tangent line at a point x = a on the function f(x) to make a linear approximation of the function.

We will designate the equation of the linear approximation as L(x). The linear approximation equation is given as:
L(x) = f(a) + f’(a)(x – a)
Where f(a) is the value of our function at the point x = a, and f’(a) is the value of the derivative of our function at the point x = a.

The image above shows an example of a linear approximation. The green line is the function f(x), and the red line is the tangent line that is used as a linear approximation. As we can see, the straight line approximates what the curve looks like at the point a.

When to use a Linear Approximation

A linear approximation can simplify the behavior of complicated functions. The closer we are to our point x = a, the more accurate the linear approximation will be. As we get farther away from the point x = a, the approximation becomes less accurate.

For a simple curve like the one shown above, a linear approximation predicts the direction of the curve. It does not predict the concavity of the curve. The linear approximation is the same thing as a tangent line to the curve at the point a.

Example Problem

Find the linear approximation of f(x) = cos(x) at the point x = 1.3. What is the y-intercept of the linear approximation?
Solution:
1.) First, let’s take the derivative of the function.
f’(x) = -sin(x)
2.) Now we can find the slope at point a.
f’(a) = -sin(1.3) = -0.9636
3.) Next, we need to find the original function’s value at point a.
f(a) = cos(1.3) = 0.2675
4.) Plugging everything into the linear approximation equation, we get:
L(x) = f(a) + f’(a)(x – a)
L(x) = 0.2675 + (-0.9636)(x – 1.3)
L(x) = – 0.9636x + 1.5202
5.) To find the y-intercept, we need to find the value of L(x) at x = 0.
L(0) = – 0.9636(0) + 1.5202
L(0) = 1.5202
6.) The linear approximation is L(x) = –0.9636x + 1.5202 and the y-intercept is at y = 1.5202.

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