Why linear approximation

For example, the function g x is given by the equation. It very often happens in applications that a model produces equations that are extremely difficult or impossible to solve. However, some of the factors are more important than others. But you don't want to ignore the factor completely, so the next thing to try is a linear approximation. This differential equation is NOT solvable exactly. If the equation for something as simple as a pendulum can't be solved exactly, imagine anything that is remotely complicated.

Linear approximation is basically the simplest kind of approximation other than constant approximation. Let me expand upon this example of solving algebraic equations. When we do it with linear approximation, the result is called Newton's method. This works very well provided the initial guess is good. There are situations where we have a problem with a small parameter, and we want to approximate the solution to the problem in terms of the solution to the problem where the parameter is zero, plus correction terms.

Sometimes this can be done exactly as stated; such problems are called regular perturbation problems. Other times there is no solution to the problem with the parameter being zero; these problems are called singular perturbation problems.

Both of these come up quite frequently in applications. There are also truly "analytic" applications, where iterated approximation allows us to demonstrate something exact, such as the existence of a solution to a problem. Real examples of this are probably above your current level, so I won't comment further unless requested. I just didn't want to say something blatantly false.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Asked 6 years, 4 months ago. Active 6 years, 4 months ago. Get access to all the courses and over HD videos with your subscription.

Get My Subscription Now. Please click here if you are not redirected within a few seconds. Home » Derivatives » Linear Approximation. This lesson is about using the tangent line to approximate another point on our curve.

Now, let's say I want to approximate f 4. If you were to plug this into the original function, then you would get 4. This would be really hard to compute without a calculator. However, using linear approximation , we can say that. We just approximated the f 4. Now let's actually see how close we were to the exact value of f 4. Notice that f 4. So we are really close!

We were only off when we've got to the second decimal place! Now so far, these questions gave us a function and a point to work with. What if none of these were given at all? What if the question only tells us to estimate a number? How would we do it? We would need to use the linear approximation. This means we have to make them ourselves. This leads us to do the following steps:. In fact, you get a bunch of decimal numbers. So we have to try something else. Going back to the linear approximation formula we have:.

If you want more practice problems about linear approximation , then I recommend you look at this link here. We know that linear approximation is just an estimation of the function's value at a specified point. However, how do we know that if our estimation is an overestimate or an underestimate? We calculate the second derivative and look at the concavity. If the second derivative of the function is greater than 0 for values near a, then the function is concave up.

This means that our approximation will be an underestimate. Notice that f x is concave upward and the tangent line is right under f x. Let's say were to use the tangent line to approximate f x. Then the y values of the tangent line are always going to be less than the actual value of f x.

Hence, we have an underestimate. Now if the second derivative of the function is less than 0 for values near a, then the function is concave down.

This means that our approximation will be an overestimate. Notice that f x is concave downward and the tangent line is right above f x.

Again, let's say that we are going to use the tangent line to approximate f x. Then the y values of the tangent line are always going to be greater than the actual value of f x. Hence, we have an overestimate. So if you ever need to see if your value is an underestimation or an overestimation, make sure you follow these steps:. If we linear approximate f 4. Now look at the second derivative. When x is positive, we see that. We know that if the function is concave down, then the tangent line will be above the function.

Hence, using the tangent line as an approximation will give an overestimated value. Not only can we approximate values with linear approximation, but we can also approximate with differentials. To approximate, we use the following formula. Since we are dealing with very small changes in x and y, then we are going to use the fact that:.

This approximation is very useful when approximating the change of y. Keep in mind back then they didn't have calculators, so this is the best approximation they could get for functions with square roots or natural logs.

Suppose x changes from 0 to 0. However, most of the time we want to estimate a value of the function, and not the change of the value.

Hence we will add both sides of the equation by y, which gives us:. This equation is a bit hard to read, so we are going to rearrange it even more.

How do we use this formula? I recommend following these steps:. Now we have learned a lot about linear approximation , but what else can we do with it? We can actually use the linear approximation formula to prove a rule known as L'Hospital's Rule.