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. Here is how the proof works.
Recall that the linear approximation formula is:. Realize that the approximation becomes more and more accurate as we pick x values that are closer to a. Now we are going to put this aside and use it later, and actually look at l'hopital's rule. We are going to assume a couple things here. We always want to apply l'hoptial's rule when we encounter indeterminate limits. There are two types of indeterminate forms. These indeterminate forms would be:.
A lot of people make the mistake of using l'hopital's rule without even checking if it is an indeterminate limit. So make sure you check it first! Otherwise, it will not work and you will get the wrong answer.
Here is a guide to using l'hopital's rule:. Now that question was a little bit easy, so why don't we take a look at something that is a bit harder. This is another indeterminate form. So we have to go back to step 3 and apply l'hoptial's rules again. Back to Course Index. You can still navigate around the site and check out our free content, but some functionality, such as sign up, will not work.
If you do have javascript enabled there may have been a loading error; try refreshing your browser. Home Calculus Derivative Applications. That's the lesson. That's the last lesson. Thus, the linear approximation formula is an application of derivatives. Let us learn more about this formula in the upcoming sections.
As we discussed in the previous section, the linear approximation formula is nothing but the equation of a tangent line. As we know the slope of this tangent is the derivative f ' a , its equation using the point-slope form is:.
Round your answer to 4 decimals. Since In fact, when we are asked to use differentials to estimate a given number, such as the square root or exponential, we are, in essence, being asked to use linearization to approximate the value.
In other words, with the help of tangent lines and a bit of calculus, we will become human calculators! And finding the differential for exponential functions follows almost the same process as seen for the linear approximation of square roots. So, together we will look at the overall process and idea behind linear approximation i.
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