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Derivatives - Tangent Line Approximation

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Tangent Line Approximation

There are only two things we need to remember about the tangent line to f at a.

  • The tangent line and f have the same y-value at a.  
    That is, the point (a, f(a)) is on f and also on the tangent line to f at a.

  • The tangent line and f have the same slope at a.  
    That is, the slope of the tangent line to f at a is f'(a).

That's it. If we can we remember those two things we'll be good when it comes to tangent lines.

  • Finding Tangent Lines

    If we remember two things, we can write the equation for the tangent line to f at a given the formula for f and the value a where we want the tangent line to go.

    • We can calculate f(a).
    • We can calculate f ' (a).

    We can write the equation of a line given a point and a slope, so once we have a point and a slope for the tangent line we're all set to go.

    We've left out one thing in this discussion of tangent lines: the magic formula. Most textbooks have a magic formula that produces the equation for the tangent line.

    We have lots of reasons for leaving this formula out. We don't need it. It takes extra memory that could be better spent remembering the limit definition of the derivative. It can be confusing.

    However, for the sake of completeness, we'll show the magic formula. But we're doing it our way.

    There's one special case that neither finding the equation of a line nor knowing the magic formula will help with. If f ' (a) is undefined and infinite, then we have a vertical tangent line. 

    The equation of such a tangent line is x = a like any vertical line. Knowing this will probably be more important for finding derivatives of parametric functions than it will be right now.

    We can also work backwards to figure out information about the function given information about its tangent line.

  • Using Tangent Lines to Approximate Function Values

    "Approximation" is what we do when we can't or don't want to find an exact value. We're going to approximate actual function values using tangent lines.

    We pointed out earlier that if we zoom in far enough on a continuous function, it looks like a line. For example, take the function f(x) = x2 and zoom in around x = 1. If we zoom in enough near x = 1, the function f looks like a line.

    If we graph the function and its tangent line at 1, we'll see that as we zoom in around x = 1, the function f looks like its tangent line.

    If we zoom back out a little bit, the function doesn't look quite so much like a line. However, the function and its tangent line are still "close together."

    This means, for example, that the y-value on the tangent line at x = 1.1 is "close" to the y-value on the function f(x) = x2 when x = 1.1.

    We found earlier that the tangent line to f(x) = x2 at 1 has the equation:

    y = 2x – 1.

    If we don't feel like calculating the actual value f(1.1), we can instead plug 1.1 into the tangent line equation and see what comes out:

    2(1.1) – 1 = 2.2 – 1 = 1.2.

    This is a good approximation of f(1.1):

    If we then go and calculate the exact value of the function, we find

    f(1.1) = 1.21.

    This means our approximation was only 0.01 off.

    Why bother? Approximation is supposed to make life easier, so why should we go to all that work of finding the equation of a line and finding the y-value of the line when x = 1.1 instead of calculating f(1.1) and being done with it?

    In that example, we could calculate f(1.1) exactly, but we can't do that for every function. Try doing this with a function like ex or ln(x). Without a calculator, evaluating those functions for most values of x will get pretty hairy.

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