Tangent line là gì

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Try it now 0:04 What is a Tangent Line? 0:55 Mathematical Definition 2:32 Equation of a Tangent Line 3:39 An Example 4:29 Lesson Summary

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Instructor: Jasmine Cetrone

Jasmine has taught college Mathematics and Meteorology and has a master”s degree in applied mathematics & atmospheric sciences.

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In this lesson, we explore the idea & definition of a tangent line both visually & algebraically. After learning how khổng lồ calculate a tangent line to lớn a curve sầu, you will find a short quiz to lớn kiểm tra your knowledge.

What Is a Tangent Line?

Let”s say we”re on a rollercoaster…in space! We”re held khổng lồ the traông chồng by the wheels of the cart, but if the cart were to lớn suddenly disconnect from the trachồng, we would soar off the traông chồng in a straight line because of the lachồng of gravity. Of course, it would be nice lớn be rescued from space, so what line would the rescuers follow in order lớn find us? As it turns out, they would find our cart along a very special line called the tangent line, like the one you are looking at now:

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A tangent line is a straight line that just barely touches a curve at one point. The idea is that the tangent line & the curve are both going in the same direction at the point of liên hệ. If we have a very wavy curve, the tangent line and the curve sầu don”t really seem khổng lồ have much in common because the tangent line is perfectly straight. However, as we zoom in closer và closer khổng lồ the point where the tangent line touches the curve, we can see that they have sầu more in comtháng than we thought, & they vị look quite similar!

Mathematical Definition

Now that we have sầu a conceptual idea of what a tangent line is, we need to lớn understvà how to lớn define one mathematically. There are two important elements to lớn finding an equation that defines a tangent line: its slope and its point of tương tác with a curve sầu. A line”s slope is its steepness, or rate of change both horizontally và vertically as it travels away from the origin.

To find the slope of a tangent line, we actually look first lớn an equation”s secant line, or a line that connects two points on a curve. To find the equation of a line, we need the slope of that line. With a tangent line, that can be tricky, but with a secant line, because we have two points, it”s no problem!

The slope of this secant line, which passes through the points (a , f(a)) và (a + h , f(a + h)) shown in the formula below. You might recognize this formula from precalculus; it”s called the difference quotient:

slope of secant line = / h

So, how does this help us with the tangent line? Well, imagine that we took that second point (a + h , f(a + h)) & brought it closer khổng lồ our first point. The closer it gets to the first point, the more the secant line starts lớn resemble the tangent line! We bring it closer và closer và closer… which is the mathematical idea of a limit. As h approaches zero, this turns our secant line inlớn our tangent line, and now we have a formula for the slope of our tangent line! It is the limit of the difference quotient as h approaches zero.

Assuming you are familiar with the basics of calculus, you will recognize this as the definition of the derivative of our function f(x) at x = a, denoted in prime notation as f “(a). The derivative sầu of a function is the instantaneous rate of change of the function & the slope of the line tangent lớn the curve.

Equation of the Tangent Line

Now that we have the slope of the tangent line, all we would need is a point on the tangent line lớn complete the equation of our line. That”s easy, because we know that our tangent line went through the point (a , f(a)). Let”s now build the equation of our line using point-slope form of a line:

y – y1 = m(x – x1), where (x1, y1) is a known point on the line, and m is the slope of the line

The equations are valid for almost all points on a curve sầu y = f(x):

y – f(a) = f”(a)(x – a) y = f(a) + f”(a)(x – a)

There are some exceptions:

In the special case where a tangent line is vertical, its slope would be undefined and we wouldn”t be able to use the equation from before. In this case, we would use the equation of a vertical line that goes through the point (a , f(a)), which would simply be the equation x = a.

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If the function is discontinuous where x = a (as in any holes, breaks, or jumps in the graph), the function doesn”t have a tangent line at that point, or finally If the function has a sharp corner or edge at x = a, the function does not have a line tangent lớn it at that point. Tangent lines only exist where the function”s curve sầu is smooth.

Example

Let”s find the equation of the line tangent lớn the curve sầu of the function f(x) = x^2 when x = 1. We”re already given the x-value of the point (x = 1), but lớn determine the corresponding y-value, let”s plug in x = 1: f(1) = (1)^2 = 1. So, we know the point is (1,1).

Next let”s find the slope of the line, which would be the derivative at x = 1:

f”(x) = 2x & f”(1) = 2

So the equation of our line becomes:

y = f(1) + f”(1)(x – 1), which simplifies khổng lồ y = 1 + 2(x – 1), which simplifies further lớn y = 2x – 1

The graph of y = x^2 and y = 2x – 1 confirms visually that we have calculated the tangent line correctly, và we”re done!

Lesson Summary

Let”s take a couple of moments to đánh giá what a tangent line is and what its equation is. A tangent line is a straight line that just barely touches a curve at one point. The idea is that the tangent line & the curve are both going in the exact same direction at the point of contact. The slope, or the steepness, of the tangent line is determined by the function”s instantaneous rate of change at that point. The slope of the line is found by creating a derivative function based on a secant line”s approach lớn the tangent line. A secant line is a line that connects two points on a curve sầu.

For smooth, continuous curves with non-vertical slopes, we can calculate the tangent line using the formula:

y = f(a) + f “(a)(x – a)

If the curve sầu has a vertical tangent line, the equation reduces khổng lồ x = a, and if the curve has a break or a sharp corner, then the curve sầu has no tangent line at that point.