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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 và Meteorology và has a master"s degree in applied mathematics and atmospheric sciences.

Xem thêm: B Clo Vừa Thể Hiện Tính Oxi Hóa Trong Phản Ứng Hóa Học Hay, Chi Tiết

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

What Is a Tangent Line?

Let"s say we"re on a rollercoaster…in space! We"re held khổng lồ the traông xã by the wheels of the cart, but if the cart were to lớn suddenly disconnect from the trachồng, we would soar off the track in a straight line because of the lachồng of gravity. Of course, it would be nice to be rescued from space, so what line would the rescuers follow in order to 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:


A tangent line is a straight line that just barely touches a curve sầu at one point. The idea is that the tangent line & the curve sầu are both going in the same direction at the point of tương tác. If we have a very wavy curve sầu, the tangent line & the curve don"t really seem to 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 more in common than we thought, & they bởi look quite similar!

Mathematical Definition

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

To find the slope of a tangent line, we actually look first to 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)) and (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)) and brought it closer khổng lồ our first point. The closer it gets lớn the first point, the more the secant line starts lớn resemble the tangent line! We bring it closer và closer and closer… which is the mathematical idea of a limit. As h approaches zero, this turns our secant line inkhổng lồ 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 sầu 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 to lớn the curve.

Equation of the Tangent Line

Now that we have sầu the slope of the tangent line, all we would need is a point on the tangent line khổng lồ 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 size of a line:

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

The equations are valid for almost all points on a curve 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 & 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. 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 khổng lồ it at that point. Tangent lines only exist where the function"s curve sầu is smooth.


Let"s find the equation of the line tangent khổng lồ the curve 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 và f"(1) = 2

So the equation of our line becomes:

y = f(1) + f"(1)(x - 1), which simplifies lớn y = 1 + 2(x - 1), which simplifies further to lớn y = 2x - 1

The graph of y = x^2 & y = 2x - 1 confirms visually that we have sầu calculated the tangent line correctly, and we"re done!

Lesson Summary

Let"s take a couple of moments to lớn Đánh Giá what a tangent line is & what its equation is. A tangent line is a straight line that just barely touches a curve sầu at one point. The idea is that the tangent line & the curve sầu are both going in the exact same direction at the point of liên hệ. 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 khổng lồ 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 has a vertical tangent line, the equation reduces to lớn x = a, và if the curve has a break or a sharp corner, then the curve sầu has no tangent line at that point.