Jasmine has taught college Mathematics and Meteorology and has a master's degree in applied mathematics and atmospheric sciences.
In this lesson, we explore the idea and definition of a tangent line both visually and algebraically. After learning how to calculate a tangent line to a curve, you will find a short quiz to test your knowledge.
What Is a Tangent Line?
Let's say we're on a rollercoaster…in space! We're held to the track by the wheels of the cart, but if the cart were to suddenly disconnect from the track, we would soar off the track in a straight line because of the lack of gravity. Of course, it would be nice to be rescued from space, so what line would the rescuers follow in order to 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 at one point. The idea is that the tangent line and the curve are both going in the same direction at the point of contact. If we have a very wavy curve, the tangent line and the curve don't really seem to have much in common because the tangent line is perfectly straight. However, as we zoom in closer and closer to the point where the tangent line touches the curve, we can see that they have more in common than we thought, and they do look quite similar!
Now that we have a conceptual idea of what a tangent line is, we need to understand how to define one mathematically. There are two important elements to finding an equation that defines a tangent line: its slope and its point of contact with a curve. A line's slope is its steepness, or rate of change both horizontally and 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 = [f(x + h) - f(x)] / 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 to our first point. The closer it gets to the first point, the more the secant line starts to resemble the tangent line! We bring it closer and closer and closer… which is the mathematical idea of a limit. As h approaches zero, this turns our secant line into 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 of a function is the instantaneous rate of change of the function and the slope of the line tangent to 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 to 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 y = f(x):
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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.
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 to it at that point. Tangent lines only exist where the function's curve is smooth.
Let's find the equation of the line tangent to 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 to 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 and f'(1) = 2
So the equation of our line becomes:
y = f(1) + f'(1)(x - 1), which simplifies to
y = 1 + 2(x - 1), which simplifies further to
y = 2x - 1
The graph of y = x^2 and y = 2x - 1 confirms visually that we have calculated the tangent line correctly, and we're done!
Let's take a couple of moments to review 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 and 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 to the tangent line. A secant line is a line that connects two points on a curve.
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 x = a, and if the curve has a break or a sharp corner, then the curve has no tangent line at that point.
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