The Equation of Tangent Lines and Local Linearity

Tangent Lines

We have been using the word "tangent" repeatedly. Now we are going to give an equation for the tangent line to a curve at a point and discuss how to determine whether a function is locally linear by examining its graph.
We're going to create a function, fof x. We'll call it the slope of the tangent line. The derivative at any given point on this function will be the slope of the tangent line drawn to that point and we'll find out what that equation 1s.

In mathematics, the equation of a line is written as y = mx + b. It's better to write this equation in point-slope form: y – y is equal to m, where m is the slope and y is the y-intercept.
The derivative of a function at a specific point, that is, the slope of the tangent line to the curve at that point, can be found by taking the derivative of the function evaluated at the x-coordinate of that point, then subtracting the x-coordinate itself.

The following example demonstrates that the slope of a tangent line can be calculated by evaluating the derivative at a given point. We'll start with y equals x cubed minus 4x. To find the derivative, we need two things: an English expression for the slope and an output value for x.
Since the derivative is found using an equation, we'll use the value 2 for x in our calculations. This yields O as our output value; therefore, doing this math with other values will yield different results.

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[Graph] point-slope
y-f(a)=f^1(a)(x-a)
y=f(a)+f^1(a)(x-a)
y=x^3 -4x write slope of the tangent line
y=2^3 -4(2)=0 x=2
For the tangent line, we have y prime equals 3x squared minus 4. If you evaluate y prime at 2, you get 3 times 2 squared minus 4. That equals 8. Using our equation, that would be y is equal to 0 plus 8 times x minus 2. So that is the equation of the tangent line. You could reduce this to get to y equal mx plus b but that's not recommended because most multiple choice would actuallv have 8 times x minus 2 as an answer choice.
y=3x^2 -4
y^1(2)=3-(2)^2 -4=8 y=0+8(x-2)
And let's visualize this.
[Graph]
Let us consult our graphing calculator. The expression "(8)(-)2" represents the tangent of the curve at that point. A closer look reveals that the derivative of this expression is indeed equal to the original expression within the given accuracy, which confirms our conclusion.
Let's zoom in on the graph of y=x. As x gets smaller, y approaches 2.
For centuries, mathematicians struggled to find an exact solution to the tangent line problem. In the 18th century, they finally came up with the derivative, which approximates nearby values of a function.