Find an equation of the tangent line to the graph of y=f(x) at the given x . f(x)=(1)/(x^3), x=3

Name: Problem 40, Chapter 1: Calculus and Its Applications
Uploaded: 2021-08-31T18:53:06-08:00
Duration: 3 min 43 s
Channel: Carson Merrill
Description: Find an equation of the tangent line to the graph of y=f(x) at the given x . f(x)=(1)/(x^3), x=3

Find an equation of the tangent line to the graph of y=f(x)...

Key Concepts

Derivative

The derivative of a function measures the rate at which the function's value changes at a particular point. It is defined as the limit of the difference quotient and is fundamental in understanding the slope of the tangent to the graph of the function.

Tangent Line

A tangent line to a curve at a given point is a straight line that just 'touches' the curve at that point. It has the same slope as the curve at that point, providing a linear approximation of the function in the neighborhood of the point.

Point-Slope Form

The point-slope form is an equation of a line that uses the slope and a specific point on the line. It is given by y - y? = m(x - x?), where (x?, y?) is the point on the line and m is the slope, making it particularly useful in writing the equation of a tangent line.

Power Rule

The power rule is a basic rule in differentiation used for functions of the form x?. It states that the derivative of x? is n*x^(n-1), which simplifies finding the derivative of power functions.

Transcript

00:01 For the following problem, we're going to find an equation of the tangent line to the graph of y equals f of x at the given x.

00:07 So it's going to be f of x equals x cubed at x equals negative 2.

00:13 So really what we're going to focus on is finding the slope of the graph because once we find the slope, it's fairly easy to find the equation of the tangent line.

00:23 We just have to pick the point where x equals negative 2, which in this case would be negative 8 on the graph.

00:28 So we've already discussed that f of x when it equals x cubed, the derivative is going to be 3x squared or the slope at any point.

00:37 So since we're plugging in a negative 2, we'll see that the slope will be 12.

00:43 And then if we're considering other values, so in this case we now have, we now have x squared and negative 1⁄2.

00:57 So if this is x squared, as we've already discussed as well, the slope will be 2x.

01:02 If we plug in a negative 1 half, we'll end up getting a negative 1.

01:08 Then we'll have 3x plus 1.

01:12 We know in this case that the slope is the same everywhere, so the rate of change is just going to be 3.

01:17 That'll be the slope at any point.

01:20 And then if we have f of x equals 5, the equation of the tangent line, since this is just a straight line, the equation of the tangent line will be it'll be the slope will be zero and then we'll just have five as the value there and that'll be the equation for our tangent line now let's consider f of x equals square root of x we're going to go through each of these types of problems then we see either through the difference quotient or some methods that we learn this is going to be one over root 2x but more specifically if we zoomed in at one 9 we would end up getting 1 over 2 times 1 3rd, so we'd get 3 halves, in fact.

02:10 Then we have a few more...

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