Find an equation for the tangent to the curve at the given point. Then sketch the curve and tang

step 1 To find the equation of a tangent line, we need two things. You need a point, which you're given

Find an equation for the tangent to the curve at the given...

Key Concepts

Graphing Functions and Tangents

Graphing involves plotting the function as well as its tangent line to visually examine their relationship at the point of tangency. This process helps illustrate how well the tangent line approximates the curve near that point and provides insight into the function's local linear behavior.

Differentiation of Power Functions

Differentiating power functions involves applying the power rule, which states that the derivative of x^n is n*x^(n-1). This rule is essential for calculating the derivative of functions that are written in terms of exponents, including reciprocals and powers.

Derivative

The derivative represents the instantaneous rate of change of a function and provides the slope of the tangent line to the curve at any given point. It is fundamental in calculus for understanding how a function changes and for determining the behavior of the function locally.

Tangent Line

A tangent line is a straight line that touches a curve at a single point and has the same slope as the curve at that point. It is used to approximate the function near the point of tangency and is calculated using the derivative of the function.

Point-Slope Form

The point-slope form is a method for writing the equation of a line when a point on the line and its slope are known. The form is expressed as y - y? = m(x - x?), where (x?, y?) is the given point and m is the slope.

Transcript

00:01 To find the equation of a tangent line, we need two things.

00:04 You need a point, which you're given.

00:06 And you need a slope, which we're going to use our limit formula.

00:18 Now, our x value comes from the order of pair, so our x value is negative 2.

00:22 So we'd have f of negative 2 plus h, which i'm going to write as h minus 2, minus f of negative 2, all over h.

00:36 So for f of h minus 2, it'll be 1 over h minus 2 cubed, minus, minus f of negative 2, which we get from the ordered pair, because if you plug in negative 2, you get negative 1 8th.

01:00 And you see i have minus a negative, so i'm just going to make that a plus, and that's over h.

01:11 And then let's see, we're going to need a common denominator, which can be eight times the h minus 2 cubed.

01:28 So in our blue fraction, we would have to multiply that fraction by 8 over 8, giving us 8.

01:37 And in the other fraction, we have to multiply by the h minus 2 cubed, top and bottom, giving us h minus 2 cubed, top and bottom, giving us h minus 2 cubed.

01:49 So writing it as one function, we have 8 plus h minus 2 cubed over the common denominator, which the two denominators are going to mush together if you want to call it that.

02:07 8h times h minus 2 cubed.

02:11 Now we're going to have to cube that h minus 2.

02:15 So i'm going to foil two of them first, which would give h squared minus 4h plus 4.

02:30 And then we're going to distribute, whoops, the h into all three, and then we're going to distribute the negative 2 into all three.

02:56 Okay, so let's do some canceling here.

02:59 Those will cancel.

03:03 And that's it.

03:06 So we have h -cubed minus 6h squared plus 12h over 8h times h minus 2 cubed.

03:20 And we're going to factor an h out of the numerator.

03:26 So you can cancel with the h there in the denominator.

03:32 And now we can do the h approaches 0.

03:35 So if we plug in a 0 for h in a numerator, we get 12.