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

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

Key Concepts

Graphical Interpretation

Graphically interpreting a function along with its tangent line provides a visual understanding of how the tangent approximates the function locally. This aids in visualizing the concept of instantaneous rate of change and the behavior of the function around the point of tangency.

Power Rule

The power rule is a basic differentiation technique used to find the derivative of functions of the form x^n, where n is any real number. It simplifies the process of determining the slope of a curve for polynomial and related functions.

Derivative

The derivative of a function at a point gives the slope of the tangent line to the curve at that point. It is a fundamental concept in calculus used to measure the rate at which the function's value is changing with respect to changes in the input variable.

Tangent Line

A tangent line to a curve at a specific point is a straight line that touches the curve exactly at that point and has the same instantaneous slope as the curve. It is used as a linear approximation of the function near that point.

Transcript

00:01 In this problem, we're asked to find the equation of the tangent line of the graph of y equals 1 over x squared at the point negative 1 comma 1.

00:07 The first rewrite this as a function that'll make it a little bit easier to deal with.

00:11 So that x equals negative 1.

00:14 So this negative 1 is coming from the x coordinate of the point we're given.

00:18 So remember, the equation for the slope of a tangent line is as a limit, the limit as h goes to 0 of f of x plus h minus f of x all over age.

00:28 In this case, again, we're given our x.

00:31 Our x is just negative 1.

00:34 So, this is the limit as h goes to 0 of, so f of negative 1 plus h.

00:41 Here, we'll write that at it, negative 1 plus h minus f of negative 1 all over h, which is the limit as h goes to 0 of 1 over negative 1 plus h squared minus 1 over negative 1 squared all over h.

01:01 So all we've done is we took the 1 plus h and we plugged it into our x, our equation, from our equation for f of x.

01:09 That's how we got the negative 1 plus h squared in the denominator here.

01:15 All right.

01:15 Now we just need to evaluate this limit.

01:18 Limit as h goes to zero.

01:19 Let's do a little bit of simplification here.

01:22 So this side is 1 over h minus 1.

01:26 Oh, not h plus 1, h minus 1 squared, minus 1, 1, 1 over negative 1, 2.

01:34 Squared as just one all over h.

01:37 Let's combine these fractions in the numerator.

01:40 That'll make our lives a little bit easier here.

01:44 So let's rewrite this one as, oh, sorry, i went to change my color here.