Sketch the graph of y=1 /(x+1) and then find the equation of the tangent line at (1, (1)/(2)) (s

Sketch the graph of y=1 /(x+1) and then find the equation of...

Key Concepts

Rational Functions

Rational functions are functions that can be expressed as the ratio of two polynomials. They often have unique characteristics such as asymptotes and discontinuities that distinguish their graphs from those of polynomial functions. Understanding rational functions involves analyzing their behavior, domains, and critical features such as intercepts and points of discontinuity.

Graph Transformations

Graph transformations involve shifting, stretching, or reflecting the basic graph of a function to obtain the graph of a more complex function. For rational functions, these transformations help in determining how the graph of a simple function, like 1/x, is modified when additional constants are introduced in the numerator or denominator.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. In the context of rational functions, vertical asymptotes typically occur where the denominator is zero, and horizontal asymptotes indicate the end behavior of the function as the independent variable tends to positive or negative infinity. Recognizing these asymptotes is crucial for accurately sketching the graph of the function.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change or slope of the function at any given point. The derivative is essential for determining the slope of the tangent line to the graph of a function, providing insights into the local behavior and dynamics of the function.

Tangent Lines

Tangent lines to a function at a particular point touch the graph at that point and have the same slope as the function does there. Finding the equation of a tangent line involves calculating the derivative to obtain the slope and then using the point-slope form of the line equation. This concept is vital in approximations and analyzing instantaneous rates of change.

Transcript

00:01 To find the equation of the tangent line, we first need to find the slope of the tangent line.

00:06 We can roughly estimate the slope by drawing in a tangent line, but it's not really clear what negative value that slope would have.

00:15 So to find the slope of the tangent line, we'll use our limit definition, which says take the limit as h approach at zero of our function evaluated at c plus h minus our function evaluated at c, all divided by h.

00:30 And note that c is the value that we're interested in, the x value.

00:36 And we already know f of c because we're told that we get out one half.

00:41 So let's set this up for the function given and for the target value c that we're given.

00:48 So what is the function evaluated at c plus h? well, that would be 1 over c plus h plus 1.

00:56 But since c is 1 itself, we might as well write 1 plus h.

01:01 Plus 1, or even simpler 2 plus h.

01:06 Now we're going to subtract f at c.

01:09 That means plug the value 1 into the function, and we already know that's 1 half, all divided by h.

01:16 We can't quite make h go to 0, because we'll get 0 divided by 0, which really doesn't help.

01:22 So instead, let's get a common denominator among these two terms.

01:26 So the first fraction will give numerator and denominator a 2.

01:31 And the second fraction will give numerator and denominator a 2 plus h.

01:35 And we still have all divided by h.

01:41 So now that we have a common denominator of 2, 2 plus h, you can subtract the numerators, and we still have all divided by h.

01:53 Note i'm writing a limit every time, even though we haven't calculated yet, that's important.

01:58 So let's look at our new numerator...

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