Equations of tangent lines by definition ( 2 ) a. Use...

Equations of tangent lines by definition ( 2 )
a. Use definition (2) (p. 129 ) to find the slope of the line tangent to the graph of $f$ at $P$.
b. Determine an equation of the tangent line at $P$.
$$f(x)=\frac{x}{x+1} ; P(-2,2)$$

Key Concepts

Tangent Line

A tangent line to a function at a point is a straight line that touches the function only at that point (in an idealized mathematical sense) and has the same instantaneous rate of change as the function at that point. It serves as the best linear approximation to the function near the given point.

Difference Quotient

The difference quotient is an expression of the form (f(x) - f(a))/(x - a), which is used in the definition of the derivative. By taking the limit of this quotient as x approaches a, one obtains the derivative, which is essential for finding the slope of the tangent line.

Definition of Derivative

The derivative of a function at a given point is defined as the limit of the difference quotient as the interval approaches zero. This limit represents the instantaneous rate of change of the function and is fundamental in determining the slope of the tangent line to the function at that specific point.

Point-Slope Form

The point-slope form is a method for writing the equation of a straight line given its slope and a point on the line. It is expressed as y - y1 = m(x - x1), and is used to formulate the equation of the tangent line once the slope (derived from the derivative) and the point of tangency are known.

Transcript

00:01 Hello, today we're going to take a given function and a specified point.

00:04 We're going to use the alternative definition of for finding the slope.

00:08 We're going to find the slope of the line at point p as well to determine the equation of tangent line at point p.

00:14 So right off the bat, we're going to specify our a and declare it such that it equals to negative 2 and that it also equals to our x not because our point is our x not.

00:27 Why not? so going to the alternative definition, we know that we have to plug in a plus h for x and you might be looking at this function going well which x do i plug it in for and the answer is both so what we're going to do is we're going to have our negative two plus h all over negative two plus h plus one minus our negative two all over negative two plus one divided by h with the limit as h approaches 0.

01:13 So making this a little nicer to look at, i'll just isolate the numerator for now, and we're going to get negative 2 plus h all over h minus 1.

01:31 And then this is going to be subtracted from our negative 2 all over negative 1, which then just simply goes to so minus 2.

01:49 Now, we're going to to look at it and see that it has a we're going to need a common denominator in that numerator.

01:56 So let's use our first trick.

01:59 Do h minus 1 over h minus 1.

02:03 And then this gives us negative 2 plus h all over h minus 1 minus 2h minus 2 all over h minus 1.

02:16 So we can combine the numerator in essence so that we get it over h minus 1 and we add 2h plus h.

02:23 So that's 3h, oh sorry, so we have negative, so we need to make that negative and positive.

02:33 So negative 2h plus h, that be a negative h, and then we have 2 minus 2, so that's 0.

02:42 So we have negative h over h minus 1...

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