Equations of tangent lines by definition (1) a. Use...

Equations of tangent lines by definition (1) a. Use definition ( 1 ) ( $p .$ I28 ) to find the slope of the line tangent to the graph of $f$ at $P$. b. Determine an equation of the tangent line at $P$. c. Plot the graph of $f$ and the tangent line at $P$. $$f(x)=\frac{4}{x^{2}} ; P(-1,4)$$

Equations of tangent lines by definition (1)
a. Use definition ( 1 ) ( $p .$ I28 ) to find the slope of the line tangent to the graph of $f$ at $P$.
b. Determine an equation of the tangent line at $P$.
c. Plot the graph of $f$ and the tangent line at $P$.
$$f(x)=\frac{4}{x^{2}} ; P(-1,4)$$

Key Concepts

Limit Definition of the Derivative

This concept involves finding the derivative of a function at a point using the limit of the difference quotient. It involves computing the limit as the interval approaches zero, which gives the instantaneous rate of change or the slope of the tangent line at that point.

Tangent Line

The tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve. Understanding the tangent line is essential for approximating the behavior of functions locally.

Point-Slope Form of a Line

This is a method for writing the equation of a line when a point on the line and the slope are known. It is particularly useful in determining the equation of the tangent line once the slope is computed.

Graphical Interpretation of Functions

Graphing a function along with its tangent line aids in visualizing the behavior of the function at a specific point. It reinforces the connection between the algebraic derivation of the tangent and its geometric significance.

Transcript

00:01 Hello, so today we're going to take a given function and a specified point p, and we're going to find the slope of the tangent line of the function at p, determine the equation of the tangent line at p, and then we're going to plot both the function and the tangent line.

00:13 So first we're going to take the original function, f of x, and we're going to take the derivative of it to get the slope to the tangent line at the specified point.

00:25 So let's first put this into a little more simplified form.

00:29 That's easier to take the derivative of.

00:32 So, 4 times x to the negative second power.

00:36 The reason we can do this is because 1 over x squared is the same as writing x to the negative second power.

00:43 So let's take the derivative of that.

00:47 So the derivative of this is 4 times negative 2 times x to the negative 3 power, which equals negative 8 over x cubed.

00:59 And to get the slope of the function at the specified point p, what we have to do is take the x value and plug it in.

01:11 So we get f prime of negative 1 equals negative 8 all over negative 1 to the third power.

01:22 So when you take something to the third power, it retains its negative.

01:27 So negative 8 all over negative 1 equals.

01:30 So that's the slope of the tangent line at our given point.

01:39 Next, let's determine the equation of tangent line.

01:42 To do this, we need our equation of line, which is y minus y -not equals the slope times x minus x -not.

01:55 So it's all for this.

01:56 We take our given point and our slope value that we just calculated and we plug it into the equation.

02:02 So this gives us y minus 4 equals 8 times x minus a negative 1.

02:11 And that's basically the same as saying x plus 1.

02:15 So this simplifies down to 8x plus 1 equals y minus 4.

02:22 Adding 4 to both sides, or sorry, that's plus 8 apologies.

02:27 So adding for to both sides, we get y equals 8x plus 12.

02:34 And that's the equation of our line at the given point.

02:38 So let's answer to part b.

02:41 Next, we're going to tabulate our values to make it easier plot.

02:45 So we have our domain of specified x values.

02:49 We have our original function, and we have our tangent lines of the function at the given point.

02:58 Let's make a table.

03:00 So let's use the values 0, 1, 2, negative 1, and negative 2.

03:08 So plugging these in, we can solve such that f of 0 equals 4 over 0 squared for our original function, and that gives us undefined...

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