The Slope of the Curve, Derivatives

In summary, the conversation discussed finding the derivative of f(x)=x^2, finding the slope of the curve at x=3, and writing an equation for the tangent line. The derivative of f(x)=x^2 is 2x and the slope can be found by using the derivative at the given point. The equation of the tangent line is y-y_0=f'(x_0)(x-x_0) where x_0=3 and y_0=9.
  • #1
MorganJ
32
0
1. Find the derivative f' of f(x)=x^2. Then find the slope of the curve y=f(x) at x=3 and write an equation for the tangent line.


2. I know that the derivative of x^2 is 2x and I'm guessing that y=x^2, therfore, (3)^2 = 9 which is the value of y. So now the point is (3,9). How do I find the slope? Do I do rise over run which is 9/3 which is 3?

3. I already know the tangent line equation but the slope is getting me confused. I know slope is y=mx+b But what if this is with derivatives?
 
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  • #2
3. I already know the tangent line equation but the slope is getting me confused. I know slope is y=mx+b But what if this is with derivatives?

No. The equation of the line is y = mx + b. m is the slope and b is the y intercept.

To find the slope of the tangent line, note that the derivative of a function at a point gives the slope of the tangent line. Then you have one point (3,9) and the slope, so you should be able to find the equation of the line.
 
  • #3
The equation of the tangent line is:

[tex]y-y_0=f'(x_0)(x-x_0)[/tex]

In this case x0=3 and y0=9

Regards.
 

1. What is the slope of a curve?

The slope of a curve is a measure of its steepness at a particular point. It represents how much the curve is changing at that point.

2. How is the slope of a curve calculated?

The slope of a curve can be calculated by finding the derivative of the curve at a specific point. This involves finding the rate of change of the curve at that point.

3. What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is a fundamental tool in calculus and is used to find the slope of curves, as well as many other applications.

4. Why is the slope of a curve important?

The slope of a curve is important because it provides information about the behavior of the curve. It can tell us if the curve is increasing or decreasing, how steep it is at a specific point, and can also be used to find maximum and minimum points on the curve.

5. What are some real-world applications of derivatives and the slope of a curve?

Derivatives and the slope of a curve have many real-world applications, including in physics, engineering, economics, and finance. They can be used to analyze motion, optimize systems, and make predictions about future trends.

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