How do you find the slope of a curve at a point?

Question

Charles Arocha · Accepted Answer

If the curve is the graph of a function #f(x)#, then the slope of the curve at the point #x=a# can be found by #f'(a)#.

Luke Alvarez · Answer

undefined To find the slope of a curve at a point, you can use calculus. Specifically, you would calculate the derivative of the function representing the curve with respect to the variable in question (typically denoted as $ f'(x) $ or $ \frac{dy}{dx} $). Once you have the derivative, you evaluate it at the given point to find the slope of the curve at that point. The derivative gives you the rate of change of the function at any given point, which corresponds to the slope of the curve at that point.

HIX Tutor · Answer

undefined Great question! Let's break it down step by step:

1. **What is a Curve?**  
   A curve is like a line that is not straight. It can bend and change direction.

2. **What is Slope?**  
   Slope tells us how steep a line is. For curves, we want to find the slope at a certain point.

3. **Using Tangent Line:**  
   To find the slope at one point on the curve, think of a straight line that just touches the curve at that point. This line is called a tangent line.

4. **Find the Function:**  
   Do you have a math function for the curve? It might look like $y = f(x)$.

5. **Choose the Point:**  
   Which point on the curve do you want to find the slope for? You need the $x$ value of that point.

6. **Use Derivative:**  
   The slope of the tangent line at that point is found using something called a derivative. A derivative tells us how fast things are changing.

7. **Calculate the Derivative:**  
   If you can, find the derivative of the function. This gives you a new function.

8. **Plug in the $x$ Value:**  
   Take that $x$ value from your chosen point and put it into the derivative function.

9. **Get the Slope:**  
   The result is the slope of the curve at that point!

Does this make sense?