For the function f(x)=4−x2
, find the slopes of the tangent lines at x=−3
, x=0
, and x=1
.


To find the slopes of the tangent lines at specific points for the function \( f(x) = 4 - x^2 \), we need to calculate the derivative \( f'(x) \). The derivative of a function gives the slope of the tangent line at any point on the function.

### Step 1: Differentiate the function
The function is \( f(x) = 4 - x^2 \). Using basic differentiation rules:

\[
f'(x) = \frac{d}{dx}(4 - x^2) = 0 - 2x = -2x
\]

So, the slope of the tangent line at any point \( x \) is \( f'(x) = -2x \).

### Step 2: Calculate the slope at each point

1. **At \( x = -3 \):**
   \[
   f'(-3) = -2(-3) = 6
   \]
   The slope of the tangent line at \( x = -3 \) is \( 6 \).

2. **At \( x = 0 \):**
   \[
   f'(0) = -2(0) = 0
   \]
   The slope of the tangent line at \( x = 0 \) is \( 0 \).

3. **At \( x = 1 \):**
   \[
   f'(1) = -2(1) = -2
   \]
   The slope of the tangent line at \( x = 1 \) is \( -2 \).

### Conclusion:
- The slope at \( x = -3 \) is \( 6 \).
- The slope at \( x = 0 \) is \( 0 \).
- The slope at \( x = 1 \) is \( -2 \).

Would you like further details or have any other questions?

Here are some related questions:
1. What is the general method for finding tangent lines to any function?
2. How do we find the equation of a tangent line at a specific point?
3. How does the second derivative help analyze the curvature of the function?
4. Can you explain the geometric meaning of the derivative?
5. How does a function's derivative relate to its increasing or decreasing behavior?

**Tip:** The slope of a tangent line gives the instantaneous rate of change of the function at a specific point!

For the function f(x) = 4 − x^2, find the slopes of the tangent lines at x = −3, x = 0, and x = 1.

Learn how to find the slopes of the tangent lines to the quadratic function f(x) = 4 − x^2 at x = −3, x = 0, and x = 1. This problem involves differentiating the function to find the derivative and applying it to specific points. Suitable for high school students in Grades 11-12 studying calculus.

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