find the equation for the normal line of x=1 for f(x)=7-3x-x^2

To find the equation of the normal line to the curve \( f(x) = 7 - 3x - x^2 \) at \( x = 1 \), we will follow these steps:

### Step 1: Find the point on the curve at \( x = 1 \)
Substitute \( x = 1 \) into \( f(x) \) to get the y-coordinate:
\[
f(1) = 7 - 3(1) - (1)^2 = 7 - 3 - 1 = 3
\]
So, the point of interest on the curve is \( (1, 3) \).

### Step 2: Find the slope of the tangent line
The derivative of \( f(x) \) gives the slope of the tangent line at any point:
\[
f'(x) = \frac{d}{dx}(7 - 3x - x^2) = -3 - 2x
\]
At \( x = 1 \), the slope of the tangent line is:
\[
f'(1) = -3 - 2(1) = -5
\]

### Step 3: Find the slope of the normal line
The slope of the normal line is the negative reciprocal of the slope of the tangent line. Since the slope of the tangent is \( -5 \), the slope of the normal line is:
\[
\text{slope of normal} = \frac{1}{5}
\]

### Step 4: Write the equation of the normal line
Using the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) = (1, 3) \), we get:
\[
y - 3 = \frac{1}{5}(x - 1)
\]
Simplifying this:
\[
y = \frac{1}{5}(x - 1) + 3 = \frac{1}{5}x - \frac{1}{5} + 3 = \frac{1}{5}x + \frac{14}{5}
\]

Thus, the equation of the normal line is:
\[
y = \frac{1}{5}x + \frac{14}{5}
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you find the equation of the tangent line to \( f(x) \) at \( x = 1 \)?
2. What is the geometric meaning of a normal line?
3. How do you interpret the negative reciprocal when finding the slope of the normal line?
4. Can we find the normal line equation at any other point on the curve?
5. How would the process change if the function were a cubic instead of a quadratic?

**Tip:** The slope of the normal line is always the negative reciprocal of the tangent line's slope, making them perpendicular at the point of contact.

Find the equation for the normal line of x=1 for f(x)=7-3x-x^2

Learn how to find the equation of the normal line to the curve f(x) = 7 - 3x - x^2 at x = 1. This calculus problem includes finding the slope of the tangent line, using derivatives, and calculating the equation of the normal line. Suitable for students in Grades 11-12 or college-level calculus.

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