Find the equation of the tangent to y = x^2  at the point where x = 3

To find the equation of the tangent to the curve \( y = x^2 \) at the point where \( x = 3 \), we can follow these steps:

### 1. Find the derivative of the function:
The derivative of \( y = x^2 \) with respect to \( x \) gives us the slope of the tangent line at any point on the curve.

\[
\frac{dy}{dx} = 2x
\]

### 2. Determine the slope at \( x = 3 \):
Substitute \( x = 3 \) into the derivative to find the slope of the tangent at that point.

\[
\text{slope} = 2(3) = 6
\]

### 3. Find the coordinates of the point on the curve:
When \( x = 3 \), we can find the \( y \)-coordinate by substituting \( x = 3 \) into the original equation \( y = x^2 \).

\[
y = 3^2 = 9
\]

So, the point of tangency is \( (3, 9) \).

### 4. Use the point-slope form of the equation of a line:
The point-slope form of the equation of a tangent line is given by:

\[
y - y_1 = m(x - x_1)
\]

where \( m \) is the slope and \( (x_1, y_1) \) is the point of tangency. Substituting \( m = 6 \), \( x_1 = 3 \), and \( y_1 = 9 \), we get:

\[
y - 9 = 6(x - 3)
\]

### 5. Simplify the equation:
\[
y - 9 = 6x - 18
\]
\[
y = 6x - 9
\]

Thus, the equation of the tangent to the curve \( y = x^2 \) at \( x = 3 \) is:

\[
y = 6x - 9
\]

Would you like more details, or do you have any questions?

Here are 5 related questions to expand on this concept:
1. How do you find the tangent line at any general point \( x = a \) on \( y = x^2 \)?
2. What is the geometric significance of the slope of the tangent line?
3. How would the process change if the curve was \( y = x^3 \)?
4. How can we determine the normal line to the curve at \( x = 3 \)?
5. Can you find the tangent to a circle using a similar approach?

**Tip:** When finding the tangent line to a curve, always differentiate the function first to get the slope formula.

Find the equation of the tangent to y = x^2 at the point where x = 3

Discover how to find the equation of the tangent to the curve y = x^2 at the point where x = 3 using derivatives and the point-slope form. This step-by-step guide covers key calculus concepts and is ideal for students in Grades 10-12.

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