Step 1: Understanding the Concept:
To find the equation of a tangent line to a curve at a given point, we need two things: the point itself (which is given) and the slope of the curve at that point. The slope is found by evaluating the derivative of the curve's function at the given point.
Step 2: Key Formula or Approach:
1. Find the derivative of the function, $\frac{dy}{dx}$.
2. Evaluate the derivative at the given point $(x_1, y_1)$ to get the slope, $m$.
3. Use the point-slope form of a line to find the equation of the tangent: $y - y_1 = m(x - x_1)$.
Step 3: Detailed Explanation:
The curve is given by the function $y = x^3$.
The point is $(x_1, y_1) = (1, 1)$.
First, find the derivative of the function:
\[ \frac{dy}{dx} = \frac{d}{dx}(x^3) = 3x^2 \]
Next, evaluate the derivative at $x=1$ to find the slope of the tangent at that point:
\[ m = \left. \frac{dy}{dx} \right|_{x=1} = 3(1)^2 = 3 \]
Now we have the slope $m=3$ and the point $(1, 1)$. Use the point-slope form:
\[ y - y_1 = m(x - x_1) \]
\[ y - 1 = 3(x - 1) \]
Expand and rearrange the equation into the general form $Ax + By + C = 0$:
\[ y - 1 = 3x - 3 \]
\[ 0 = 3x - y - 3 + 1 \]
\[ 3x - y - 2 = 0 \]
Step 4: Final Answer:
The equation of the tangent to the curve $y=x^3$ at the point (1, 1) is $3x - y - 2 = 0$. Therefore, option (C) is correct.