AP Calculus AB Exam Question:

Find the equation of the tangent line to the curve defined by the equation $x^2 - 2xy + y^2 = 4$ at the point $(1, 1)$.

Answer:

Step 1: Start by differentiating both sides of the equation $x^2 - 2xy + y^2 = 4$ implicitly with respect to $x$. Using the chain rule, we differentiate each term individually:

Simplifying each term gives:

Step 2: Group like terms:

Step 3: Factor out the common factor of $\frac{dy}{dx}$:

Simplifying further:

Step 4: Divide both sides by $2x$ to solve for $\frac{dy}{dx}$:

Simplifying:

Step 5: Now that we have the derivative $\frac{dy}{dx}$, we can find the slope of the tangent line.

At the point $(1, 1)$, the slope of the tangent line is:

Step 6: Using the point-slope form of a line $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the given point and $m$ is the slope we just found, substitute the values $(x_1, y_1) = (1, 1)$ and $m = -1$:

Step 7: Simplify and rewrite the equation in slope-intercept form $y = mx + b$:

Therefore, the equation of the tangent line to the curve $x^2 - 2xy + y^2 = 4$ at the point $(1, 1)$ is $y = -x + 2$.