Final answer:
The equation of the tangent line to the curve y = x² that is parallel to y = 3x is y = 3x - 9/2.
Step-by-step explanation:
To find the equation of the tangent line to the curve y = x² that is parallel to the line y = 3x, we need to determine the slope of the tangent line to y = x² at a given point. Since the derivative of x² is 2x, the slope of the tangent line at any point (x, x²) on the curve is 2x. In order for the tangent line to be parallel to y = 3x, its slope must be 3. Therefore, we need to find the value of x for which 2x = 3.
Solving 2x = 3 for x, we get x = 3/2. Plugging this value of x into the original equation y = x², we can find the corresponding y-coordinate. Substituting x = 3/2 into y = (3/2)², we get y = 9/4.
So, the point of tangency is (3/2, 9/4). Using the point-slope form of the equation of a line, y - y₁ = m(x - x₁), where (x₁, y₁) is the point and m is the slope, we will substitute the values to calculate the equation of the tangent line. Plugging in the values, we get y - 9/4 = 3(x - 3/2), which simplifies to y = 3x - 9/2.