Final answer:
To find the equation of the tangent line to the curve y = x² that is parallel to y = 3x, find the point of intersection of the curve and the line. The point of intersection is (3/2, 9/4). Then, use the point-slope form of the equation of a line to find the equation of the tangent line, which is y = 3x - 9/4.
Step-by-step explanation:
To find an equation of the tangent line to the curve y = x² that is parallel to the line y = 3x, we need to find the point where the tangent line intersects the curve. Since we want the tangent line to be parallel to y = 3x, the slopes of the tangent line and the curve should be equal.
The slope of the curve y = x² is given by dy/dx = 2x. So, to find the x-coordinate of the point of intersection, we set the slopes equal: 2x = 3. Solving for x, we get x = 3/2.
Substituting x = 3/2 into the equation of the curve y = x², we find y = (3/2)² = 9/4.
Therefore, the point of intersection is (3/2, 9/4). Now we can use the point-slope form of the equation of a line to find the equation of the tangent line. Since the slope is parallel to y = 3x, the slope of the tangent line is also 3.
Using the point-slope form, we have y - (9/4) = 3(x - 3/2).
Simplifying this equation, we have y - 9/4 = 3x - 9/2.
Adding 9/4 to both sides, we get y = 3x - 9/2 + 9/4.
Combining like terms, the equation of the tangent line is y = 3x - 9/4 + 9/4.
Finally, simplifying, we have y = 3x - 9/4, which is the equation of the tangent line.