Final answer:
To find the equation for the line that is tangent to the curve y = 3x³ - 3x at the point (1,0), we need to find the slope of the curve at that point. The slope of the curve at a specific point can be found by taking the derivative of the equation of the curve and evaluating it at that point.
Step-by-step explanation:
To find the equation for the line that is tangent to the curve y = 3x³ - 3x at the point (1,0), we need to find the slope of the curve at that point. The slope of the curve at a specific point can be found by taking the derivative of the equation of the curve and evaluating it at that point. Let's do that:
First, find the derivative of y = 3x³ - 3x:
dy/dx = 9x² - 3
Next, evaluate the derivative at the point (1,0) to find the slope of the curve at that point:
dy/dx = 9(1)² - 3 = 9 - 3 = 6
So, the slope of the curve at the point (1,0) is 6. Since the line that is tangent to the curve at a specific point has the same slope as the curve at that point, the equation for the line that is tangent to the curve at (1,0) is given by y = 6x + c, where c is a constant. Now, we can plug in the coordinates of the point (1,0) into the equation to find the value of c:
0 = 6(1) + c
c = -6
Therefore, the equation for the line that is tangent to the curve y = 3x³ - 3x at the point (1,0) is y = 6x - 6.