Final answer:
To find the equation of the tangent line to the curve at the point (1, 3e²), calculate the derivative to get the slope at that point and then use the point-slope form to write the equation. The slope is 6e and the equation is y - 3e² = 6e(x - 1).
Step-by-step explanation:
To find the tangent line to the curve y = 3ex² at the point (1, 3e²), we first need to calculate the derivative of y with respect to x to get the slope of the tangent line at that point. The derivative, denoted as y', is given by using the chain rule:
y = 3ex²
y' = d/dx(3ex²) = 3ex² × 2x = 6xex²
Plugging in x = 1, we find:
y'(1) = 6×1e1² = 6e
This is the slope of the tangent line at x = 1. Now we use the point-slope form of a line to write the equation of the tangent line:
y - y1 = m(x - x1)
Where (x1, y1) is the point (1, 3e²) and m is the slope 6e.
y - 3e² = 6e(x - 1)
That is the equation of the tangent line at the given point.