The maximum slope of the curve defined by the equation y = x²e^(-3x) occurs at x = 1/3. The slope is calculated using the formula slope = (y2 - y1) / (x2 - x1), with the origin (0,0) as one point. The confusion arises from misinterpreting the graph, where the local minimum at x = 1/3 may lead to incorrect conclusions about the slope's behavior. Properly analyzing the derivative of the function confirms the maximum slope at this point.
PREREQUISITESStudents studying calculus, particularly those focusing on derivatives and optimization problems, as well as educators looking for examples of slope analysis in exponential functions.
zJakeAdam said:I mean, the graph just seems to go continuous and so wouldn't that mean their is no maximum slope?
Undoubtedly0 said:Greetings and welcome!
Interesting, are you sure you graphed it correctly? Consider the formula for the slope of a line between two points:
slope = \frac{y_2 - y_1}{x_2 - x_1}.
Since we know that the line in question goes through the origin, we can say (x1,y1) = (0,0). Now consider some point on the curve, (x2, y2) = (x, x2e-3x). This should make sense because when we are x units to the right, the curve is x2e-3x units up. Try plugging these values into the slope formula and then maximizing the slope.