41. Among all triangles in the first quadrant formed by the x-axis, the y-axis, and tangent lines to the graph of y=3x-x^2, what is the smallest possible area?

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Identify the function y = 3x - x^2 and note that the problem involves finding the area of triangles formed by tangent lines to this curve in the first quadrant.

Find the derivative of the function y = 3x - x^2 to determine the slope of the tangent line at any point (a, 3a - a^2). The derivative is dy/dx = 3 - 2x.

The equation of the tangent line at point (a, 3a - a^2) is y - (3a - a^2) = (3 - 2a)(x - a). Simplify this to find the y-intercept and x-intercept of the tangent line.

The x-intercept occurs when y = 0, and the y-intercept occurs when x = 0. Use these intercepts to determine the base and height of the triangle formed by the tangent line, the x-axis, and the y-axis.

Express the area of the triangle as a function of a, A(a) = (1/2) * base * height. Differentiate A(a) with respect to a, set the derivative equal to zero, and solve for a to find the value that minimizes the area.

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주요 개념

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Tangent Line to a Curve

A tangent line to a curve at a given point is a straight line that just touches the curve at that point. It has the same slope as the curve at the point of tangency. For the function y = 3x - x^2, the slope of the tangent line at any point (a, 3a - a^2) is given by the derivative, which is 3 - 2a.

Area of a Triangle

The area of a triangle can be calculated using the formula: Area = 0.5 * base * height. In this context, the triangle is formed by the x-axis, y-axis, and the tangent line. The base and height are determined by the x and y intercepts of the tangent line, which can be found using the point-slope form of the line equation.

Optimization in Calculus

Optimization involves finding the maximum or minimum value of a function. In this problem, we need to minimize the area of the triangle. This requires setting up an equation for the area in terms of a single variable, differentiating it, and finding critical points to determine the minimum area using the first or second derivative test.