Equal area property for parabolas Let f ( x ) = ax 2 + bx + c be an arbitrary quadratic function and choose two points x = p and x= q . Let L 1 be the line tangent to the graph of f at the point ( p , f ( p )) and let L 2 be the line tangent to the graph at the point ( q , f ( q )). Let x = s be the vertical line through the intersection point of L 1 and L 2 . Finally, let R 1 , be the region bounded by y = f ( x ), L 1 , and the vertical line x = s , and let R 2 be the region bounded by y = f ( x ), L 2 , and the vertical line x = s . Prove that the area of R 1 equals the area of R 2 .
Equal area property for parabolas Let f ( x ) = ax 2 + bx + c be an arbitrary quadratic function and choose two points x = p and x= q . Let L 1 be the line tangent to the graph of f at the point ( p , f ( p )) and let L 2 be the line tangent to the graph at the point ( q , f ( q )). Let x = s be the vertical line through the intersection point of L 1 and L 2 . Finally, let R 1 , be the region bounded by y = f ( x ), L 1 , and the vertical line x = s , and let R 2 be the region bounded by y = f ( x ), L 2 , and the vertical line x = s . Prove that the area of R 1 equals the area of R 2 .
Solution Summary: The author explains that the area of R_1 is equal to the amount of the value of a quadratic function.
Equal area property for parabolas Let f(x) = ax2 + bx + c be an arbitrary quadratic function and choose two points x = p and x= q. Let L1 be the line tangent to the graph of f at the point (p, f(p)) and let L2 be the line tangent to the graph at the point (q, f(q)). Let x = s be the vertical line through the intersection point of L1 and L2. Finally, let R1, be the region bounded by y =f(x), L1, and the vertical line x = s, and let R2 be the region bounded by y = f(x), L2, and the vertical line x = s. Prove that the area of R1 equals the area of R2.
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