Arctangent family
In the diagram above, the black curves show some members of the family The red curve intersects all members of this family perpendicularly.
If the red curve is described by the function , and , how much is ?
The answer is 78.
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1 solution
The condition of perpendicularity can be written as f c ′ ( x ) ⋅ g ′ ( x ) = − 1 , wherever f c ( x ) = g ( x ) . The derivative of the arc tangent function is known: 1 + x 2 1 ⋅ g ′ ( x ) = − 1 ∴ g ′ ( x ) = − x 2 − 1 . Integrating, we get g ( x ) = − 3 1 x 3 − x + k , but since g ( 0 ) = 0 the integration constant should be k = 0 .
Finally, g ( − 6 ) = − 3 1 ( − 6 ) 3 − ( − 6 ) = 3 1 ⋅ 2 1 6 + 6 = 7 2 + 6 = 7 8 .