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Mathematics-Online problems:

Interactive Problem 203: Determination of Coefficients of Parabolas, Tangent Lines and Areas

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
The parabolas $ p$ and $ q$ intersect at the point $ (0,0)$ orthogonally and are tangent to the parabola $ r$ opening downward at the points $ (-1,0)$ and $ (2,0)$.

\includegraphics{TdM_1996_1}

Determine $ p,$ $ q,$ and $ r,$ as well as the slopes of the tangent lines at the points of intersection.
Find the area $ A$ of the shaded region.

Answer:

Parabolas:

$ p(x) = $$ \:x(x+1) $
$ q(x) = $$ \:x(x-2) $
$ r(x) = $ $ \:(x+1)(x-2)$

Points of intersection (other than points of tangency): $ (0,0),$ $ \big($$ ,$$ \big)$

Slopes of the tangent lines at the points of intersection:

$ x$ $ p^\prime(x)$ $ q^\prime (x)$
0
?


Area $ A$:

(The results should be correct to three decimal places.)


   

(From: Day of Mathematics 1996)

see also:

  automatically generated: 8/11/2017