Reference no: EM131004364
Problem 1. Between the new film The Martian (potential spoiler alert!) and NASA's recent announcement that liquid water still flows on Mars, Brutus Buckeye has "Mars fever". Looking through Brutus's new telescope Brutus is able to sketch the following surface features of Mars:
After consulting with several astronomers, Brutus is able to deduce that the implicit equation
sin3 (x2 + y) = 1/8
gives the graph of this sketch. One interesting fact Brutus discovered from the astronomers is that knowing the value of dy/dx gives clues about the changes in the Martian landscape.
(a) Using implicit differentiation of the equation
sin3 (x2 + y) = 1/8
to find dy/dx.
(b) Using dy/dx find the slope of the tangent line at (√π /12, π /12).
(c) Find the equation of the tangent line at (√π /12, π /12).
Problem 2. The next day, Brutus looks through the telescope again and sketch new area of the Martian landscape:
Again Brutus is able to deduce that the implicit equal
sin (x2 + y) = x2 - 2x
gives the graph of this new sketch.
(a) Using implicit differentiation of the equation
sin (x2 + y) = x2 - 2x
to find dy/dz.
(b) Unfortunately, the astronomers tell Brutus that all of the Ohio supercomputers are scheduled for the next three month. This makes Brutus sad. Undeterred, Brutus figure out a way around the problem:
i. Solve for y in the equation
sin (x2 + y) = x2 - 2x
(ii) Use your explicit equation for y in terms of x to find dy/dx.
(iii) Use your explicit equation for y in terms of x to estimate point on the graph and find the value of dy/dx evaluated at this point.
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