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METU NCC Mat 120 Spring 2009 Midterm 2

# METU NCC Mat 120 Spring 2009 Midterm 2 - (a Find the...

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Dept./Sec. Time Duration Signature : : : : : : Last Name Name Student No M E T U Northern Cyprus Campus Math 120 Calculus for functions of several variables II. Exam 20.04.2009 17: 40 120 minutes 7 QUESTIONS ON 4 PAGES TOTAL 100 POINTS 1 2 3 4 5 6 7 Q1 (15=8+7 pts.) (a) Find the plane through the points (3 , 0 , 0) , (0 , 1 , 2) , (0 , 0 , 1) (b) Find the line through the origin (0 , 0 , 0) perpendicular to the plane 2 x + 3 y - z = 16.

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Q2 (15 pts.) Sketch the graph of the quadric surface ( x - 1) 2 + 5 y 2 = 2 z . Q3 (10 pts.) Find the parametric equations of the tangent line to a space curve x = t 3 , y = 1 + t, z = 2 t at the point ( - 1 , 0 , - 2).
Q4 (15=7+8 pts.) Find the following limits if they exist. Explain your answers. (a) lim ( x,y ) (0 , 0) xy x 2 + y 2 + 1 (b) lim ( x,y ) (0 , 0) x 3 y x 4 + y 4

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Q5 (15=7+8 pts.) Consider the function f ( x, y ) =
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Unformatted text preview: (a) Find the tangent plane to the graph of the function f ( x, y ) at the point (1 , 3) . (b) Use the tangent plane appoximation to estimate the value f (1 . 1 , 2 . 9). Q6 (15 pts.) Use the Chain Rule to ﬁnd the partial derivative ∂f ∂s if f ( x, y, z ) = z ln ( x + y 2 + z 3 ) and x = 3 t-s , y = t + 2 s , z = ts . Q7 (15 pts.) Let r ( t ) = h f ( t ) , g ( t ) , h ( t ) i be a vector-valued funtion such that r ( t ) = r 000 ( t ) for all t . Show that the triple (or box)-product a ( t ) = r ( t ) · ( r ( t ) × r 00 ( t )) is a constant function. ( Hint: consider the derivative a ( t ))...
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