Midterm Exam 2

# Midterm Exam 2 - 7, f (1 . 3) ≈ 1 . 6, f (4 . 7) ≈ 2 ....

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MATH150 Midterm Exam 2 11/14/05 GIVE DETAILED AND COMPLETE SOLUTIONS. FULLY SIMPLIFY YOUR ANSWERS. NO CALCULATORS PERMITTED! 1. (12 marks) Determine the following limits: (a) lim x →∞ ln(ln x ) ln 2 x (b) lim x 0 sinh( x ) - x x 3 (c) lim x π/ 2 (cos x ) cos x 2. (12 marks) What is the maximal area a right triangle can have whose hypotenuse has a length of 2 cm? Carefully justify your answer! 3. (14 marks) Let f ( x ) = x 3 e 1 - x , deﬁned for all real numbers. (a) Find the horizontal and vertical asymptotes of f (if any). (b) Find and classify the critical and singular points (if any) and ﬁnd the intervals of increase and decrease of f . (c) Find all inﬂection points (if any) and the intervals of concavity of f . (d) Give a rough sketch of the graph of f . The graph should show all the important features of f but it does not have to be drawn to scale. You may ﬁnd the following useful: 3 1 .
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Unformatted text preview: 7, f (1 . 3) ≈ 1 . 6, f (4 . 7) ≈ 2 . 6, the absolute maximum value of f is approximately equal to 3.7. 4. (12 marks) Let f ( x, y ) = 6 x 2-4 x 3 y + y 4 . (a) Compute an equation of the tangent plane to the graph of f at the point P (-1 , 1 , 11). (b) Compute the parametric equations of the normal line to the graph of f at P . (c) Find all points on the graph of f where the tangent plane to the graph is horizontal. 5. (10 marks) Let f ( x, y ) = ( xy 2 + x 4 ( x 2 + y 2 ) 3 / 2 if ( x, y ) ± = (0 , 0) if ( x, y ) = (0 , 0) (a) Show that lim ( x,y ) → (0 , 0) f ( x, y ) does not exist. (b) Is f continuous at (0 , 0)? Justify! (c) Use the limit deﬁnition of partial derivatives to compute ∂f ∂x (0 , 0) and ∂f ∂y (0 , 0) or to show that they don’t exist....
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## This note was uploaded on 08/23/2008 for the course MATH 150 taught by Professor Hundemer during the Fall '04 term at McGill.

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