2400_midterm_1

2400_midterm_1 - MATH 2400 FIRST MIDTERM EXAM PETE L CLARK...

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MATH 2400 FIRST MIDTERM EXAM PETE L. CLARK Directions : You will have 75 minutes to complete the exam. Calculators are not permitted (nor would they be helpful...). Please attempt all problems. When in doubt, err on the side of writing a little too much rather than a little too little. 1)a) Find all real numbers x, y such that y 2 > 5 x 2 . Solution: y 2 > 5 x 2 ⇐⇒ | y | > 5 | x | . b) Sketch a graph of the set of all points in the plane satisfying y 2 > 5 x 2 . Solution: I will describe the picture to (in less than a thousand words). Con- sider the two lines y = ± 5 x . These lines divide the plane into four different regions. The desired set of points consists of the top and bottom regions and not the left and right regions. This can be seen for instance by first taking x and y to be non-negative and noting that in the first quadrant we want all points ( x, y ) with y > 5 x . Then, since the locus | y | > 5 | x | is unchanged by x 7→ − x and y 7→ − y we can simply reflect this “45 degree sector” through the x -axis, through
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