HW01-solutions - vasquez(mpv244 HW01 Schultz(56445 This...

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vasquez (mpv244) – HW01 – Schultz – (56445) 1 This print-out should have 24 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Express the number r = 2 1 1 + 3 2 in its simplest form. 1. r = 8 2. r = 12 3. r = 12 5 4. r = 24 5 5. r = 16 5 6. r = 8 5 correct Explanation: Since 1 + 3 2 = 5 2 , we see that r = 2 2 5 = 20 4 10 = 8 5 . keywords: 002 10.0 points Simplify the expression parenleftbigg 4 x 3 z 3 x 4 y 3 parenrightbigg 2 as much as possible, leaving no negative ex- ponents. 1. x 14 16 z 6 y 6 2. x 2 4 z 2 y 2 3. x 14 16 z 6 4. x 4 z 3 y 3 5. x 14 y 6 16 z 6 correct Explanation: By the Laws of Exponents, parenleftbigg 4 x 3 z 3 x 4 y 3 parenrightbigg 2 = parenleftbigg 4 z 3 x 4+3 y 3 parenrightbigg 2 = 4 2 z 3 · 2 x 7 · 2 y 3 · 2 which after simplification becomes x 14 y 6 16 z 6 . keywords: 003 10.0 points Which, if any, of the following statements are true when a, b are real numbers? A. For all a and b , radicalBig ( a + b ) 2 = a + b . B. For all positive a and b , a + b = radicalBig a + 2 ab + b . C. For all positive a and b . a b a + b = a + b . 1. A only 2. B only correct
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vasquez (mpv244) – HW01 – Schultz – (56445) 2 3. all of them 4. none of them 5. C only 6. B and C only 7. A and C only 8. A and B only Explanation: A. FALSE: radicalBig ( x + y ) 2 = | x + y | , and since radicalbig ( · ) is always non-negative, the right hand side has to be non-negative. But if a, b can be positive or negative, an absolute value sign is then needed on the right. B. TRUE: by the known product, ( x + y ) 2 = x 2 + 2 xy + y 2 . On the other hand, radicalBig ( x + y ) 2 = | x + y | , so if x + y > 0, x + y = radicalbig x 2 + 2 xy + y 2 . But if a, b are positive we can set x = a and y = b . The result follows since x and y are then positive. C. FALSE: by the known difference of squares factorization, x 2 y 2 = ( x y )( x + y ) . But if a, b are positive we can set x = a and y = b . Thus, after division, a b a + b = a b , contrary to the assertion. keywords: square root, properties of square root, PlaceUT, TrueFalse, T/F, 004 10.0 points Find the domain and range of the function h ( x ) = radicalbig 36 x 2 . 1. domain = [0 , 6] , range = [0 , 6] 2. domain = [0 , 6] , range = [6 , 6) 3. domain = ( −∞ , ) , range = [0 , ) 4. domain = [ 6 , 6] , range = [0 , 6] cor- rect 5. domain = ( 6 , 6) , range = (0 , 6) 6. domain = [ 6 , 6] , range = [ 6 , 6] Explanation: If we write y = radicalbig 36 x 2 , then x 2 + y 2 = 36 whose graph is a circle centered at the origin having radius 6. Thus the graph of h is the upper semi-circle shown in 6 6 6 Consequently, h has domain = [ 6 , 6] , range = [0 , 6] . keywords: range, range of function, circle, domain, domain of function,
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vasquez (mpv244) – HW01 – Schultz – (56445) 3 005 10.0 points There is a linear relationship T F = mT C + b between the temperature T F on the Fahren- heit scale and its equivalent T C on the Centi- grade scale. As the thermometers 32 212 0 100 Freezing Boiling Fahrenheit Centigrade show, water freezes at 32 F and boils at 212 F, whereas it freezes at 0 C and boils at 100 C.
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