soltest2math1501

soltest2math1501 - Answer Note that 36 days is 1(10 of a...

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Georgia Tech FALL 2009 Heinrich Matzinger SOLUTION FOR QUIZ 2 FOR MATH 1501: CALCULUS I 1) Explain why for f ( x ) = x 2 , we have f ( x ) = 2 x . Answer: in the book! 2) Show that ( f ( x ) · g ( x )) = f ( x ) · g ( x ) + f ( x ) · g ( x ) . Answer: In the book! 3) Let f ( x ) = x 2 + 3 x . What is f ( x ) equal to? Answer: f ( x ) = 2 x + 3. 4) Calculate f ( x ) where f ( x ) = sin( x ) x 3 . We assume that x is not equal to 0. Answer: f ( x ) = x 3 cos( x ) - 3 x 2 sin( x ) x 6 = x cos( x ) - 3sin( x ) x 4 . 5) Let f ( x ) = 1 x . What is f ( x ) equal to? Again we assume x di±erent from 0. Answer: f ( x ) = - 1 x 2 . 6) Let f ( x ) = x + 1. What is f ( x ) equal to? We assume x is strictly larger than - 1. Answer: f ( x ) = 1 2 x + 1 . 7) Let f ( x ) = x 0 . 1 . What is f ( x ) equal to? Answer: f ( x ) = 0 . 1 x 0 . 9 . 8) Assume that f ( t ) denotes the number of radioactive atoms in a contaminated cite at time t expressed in million. Assume that now is t = 0 and that at the present moment there are 42 million radioactive atoms in that cite. Hence f (0) = 42. Assume that the curve f ( t ) is nice and smooth and that f (0) = - 2. The time unit is the year. What is in 36 days after t = 0 approximately the number of atoms going to be equal to in that cite? Why?
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Unformatted text preview: Answer: Note that 36 days is 1 / (10) of a year, the unit used here. So, the change will be about f ′ (0) · . 1 =-. 2 and hence after 36 days, there should be about 41 . 8 Millions. 9) Let f ( x ) be equal to f ( x ) = cos( x ) x . Calculate f ′ ( x ). Answer: f ′ ( x ) =-x sin( x )-cos( x ) x 2 . 10) Calculate the limit lim x → sin(2 x 3 + 3 x 2 ) x 3 + x Answer: We have sin(2 x 3 + 3 x 2 ) x 3 + x = sin(2 x 3 + 3 x 2 ) 2 x 3 + 3 x 2 · 2 x 3 + 3 x 2 x 3 + x . But the limit of sin(2 x 3 + 3 x 2 ) 2 x 3 + 3 x 2 as x goes to 0 is 1. (When x goes to 0, we have that 2 x 3 + 3 x 2 also goes to 0. ..) Also, the limit 2 x 3 + 3 x 2 x 3 + x = 2 x 2 + 3 x x 2 + 1 is 0 / 1 = 0 as x → 0. Hence, the answer is 1 · 0 = 0....
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soltest2math1501 - Answer Note that 36 days is 1(10 of a...

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