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Unformatted text preview: 016A S2 Homework 1 Solution Jaeyoung Park * September 2, 2007 0.1 *10 Use intervals to describe the real numbers satisfying the in equality x  1 and x &lt; 8. Solution You may change the expression little bit: 1 x &lt; 8, which is equivalent. Using interval notation, we can describe this inequality as [ 1 , 8) 0.1 *20 If f ( x ) = x 2 + 4 x + 3, find f ( a 1) and f ( a 2). Solution To find f ( a 1), we substitute a 1 for every x appears in the formula defining f ( x ). So we get, f ( a 1) = ( a 1) 2 + 4( a 1) + 3 This expression can be simplified using ( a 1) 2 = a 2 2 a + 1. So we get f ( a 1) = [ a 2 2 a + 1] + [4 a 4] + 3 = a 2 + 2 a Similarly for f ( a 2). Substituting a 2 for x , we get f ( a 2) = ( a 2) 2 + 4( a 2) + 3 Using the fact that ( a 2) 2 = a 2 4 a + 4, simplify this expression to get f ( a 2) = [ a 2 4 a + 4] + [4 a 8] + 3 = a 2 1 * jaypark at m a t h . b e r k e l e y . e d u. GSI for 16A 205,211,213 1 0.1 *22 (a) Suppose that b = 20. Find the response of the muscles when x = 60. Solution Since b = 20, the response of the muscle R ( x ) is given by R ( x ) = 100 x 20 + x , for x Substituting x = 60, we find R (60). R (60) = 100 60 20 + 60 = 100 60 80 = 6000 80 = 600 8 = 75 So the answer is 75%. 0.1 *22 (b) Determine the value of b if R (50) = 60 that is, if a concentration of x = 50 units produces a 60% response. Solution Substituting 50 for x , we get R (50) = 100 50 b + 50 Since R (50) = 60, we get 100 50 b + 50 = 60 Multiplying both sides by b + 50, we get an equation 60( b + 50) = 100 50 = 5000 = 60 b + 3000 = 5000 Subtracting both sides by 3000, we get 60 b = 2000 = b = 2000 60 = 100 3 You may verify this answer by plugging b = 100 3 in R ( x ) and finding the value of R (50). So the answer is b = 100 3 0.1 *25 Describe the domain of the function g ( x ) = 1 3 x Solution Implicitly, we are working with real numbers. Also, recall that we understand the intended domain to consist of all numbers for which the defining formula make sense. Since 3 x is defined if and only if 3 x 0 and dividing by zero doesnt make sense, the defining formula makes sense if and only if 3 x &gt; 0. So the domain is x &gt; 3. 2 0.1 *26 Describe the domain of the function g ( x ) = 4 x ( x +2) Solution As we did in the previous problem, we find the values which the defining formula makes sense. The formula for g ( x ) makes sense unless the denominator x ( x + 2) is zero. Finding the zero of x ( x + 2), we get x = 0 or x = 2. So the domain is all real numbers except x = 0 , 2. You may express this domain by Rr { , 2 } . 0.1 *30 Decide which curves are graphs of functions....
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This note was uploaded on 04/01/2008 for the course MATH 16A taught by Professor Stankova during the Fall '07 term at University of California, Berkeley.
 Fall '07
 Stankova
 Real Numbers

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