This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full Document
 Fall '07
 Stankova
 Real Numbers

Click to edit the document details