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Math121Quiz3Sols - QUIZ#3(SECTIONS 3.1 3.2 3.3 3.6...

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QUIZ #3 (SECTIONS 3.1, 3.2, 3.3, 3.6) SOLUTIONS MATH 121 – FALL 2003 – KUNIYUKI 105 POINTS TOTAL, BUT 100 POINTS = 100% 1) Sketch the graph of f x x x ( ) = - + 4 3 4 3 . You must: • Find and label all points at critical numbers and inflection points (if any). • Classify all points at critical numbers as relative maximum points, relative minimum points, or neither. • Find the y -intercept. • Have your graph correctly show where f is increasing / decreasing, and where f is concave up / concave down. • Show all steps, as we have done in class. (30 points) Step 1 : Domain = R . Step 2 : Find ¢ f and critical numbers (CNs). f x x x f x x x x x ( ) = - + ¢ ( ) = - = - ( ) 4 3 3 2 2 4 3 4 12 4 3 This is never DNE. It equals 0 at x = 0 and x = 3 , which are in the domain of f . The CNs are 0 and 3. Step 3 : Do a sign diagram for ¢ f and classify the points at the CNs. Test x = - 1 0 Test x = 1 3 Test x = 4 ¢ f sign - - + f Classify Points at CNs (Using 1 st DT.) Neither R.Max. nor R.Min. Pt. R.Min. Pt. Plug into f x ( ) to get y 0 0 , f ( ) ( ) 0,3 ( ) 3 3 , f ( ) ( ) 3, 24 - ( )
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¢ ( ) = ( ) ( ) - ( ) ¢ - ( ) = + ( ) + ( ) - ( ) = - ¢ ( ) = + ( ) + ( ) - ( ) = - ¢ ( ) = + ( ) + ( ) + ( ) = + f x x x f f f 4 3 1 1 4 2 Step 4 : Skeleton graph for f ; y -intercept y -intercept = f 0 3 ( ) = , the constant term from the f x ( ) rule. (We already knew that 0 3 , ( ) was on the graph of f. ) Step 5 : Find ¢¢ f and possible inflection numbers (PINs). ¢ ( ) = - ¢¢ ( ) = - = - ( ) f x x x f x x x x x 4 12 12 24 12 2 3 2 2 ¢¢ f is never DNE. It equals 0 at x = 0 and x = 2 , which are in the domain of f . The PINs are 0 and 2. Step 6 : Do a sign diagram for ¢¢ f and find inflection points (IPs). Test x = - 1 0 Test x = 1 2 Test x = 3 ¢¢ f sign + - + f graph CU » ( ) CD « ( ) CU » ( ) Inflection Points (IPs)? Yes, IP Yes, IP Plug into f x ( ) to get y 0 0 , f ( ) ( ) 0,3 ( ) 2 2 , f ( ) ( ) 2, 13 - ( )
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¢¢ ( ) = ( ) ( ) - ( ) ¢¢ - ( ) = + ( ) - ( ) - ( ) = + ¢¢ ( ) = + ( ) + ( ) - ( ) = - ¢¢ ( ) = + ( ) + ( ) + ( ) = + f x x x f f f 12 2 1 1 3 Step 7 : Sketch the graph of f .
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