Worked Examples 1. Consider the function f(x) = 1 Solution x2 – 1.1 Determine the value of: Step 1: Find the value of + i.e: f(2 h), (2 + = 1 (2 + f h) – h)2 = 1 ( 4 + 4h + h2) – (2 + = f h) –3 –4h –h2 Step 2: Find the value of i.e: f(2), = 1 (2)2 f(2) – = f(2) –3 Step 3: Now using your results from Steps 1 and 2 above, evaluate the given expression, i.e.: = = = = ; 0 h = –4 –h 1.2 Interpret your answer This answer represents of the average gradient = 1 between = 2 and = 2 + f(x) x x h –x2 2. = + 1. Given: f(x) x2 2.1 Find the average gradient between : Find the value of , i.e: Step1 f(a) the points R (a and ; f(a)) S(a = + 1 a2 + h ; f(a +h)) f(a) Step 2: Find the value of , i.e.: f(a + h) = (a + 1 = + 2ah + + 1 f(a+h) + h)2 a2 h2 Step 3: Now determine the average gradient,i.e: = = = ; 0 h = 2a + h
Worked Examples 1. Consider the function f(x) = 1 – x2 Solution 1.1 Determine the value of: Step 1: Find the value of f(2 + h), i.e: f (2 + h) = 1 – (2 + h)2 = 1 – ( 4 + 4h + h2) f (2 + h) = –3 –4h –h2 Step 2: Find the value of f(2), i.e: f(2) = 1 – (2)2 f(2) = –3 Step 3: Now using your results from Steps 1 and 2 above, evaluate the given expression, i.e.: = = = = ; h 0 = –4 –h 1.2 Interpret your answer This answer represents the average gradient of f(x) = 1 –x2 between x = 2 and x = 2 + h 2. Given: f(x) = x2 + 1. 2.1 Find the average gradient between the points R (a ; f(a)) and S(a + h ; f(a +h)) Step1 : Find the value of f(a) , i.e: f(a) = a2 + 1 Step 2: Find the value of f(a + h) , i.e.: f(a+h) = (a + h)2 + 1 = a2 + 2ah + h2 + 1 Step 3: Now determine the average gradient,i.e: = = = ; h 0 = 2a + h
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