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By Gottlob Frege
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The graph then begins curving upwards for the remainder of the graph shown. From this, we can estimate that the graph is concave up on the intervals (,1) and (2, ) , and is concave down on the interval (1,2) . The graph has inflection points at x = -1 and x = 2. Try it Now 6. Using the graph from Try it Now 4, f ( x) x 3 6 x 2 15 x 20 , estimate the intervals on which the function is concave up and concave down. Behaviors of the Toolkit Functions We will now return to our toolkit functions and discuss their graphical behavior.
37. f ( x) x 4 4 x 3 5 38. h( x) x 5 5 x 4 10 x 3 10 x 2 1 39. g (t ) t t 3 41. m( x) x 4 2 x 3 12 x 2 10 x 4 40. k (t ) 3t 2 / 3 t 42. 4 Composition of Functions Suppose we wanted to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The temperature depends on the day, and the Cost depends on the temperature.
64. 14. The input (years) has changed by 2. 50. 25 dollars per year 2 years Try it Now 1. Using the same cost of gas function, find the average rate of change between 2003 and 2008 Notice that in the last example the change of output was negative since the output value of the function had decreased. Correspondingly, the average rate of change is negative. Example 2 Given the function g(t) shown here, find the average rate of change on the interval [0, 3]. At t = 0, the graph shows g (0) 1 At t = 3, the graph shows g (3) 4 The output has changed by 3 while the input has changed by 3, giving an average rate of change of: 4 1 3 1 30 3 Example 3 On a road trip, after picking up your friend who lives 10 miles away, you decide to record your distance from home over time.
The Foundations of Arithmetic by Gottlob Frege