Section 1.3

Questions 1. What is the difference between a rate of change and an average rate of change? The rate of change describes how one quantity changes in relation to another given quantity. For instance, when x is the independent variable and y is the dependent variable. Therefore, the rate of change in y and change in x can either be positive or negative. On the other hand, the average rate of change of the function is considered to be the slope of the associated line. Normally, the average rate from 3 to 0 is 1. That is for the interval [3,0] the average rate experienced is 1 unit in the x-axis. 2. Give the formula for calculating average rate of change and answer Try it Now #1. Use the cost of gas function given in the table on the previous page to calculate the average rate of change in the cost of gas between 2003 and 2008. Average rateof changeis given by; change of output change of input = ∆ y ∆ x = y 2 y 1 x 2 x 1 Average rateof change = 3.01 1.69 5 = 0.264 dollars per year 3. Work through examples 2 and 3 on your own, then formally state the average rate of change formula using function notation. This formula is identical to the one given in the previous definition for average rate of change but just uses different notation. Average rateof changeis given by; change of output change of input = f ( b )− f ( a ) b a 4. Show your work to complete Try it Now #2. f ( x ) = x 2 x [1,9] f ( 1 ) = 1 2 1 =− 1 f ( 9 ) = 1 2 9 =− 5 changeof output changeof input = f ( b )− f ( a ) b a = 5 −− 1 9 1 = 4 8 =− 0.5
5. After practicing with the examples 5 and 6, show your work to complete Try it Now #3. This one might be challenging, so here's a demonstration video to help you. f ( x ) = x 3 + 2 [a, a+h] f ( a ) = a 3 + 2 f ( 9 ) =( a + h ) 3 + 2 a ( a + h ) 3 + 2 ( ¿¿ 3 + 2 ) a + h a = ( a + h ) 3 a 3 h changeof output changeof input = f ( b )− f ( a ) b a = ¿ 6. State the definition for marginal cost, marginal revenue and marginal profit. We will cover marginal analysis later using calculus, but you can perform an estimation using algebra. See example 7 for practice. There may be a homework problem that asks you to find marginal cost. For now, you'll just need this definition to estimate it. Marginal Cost is typically the change experienced in the total cost, especially when the quantity of items produced increases by one unit. Similarly, marginal profit and marginal revenue are the change in the profit and revenue, if the quantity of the given items increases by one. 7. What does it mean to say a function is increasing or a function is decreasing over some interval of input ? See the definition and Example 8, then answer this in your own words. Normally, a function increasing on the interval if the function values also increases as the input do the same. Again, a function will be increasing on the interval if the average rate of change of the same function is positive. 8. Let's try some technology. The graph of the function given in Try it Now #4 is shown below. Use it to estimate the local extrema of the function (the high points and low points) and determine the intervals on which the function increases (there are two) and decreases. Grab the red point and drag it to see the increasing and decreasing behavior of the function: as the x values increase from left to right, note where the red dot rises, then falls, then rises on the graph to see the intervals over which the function increases, decreases, and increases again.
Local maximum at x = -1 extrema is (-1, 28). Also the local minimum is at (5, -80).
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