Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 19. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | Applications of Linear Models Variations on a Theme 161 b Suppose you tried 220 5x for 0 x 6 C x I 250 25x for x 6. For x 6 this model correctly counts the 5 commission for the flrst six items in with the base rate but then it gives an additional 25 for each of those flrst six items. This amounts to giving a 30 commission for each of the flrst six items sold. For instance when x 7 this model gives 220 5 item 6 items 25 item 7 items instead of the correct 220 5 item 6 items 25 item 1 item . REMARK The function C x is deflned piecewise It is deflned as one function on one interval and as another function on a second interval. Because each piece is linear the function is called piecewise linear. The slope of C x corresponds to the commission rate the rate of change of salary with respect to the number of items sold. The commission rate changes at x 6. Below is a sketch of the slope function typically denoted by C . C dollars item 25 - o 5-------------o -------------1------------ - x items 6 Figure EXAMPLE The resale value of a used TI-81 calculator is a function that decreases with time as newer and more advanced models come out there is less demand for the older TI-81. Let s call P t the price in dollars of a used TI-81 at time t where t is measured in years with t 0 corresponding to January 1 1992. Suppose that on January 1 1992 the resale value was 75 and was decreasing at a rate of 10 per year and that on January 1 1995 the resale value was 51. Furthermore let s assume that although the value is always going down it is going down less and less steeply as time passes. The rationale behind this assumption might be that inflation tends to drive prices up over time and that the calculator will always have some positive value. i. Sketch a possible graph of P t incorporating all the information given. ii. What is the average rate of change of the calculator s value between t 0 and t 3 iii. What can we say about the value of the calculator on January 1 1994
