tailieunhanh - Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 35

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 35. 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 the Exponential Function 321 i. Approximate the x-values where the graph of y f x intersects the horizontal line y k. Equivalently trace along the graph until the y-coordinate is k. ii. Approximate the zeros of y f x - k. Let s use the second method to approximate the solution to 2 . Estimate the zeros of the function - 2. You should come up with approximately the money doubles approximately every Generalization Suppose we put M0 dollars in a bank at interest rate r per year compounded annually. If the interest rate is 5 then r . Assume the money is put in the bank at the beginning of the year and interest is compounded at the end of the year. Then each year the balance is multiplied by 1 r. The balance after t years is given by the function M t 0 1 r t. In particular if 5000 is put in a bank paying 4 interest per year compounded annually this formula says that M t 5000 which agrees with our answer from Example . Notice that r is and not 4 if we used r 4 instead then at the end of one year the account would have grown from 5000 to 5000 1 4 25 000 This is a nice deal if you can get it but not a reasonable answer to this problem. COMMENT When we write the equation M t 5000 and let the domain be t 0 we are modeling a discontinuous phenomenon with a continuous model. This equation will only mirror reality if t is an integer . if the bank has just paid the annual interest to the account. For instance if t e 0 1 then the interest has not yet been compounded so the balance should be exactly 5000 over this interval. At t 1 it should jump to 5200. However it is convenient to model this discrete process with a continuous function as we have seen done in examples in previous chapters. B - 5408 - 5200 - 5000 - - continuous model -------- 5 -0 actual function not continuous -1-----------1----------- t 2 3 t 0 Figure EXAMPLE Suppose we put the 5000 in a bank paying interest at an .