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Transcendental Functions
                     180
                                                  CHAPTER 6
                                   birth and is locked in at 11% interest compounded continuously. How much
                                   principal should be put into the account to yield the desired payoff?
                                   SOLUTION
                                     Let P be the initial principal deposited in the account on the day of the
                                   nephew’s birth. Using our compound interest equation (..), we have
                                                        100000 = P · e 11·21/100 ,

                                   expressing the fact that after 21 years at 11% interest compounded continuously
                                   we want the value of the account to be $100,000.
                                     Solving for P gives
                                           P = 100000 · e −0.11·21  = 100000 · e −2.31  = 9926.13.
                                   The aunt needs to endow the fund with an initial $9926.13.

                               You Try It: Suppose that we want a certain endowment to pay $50,000 in cash
                               ten years from now. The endowment will be set up today with $5,000 principal and
                               locked in at a fixed interest rate. What interest rate (compounded continuously) is
                               needed to guarantee the desired payoff?



                   6.6 Inverse Trigonometric Functions

                               6.6.1      INTRODUCTORY REMARKS

                               Figure 6.14 shows the graphs of each of the six trigonometric functions. Notice that
                               each graph has the property that some horizontal line intersects the graph at least
                               twice. Therefore none of these functions is invertible. Another way of seeing this
                               point is that each of the trigonometric functions is 2π-periodic (that is, the function
                               repeats itself every 2π units: f(x + 2π) = f(x)), hence each of these functions is
                               not one-to-one.
                                  If we want to discuss inverses for the trigonometric functions, then we must
                               restrict their domains (this concept was introduced in Subsection 1.8.5). In this
                               section we learn the standard methods for performing this restriction operation
                               with the trigonometric functions.

                               6.6.2      INVERSE SINE AND COSINE

                               Consider the sine function with domain restricted to the interval [−π/2,π/2]
                               (Fig. 6.15). We use the notation Sin x to denote this restricted function. Observe
                               that
                                                          d
                                                            Sin x = cos x> 0
                                                         dx
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