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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