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print barcode image c# Transcendental Functions in Software
Transcendental Functions Printing Quick Response Code In None Using Barcode creation for Software Control to generate, create Quick Response Code image in Software applications. Recognize QR Code JIS X 0510 In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. birth and is locked in at 11% interest compounded continuously How much principal should be put into the account to yield the desired payoff QR Code Encoder In Visual C# Using Barcode generation for VS .NET Control to generate, create Denso QR Bar Code image in Visual Studio .NET applications. QR Code ISO/IEC18004 Creator In Visual Studio .NET Using Barcode encoder for ASP.NET Control to generate, create QR Code 2d barcode image in ASP.NET applications. 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 e11 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 011 21 = 100000 e 231 = 992613 The aunt needs to endow the fund with an initial $992613 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 xed interest rate What interest rate (compounded continuously) is needed to guarantee the desired payoff Creating QR Code 2d Barcode In .NET Framework Using Barcode drawer for .NET framework Control to generate, create QR Code 2d barcode image in VS .NET applications. QR Code 2d Barcode Maker In VB.NET Using Barcode creator for .NET Control to generate, create QR Code JIS X 0510 image in Visual Studio .NET applications. Inverse Trigonometric Functions
Data Matrix ECC200 Creator In None Using Barcode creation for Software Control to generate, create Data Matrix ECC200 image in Software applications. Paint GS1128 In None Using Barcode printer for Software Control to generate, create EAN / UCC  14 image in Software applications. 661 INTRODUCTORY REMARKS
Generating Universal Product Code Version A In None Using Barcode generator for Software Control to generate, create UPC Code image in Software applications. Creating Code 128 Code Set B In None Using Barcode generation for Software Control to generate, create Code 128B image in Software applications. Figure 614 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 2 periodic (that is, the function repeats itself every 2 units: f (x + 2 ) = f (x)), hence each of these functions is not onetoone If we want to discuss inverses for the trigonometric functions, then we must restrict their domains (this concept was introduced in Subsection 185) In this section we learn the standard methods for performing this restriction operation with the trigonometric functions Code 39 Extended Creator In None Using Barcode creator for Software Control to generate, create Code39 image in Software applications. Drawing Bar Code In None Using Barcode encoder for Software Control to generate, create bar code image in Software applications. INVERSE SINE AND COSINE
Identcode Drawer In None Using Barcode drawer for Software Control to generate, create Identcode image in Software applications. EAN13 Reader In C# Using Barcode recognizer for .NET framework Control to read, scan read, scan image in .NET applications. Consider the sine function with domain restricted to the interval [ /2, /2] (Fig 615) We use the notation Sin x to denote this restricted function Observe that d Sin x = cos x > 0 dx Barcode Drawer In None Using Barcode creation for Microsoft Word Control to generate, create barcode image in Office Word applications. Matrix 2D Barcode Generation In Java Using Barcode creation for Java Control to generate, create Matrix 2D Barcode image in Java applications. CHAPTER 6 Transcendental Functions
Encoding Code 128 Code Set A In Java Using Barcode printer for Java Control to generate, create Code 128C image in Java applications. Read Data Matrix In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. 2 sec x 1 cos x _3 _2 _1 sin x
Making Barcode In None Using Barcode generator for Office Word Control to generate, create bar code image in Microsoft Word applications. Scanning ANSI/AIM Code 39 In Visual C# Using Barcode reader for VS .NET Control to read, scan read, scan image in VS .NET applications. _1 csc x _2 tan x
cot x
Fig 614
y = Sin x 1
Fig 615
Transcendental Functions
on the interval ( /2, /2) At the endpoints of the interval, and only there, the function Sin x takes the values 1 and +1 Therefore Sin x is increasing on its entire domain So it is onetoone Furthermore the Sine function assumes every value in the interval [ 1, 1] Thus Sin : [ /2, /2] [ 1, 1] is onetoone and onto; therefore f (x) = Sin x is an invertible function We can obtain the graph of Sin 1 x by the principle of re ection in the line y = x (Fig 616) The function Sin 1 : [ 1, 1] [ /2, /2] is increasing, onetoone, and onto y = Sin
_ F/2 Fig 616
The study of the inverse of cosine involves similar considerations, but we must select a different domain for our function We de ne Cos x to be the cosine function restricted to the interval [0, ] Then, as Fig 617 shows, g(x) = Cos x is a onetoone function It takes on all the values in the interval [ 1, 1] Thus Cos : [0, ] [ 1, 1] is onetoone and onto; therefore it possesses an inverse We re ect the graph of Cos x in the line y = x to obtain the graph of the function Cos 1 The result is shown in Fig 618 EXAMPLE 635
Calculate
, Sin 3
, Cos
CHAPTER 6 Transcendental Functions
y = Cos x
Fig 617
y = Cos
Fig 618
SOLUTION We have Sin
3 2 , 3 Sin 1 0 = 0, Transcendental Functions
= 4 2 2 Notice that even though the sine function takes the value 3/2 at many different values of the variable x, the function Sine takes this value only at x = /3 Similar comments apply to the other two examples We also have 5 3 1 Cos = , 2 6 Cos 1 0 = , 2 2 Cos 1 = 2 4 We calculate the derivative of f (t) = Sin 1 t by using the usual trick for inverse functions The result is d 1 1 (Sin 1 (x)) = = dx 1 x2 1 sin2 (Sin 1 x) The derivative of the function Cos 1 t is calculated much like that of Sin 1 t We nd that d 1 (Cos 1 (x)) = dx 1 x2 EXAMPLE 636

