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x 0 x Read QR Code JIS X 0510 In .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET framework applications. Draw QR Code ISO/IEC18004 In VS .NET Using Barcode creation for .NET framework Control to generate, create QR Code image in .NET framework applications. 28.24 If a plane is ying at a speed of 480 mi/h at a steady elevation of 3 miles above the ground and the pilot is sighting a location on the ground directly ahead, how fast is the sighting instrument turning when the angle between the path of the plane and the line of sight is 30 QR Code Decoder In Visual Studio .NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET applications. Generating Barcode In .NET Using Barcode drawer for VS .NET Control to generate, create bar code image in Visual Studio .NET applications. Antiderivatives
Decode Bar Code In .NET Framework Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET applications. QR Code Creation In Visual C# Using Barcode encoder for .NET framework Control to generate, create QR image in VS .NET applications. 29.1 DEFINITION AND NOTATION De nition: An antiderivative of a function f is a function whose derivative is f . EXAMPLES QR Code ISO/IEC18004 Creator In VS .NET Using Barcode generation for ASP.NET Control to generate, create QRCode image in ASP.NET applications. Encode QR In VB.NET Using Barcode encoder for VS .NET Control to generate, create QR image in VS .NET applications. (a) x 2 is an antiderivative of 2x, since Dx (x 2 ) = 2x. (b) x 4 /4 is an antiderivative of x 3 , since Dx (x 4 /4) = x 3 . (c) 3x 3 4x 2 + 5 is an antiderivative of 9x 2 8x, since Dx (3x 3 4x 2 + 5) = 9x 2 8x. (d) x 2 + 3 is an antiderivative of 2x, since Dx (x 2 + 3) = 2x. (e) sin x is an antiderivative of cos x, since Dx (sin x) = cos x. Encode Code 39 In .NET Framework Using Barcode creation for .NET Control to generate, create Code 39 Full ASCII image in VS .NET applications. UCC  12 Encoder In .NET Using Barcode drawer for VS .NET Control to generate, create UPC A image in Visual Studio .NET applications. Examples (a) and (d) show that a function can have more than one antiderivative. This is true for all functions. If g(x) is an antiderivative of f (x), then g(x) + C is also an antiderivative of f (x), where C is any constant. The reason is that Dx (C) = 0, whence Dx (g(x) + C) = Dx (g(x)) Theorem 29.1: If F (x) = 0 for all x in an interval I , then F(x) is a constant on I . The assumption F (x) = 0 tells us that the graph of F always has a horizontal tangent. It is then obvious that the graph of F must be a horizontal straight line; that is, F(x) is constant. For a rigorous proof, see Problem 29.4. Let us nd the relationship between any two antiderivatives of a function. Corollary 29.2: If g (x) = h (x) for all x in an interval I , then there is a constant C such that g(x) = h(x) + C for all x in I . Indeed, Dx (g(x) h(x)) = g (x) h (x) = 0 whence, by Theorem 29.1, g(x) h(x) = C, or g(x) = h(x) + C. According to Corollary 29.2, any two antiderivatives of a given function differ only by a constant. Thus, if we know one antiderivative of a function, we know them all. Encoding USS128 In Visual Studio .NET Using Barcode encoder for VS .NET Control to generate, create USS128 image in VS .NET applications. Painting MSI Plessey In VS .NET Using Barcode encoder for .NET Control to generate, create MSI Plessey image in Visual Studio .NET applications. Copyright 2008, 1997, 1985 by The McGrawHill Companies, Inc. Click here for terms of use.
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ECC200 Printer In C#.NET Using Barcode creator for .NET Control to generate, create Data Matrix ECC200 image in Visual Studio .NET applications. Creating USS128 In Java Using Barcode creation for Java Control to generate, create GS1 128 image in Java applications. f (x) dx stands for any antiderivative of f . Thus, Dx f (x) dx = f (x) other terminology Sometimes the term inde nite integral is used instead of antiderivative, and the process of nding antiderivatives is termed integration. In the expression f (x) dx, f (x) is called the integrand. The motive for this nomenclature will become clear in 31. EXAMPLES
(a) x3 + C. Since Dx (x 3 /3) = x 2 , we know that x 3 /3 is an antiderivative of x 2 . By Corollary 29.2, any other 3 antiderivative of x 2 is of the form (x 3 /3) + C, where C is a constant. x 2 dx = cos x dx = sin x + C sin x dx = cos x + C sec2 x dx = tan x + C 0 dx = C 1 dx = x + C (b) (c) (d) (e) (f) RULES FOR ANTIDERIVATIVES
The rules for derivatives in particular, the sumordifference rule and the chain rule yield corresponding rules for antiderivatives. RULE 1. EXAMPLE 3 dx = 3x + C
a dx = ax + C for any constant a.
RULE 2.
note
x r dx =
x r+1 + C for any rational number r other than r = 1. r+1
The antiderivative of x 1 will be dealt with in 34.
Rule 2 follows from Theorem 15.4, according to which Dx (x r+1 ) = (r + 1)x r or Dx x r+1 r+1 = xr
CHAP. 29] ANTIDERIVATIVES
EXAMPLES
(a) x dx = x 1/2 dx = x 3/2 +C =
2 3/2 x +C 3
1 dx = x3
x 3 dx =
1 x 2 1 + C = x 2 + C = 2 + C 2 2 2x
RULE 3.
af (x) dx = a
f (x) dx for any constant a. f (x) dx = a Dx
5x 2 dx = 5
This follows from Dx a EXAMPLE RULE 4. (i) (ii) For Dx
f (x) dx = af (x). x3 3 +C = 5x 3 +C 3
x 2 dx = 5
[f (x) + g(x)] dx = [f (x) g(x)] dx = g(x) dx = Dx
f (x) dx + f (x) dx
g(x) dx g(x) dx g(x) dx = f (x) g(x).

