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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
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29.1 DEFINITION AND NOTATION De nition: An antiderivative of a function f is a function whose derivative is f . EXAMPLES
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(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.
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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.
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[CHAP. 29
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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 sum-or-difference 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).