미적분 자료등록 자연로그, 지수함수, 역, 삼각함수에 대한 미적분 Report
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6. The Logarithm, the Exponential, and the Inverse Trigonometric Functions.
Definition. If x is a positive real number, we define the natural logarithm of x, denoted temporarily by L(x), to be the integral L(x) = .
Theorem 6.1. The logarithm function has the following properties:
(a) L(1) = 0
(b) L`(x) = 1/x for every x > 0
(c) L(ab) = L(a) + L(b) for every a > 0, b > 0
- t/a
Theorem. 6.2. For every real number b there is exactly one positive real number a whose logarithm, L(a) is equal to b.
Definition. We denote by e that number for which L(e) = 1.
Definition. If b > 0, b ≠ 1, and if x > 0, the logarithm of x to the base b is the number
logbx = . where the logarithms on the right are natural logarithms.
Point. ∫ 1/x dx = log x + C
Point. ∫ du / u = log u +C
Point. ∫ f`(x)dx / f(x) = log f(x) + C.
Point. L0(x) = log|x| = .
Definition. For any real x, we define E(x) to be that number y whose logarithm is x. That is, y=E(x) means that L(y) = x.
Theorem. 6.6. The exponential function has the following properties:
(a) E(0) = 1, E(1) = e.
(b) E`(x) = E(x) for every x
(c) E(a+b) = E(a)E(b) for all a and b.
Point. ax = exloga
Point. ∫ ef(x)f`(x)dx = ef(x) +C.
Point. 1/(1+ex) = 1-(ex/(1+ex))
Theorem 6.7. Assume f is strictly increasing and continuous on an interval [a,b], and let g be the inverse of f. If the derivative f`(x) exists and is nonzero at a point x in (a,b), then the derivative g`(y) also exists and is nonzero at the corresponding point y, where y=f(x). Moreover, the two derivatives are reciprocals of each other; that is, we have g`(y) = .
Point. D arcsinx = . if -1< x <1.
= arcsinx.
Point. D arccosx = .
= π/2 - arccos x
Point. D arctanx = .
= arctan x
Point. arccot x = π/2 - arctan x for all real x
arcsecx = arccos(1/x) when |x|≥ 1,
arccscx = arcsin(1/x) when |x|≥ 1.
Point. u = arctan x, tan u = x…(생략)
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