2010 II / y = e x y = log x = log e x 2. ( e x ) = e x 3. ( ) log x = 1 x 1.2 Warming Up 1 u = log a M a u = M a 0

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1 2010 II / y = e x y = log x = log e x 2. e x ) = e x 3. ) log x = 1 x 1.2 Warming Up 1 u = log a M a u = M a 0 log a 1 a 1 log a a a r+s log a M + log a N 1 0 a 1 a r a s log a MN π e = π π PIE

2 2010 II / , 3 f g) = f g + f g 1. fx) = x e x f x) = 2. fx) = x log x f x) = 3. fx) = ex x = x 1 e x f x) = 4. fx) = log x 2 3 = 2 3 log x f x) =

3 2010 II / e ) 1 1 x- y- 2 x y = log 2 x y = 2 x x 1: y = 2 x, y = log 2 x y x 1: y = 2 x, y = log 2 x

4 2010 II / ) 2 2 x- y- 2 x y = log 3 x y = 3 x x 2: y = 3 x, y = log 3 x y x 2: y = 3 x, y = log 3 x 2.2 y = a x, y = log a x a y = a x y-log a x x a a

5 2010 II / = e = e e = lim 1 + ) 1 = lim n n )n 1) e e x 2. log x := log e x e fx) = e x f 0) = 1 2) f x) = e x e x ) = e x 3) 3.3 fa + ) fa) f a) a a + a a + π= =EXP)e = log 10 M 10 ln x

6 2010 II / ) e 2.2 3) f a) x = a f a) fa + ) fa) 4) f 0) f) f0) = e 1 5) 3) f a) fa + ) fa) 6) = e a+ e a = e a e e a = e a e 1 ) 7) e a f 0) = e a. 8) f a) = e a f x) = e x // fx) = e x gx) = log x 0, 1), ) f 0) = 1 g ) = 3:

7 2010 II / gx) = log x g 1) = 1 9) g x) = 1 x ) log x = 1 x 10). 9) fx) = e x y = x gx) = log x fx) = e x i y 0, 1) e 0 = 1 ii f 0) = 1 gx) = log x i x 0, 1) log 1 = 0 ii g 1) = 1 10) fx) = e x a b y = x b a gx) = log x fx) = e x i y = e x ii f x) = a b = ex gx) = log x i y = log x e y = x ii g x) = b a = 1 e y = 1 x //

8 富山大学経済学部 2010 年度経営経済の基礎数学 II プリント第 6 回月 / 木エクセルの図で確認図 4: y = 2x, y = log2 x 図 5: y = ex, y = log x 図 6: y = 3x, y = log3 x 図 7: y = 4x, y = log4 x

9 2010 II / f g) = f g + f g 1. fx) = x e x f x) = x) e x + x e x ) = e x + x e x = e x 1 + x) = e x x + 1) // 2. fx) = x 2 e x f x) = 3. fx) = e 2x = e x e x f x) = 4. fx) = e 3x = e 2x e x f x) = 5. fx) = e 4x 3, 4 f x) = 6. fx) = {}}{ 3e 3 x e 3x 3e 3 f x) = 3x 2) = 3 2x = 6x

10 2010 II / fx) = x log x f x) = x) log x + x log x) = log x + x 1 x = log x + 1 // 8. fx) = x 2 log x f x) = 9. fx) = e x log x f x) = 10. fx) = log x α = α log x 06 f x) = α log x) = α 1 x = α x // 11. fx) = log x 1 2 = 1 2 log x f x) = 12. fx) = log x 1 2 3x 5.4 f x) =

11 2010 II / ) lim log 1 + ) 1 = 1 11) e 1 ) lim. 11 = 1 12) log 1 + ) 1 log e = log e e = 1 lim log 1 + ) 1 = 1 12e 1 = k = log 1 + k) 0 k 0 log 1 + k) ) = e 1 k k 0 log e = 1 lim e 1 ) = 1 // 5.2 e x ) = e x 3 ) e x ) = e x. e x ) = lim e x+ e x = e x e e x lim e r e s = e rs = e x e 1 ) lim e 1 ) = e x lim = e x 12 //

12 2010 II / ) log x = x III dy dx = dy du du dx ) log x) = 1 x. u = log e M e u = M fx) = log x = log e x e fx) = x 15) 15 x ) = 1 y = e u u = fx) y u u x dy du = eu du dx = f x) Cain Rule = dy dx = dy du du dx = eu f x) = e fx) f x) = 1 = 16) ) f x) = 1 e fx) 16 = 1 x 15 //

09 II 09/11/ y = e x y = log x = log e x 2. ( e x ) = e x 3. ( ) log x = 1 x 1 Warming Up 1 u = log a M a u = M log a 1 a 0 a 1 a r+s 0 a r

09 II 09/11/ y = e x y = log x = log e x 2. ( e x ) = e x 3. ( ) log x = 1 x 1 Warming Up 1 u = log a M a u = M log a 1 a 0 a 1 a r+s 0 a r 09 II 09/11/16 1 5.6 1. y = e x y = log x = log e x 2. e x ) = e x 3. ) log x = 1 x 1 Warming Up 1 u = log a M a u = M log a 1 a 0 a 1 a r+s 0 a r a s 1 a 2 f g) = f g + f g 1. fx) = x e x f x) = 2. fx)