곡선 위의 한 점에 접하는 직선이다. 곡선과 접선이 접하는 점을 접점이라 한다.
함수 y = f ( x ) y = f ( x ) x x Δ x Δ x y y Δ y Δ y Δ y Δ x Δ x Δ y
함수 y = f ( x ) y = f ( x ) x x a a b b Δ x = b − a Δ x = b − a Δ y = f ( b ) − f ( a ) Δ y = f ( b ) − f ( a ) f ( x ) f ( x ) Δ y Δ x = f ( b ) − f ( a ) b − a = f ( a + Δ x ) − f ( a ) Δ x Δ x Δ y = b − a f ( b ) − f ( a ) = Δ x f ( a + Δ x ) − f ( a )
함수 f ( x ) = x 2 f ( x ) = x 2 x x 2 2 4 4
f ( x ) f ( x ) x x 2 2 4 4 Δ x = 4 − 2 = 2 Δ x = 4 − 2 = 2 Δ y = f ( 4 ) − f ( 2 ) Δ y = f ( 4 ) − f ( 2 ) Δ y Δ x = f ( 4 ) − f ( 2 ) 2 Δ x Δ y = 2 f ( 4 ) − f ( 2 )
이때 f ( 4 ) = 4 2 = 16 f ( 4 ) = 4 2 = 1 6 f ( 2 ) = 2 2 = 4 f ( 2 ) = 2 2 = 4 f ( x ) f ( x ) Δ y Δ x = Δ x Δ y = f ( 4 ) − f ( 2 ) 2 = 2 f ( 4 ) − f ( 2 ) = 16 − 4 2 = 2 1 6 − 4 = 12 2 = 2 1 2 = 6 6
함수 y = f ( x ) y = f ( x ) x x a a a + Δ x a + Δ x Δ y Δ x = f ( a + Δ x ) − f ( a ) Δ x Δ x Δ y = Δ x f ( a + Δ x ) − f ( a ) Δ x Δ x 0 0 x = a x = a f ′ ( a ) f ′ ( a )
수식으로 정리하면 f ′ ( a ) = Δ x → 0 Δ y Δ x = Δ x → 0 f ( a + Δ x ) − f ( a ) Δ x f ′ ( a ) = Δ x → 0 lim Δ x Δ y = Δ x → 0 lim Δ x f ( a + Δ x ) − f ( a )
f ′ ( a ) f ′ ( a ) P ( a , f ( a ) ) P ( a , f ( a ) )
Δ x Δ x 0 0 Δ y Δ x Δ x Δ y d y d x d x d y
함수 f ( x ) = x 2 f ( x ) = x 2 f ′ ( 1 ) f ′ ( 1 )
f ′ ( 1 ) = Δ x → 0 Δ y Δ x = Δ x → 0 f ( 1 + Δ x ) − f ( 1 ) Δ x f ′ ( 1 ) = Δ x → 0 lim Δ x Δ y = Δ x → 0 lim Δ x f ( 1 + Δ x ) − f ( 1 )
이때 f ( 1 + Δ x ) = f ( 1 + Δ x ) = ( 1 + Δ x ) 2 = ( 1 + Δ x ) 2 = ( 1 + Δ x ) ( 1 + Δ x ) = ( 1 + Δ x ) ( 1 + Δ x ) = 1 + Δ x + Δ x + ( Δ x ) 2 = 1 + Δ x + Δ x + ( Δ x ) 2 = ( Δ x ) 2 + 2 Δ x + 1 ( Δ x ) 2 + 2 Δ x + 1 f ( 1 ) = 1 2 = 1 f ( 1 ) = 1 2 = 1 f ′ ( 1 ) = f ′ ( 1 ) = Δ x → 0 f ( 1 + Δ x ) − f ( 1 ) Δ x = Δ x → 0 lim Δ x f ( 1 + Δ x ) − f ( 1 ) = Δ x → 0 ( Δ x ) 2 + 2 Δ x + 1 − 1 Δ x = Δ x → 0 lim Δ x ( Δ x ) 2 + 2 Δ x + 1 − 1 = Δ x → 0 ( Δ x ) 2 + 2 Δ x Δ x = Δ x → 0 lim Δ x ( Δ x ) 2 + 2 Δ x = Δ x → 0 ( Δ x + 2 ) = Δ x → 0 lim ( Δ x + 2 ) = 2 2
1번 문항 정답 및 해설 보기
정답: $4$
f ( x ) f ( x ) x x 1 1 3 3 Δ x = 3 − 1 = 2 Δ x = 3 − 1 = 2 Δ y = f ( 3 ) − f ( 1 ) Δ y = f ( 3 ) − f ( 1 ) Δ y Δ x = f ( 3 ) − f ( 1 ) 2 Δ x Δ y = 2 f ( 3 ) − f ( 1 )
이때 f ( 3 ) = f ( 3 ) = 3 3 − 2 = 3 3 − 2 = 27 − 2 = 2 7 − 2 = 25 2 5 f ( 1 ) = f ( 1 ) = 1 3 − 2 = 1 3 − 2 = 1 − 2 = 1 − 2 = − 1 − 1 f ( x ) f ( x ) Δ y Δ x = Δ x Δ y = f ( 3 ) − f ( 1 ) 2 = 2 f ( 3 ) − f ( 1 ) = 25 − ( − 1 ) 2 = 2 2 5 − ( − 1 ) = 25 + 1 2 = 2 2 5 + 1 = 26 2 = 2 2 6 = 13 1 3
따라서 정답은 4 4
2번 문항 정답 및 해설 보기
정답: $2$
f ′ ( 1 ) = Δ x → 0 Δ y Δ x = Δ x → 0 f ( 1 + Δ x ) − f ( 1 ) Δ x f ′ ( 1 ) = Δ x → 0 lim Δ x Δ y = Δ x → 0 lim Δ x f ( 1 + Δ x ) − f ( 1 )
이때 f ( 1 + Δ x ) = 1 1 + Δ x f ( 1 + Δ x ) = 1 + Δ x 1 f ( 1 ) = 1 1 = 1 f ( 1 ) = 1 1 = 1 f ′ ( 1 ) = f ′ ( 1 ) = Δ x → 0 f ( 1 + Δ x ) − f ( 1 ) Δ x = Δ x → 0 lim Δ x f ( 1 + Δ x ) − f ( 1 ) = Δ x → 0 1 1 + Δ x − 1 Δ x = Δ x → 0 lim Δ x 1 + Δ x 1 − 1 = Δ x → 0 ( 1 1 + Δ x − 1 ) ( 1 + Δ x ) Δ x ( 1 + Δ x ) = Δ x → 0 lim Δ x ( 1 + Δ x ) ( 1 + Δ x 1 − 1 ) ( 1 + Δ x ) = Δ x → 0 1 − ( 1 + Δ x ) Δ x ( 1 + Δ x ) = Δ x → 0 lim Δ x ( 1 + Δ x ) 1 − ( 1 + Δ x ) = Δ x → 0 1 − 1 − Δ x Δ x ( 1 + Δ x ) = Δ x → 0 lim Δ x ( 1 + Δ x ) 1 − 1 − Δ x = Δ x → 0 − Δ x Δ x ( 1 + Δ x ) = Δ x → 0 lim Δ x ( 1 + Δ x ) − Δ x = Δ x → 0 − 1 ( 1 + Δ x ) = Δ x → 0 lim ( 1 + Δ x ) − 1 = − 1 − 1
따라서 정답은 2 2
