Tiêu đề MSTE 15 Solutions.pdf ✅

• A. By power series 1 1 − x = 1 + x + x 2 + x 3 + ⋯ − ln(1 − x) = x + 1 2 x 2 + 1 3 x 3 + ⋯ ln(1 − x) = −x − 1 2 x 2 − 1 3 x 3 − ⋯ ln[1 − (1 − x)] = −(1 − x) − 1 2 (1 − x) 2 − 1 3 (1 − x) 3 − ⋯ ln x = −(1 − x) − 1 2 (1 − x) 2 − 1 3 (1 − x) 3 − ⋯ limx→1 x 3 − 3x + 2 1 − x + ln x = limx→1 (x − 1) 2 (x + 2) (1 − x) + [−(1 − x) − 1 2 (1 − x) 2 − 1 3 (1 − x) 3 − ⋯ ] = limx→1 (x − 1) 2 (x + 2) − 1 2 (1 − x) 2 − 1 3 (1 − x) 3 − ⋯ = limx→1 x + 2 − 1 2 − 1 3 (1 − x) − ⋯ = 1 + 2 − 1 2 = −6 • B. By L’Hopital’s Rule limx→1 x 3 − 3x + 2 1 − x + ln x = limx→1 3x 2 − 3 −1 + 1 x = limx→1 6x − 1 x 2 = 6(1) − 1 1 2 = −6 SITUATION 2. Find the first derivative of the following functions ▣ 5. 2 sin2 x + cos 2x [SOLUTION] y = 2 sin2 x + (1 − 2 sin2 x) y = 1
y ′ = 0 ▣ 6. 5 3x [SOLUTION] y = 5 3x y ′ = (5 3x ) ln 5 (3) y ′ = (3 ln 5)5 3x ▣ 7. y = x x [SOLUTION] y = x x ln y = ln x x ln y = x ln x y ′ y = (1) ln x + x ( 1 x ) y ′ y = ln x + 1 y ′ = y(ln x + 1) y′ = x x (ln x + 1) ▣ 8. Find the slope of the curve y = x 3 + 3x + 14 where the curve crosses the x-axis. [SOLUTION] When the curve crosses the x-axis, x 3 + 3x + 14 = 0 x = −2 y ′ = 3x 2 + 3 y ′ = 3(−2) 2 + 3 y ′ = 15 ▣ 9. Find the slope of y(3x − y 2 ) = 10 at the point (3,2). [SOLUTION] Check if the point is within the curve. 2(3(3) − 2 2 ) = 10 10 = 10 → ok y(3x − y 2 ) = 10 3xy − y 3 = 10 3(xy ′ + y) − 3y 2y ′ = 0 3xy ′ + 3y − 3y 2y ′ = 0 (3x − 3y 2 )y ′ = −3y
y ′ = −3y 3x − 3y 2 At (3,2), y ′ = −3(2) 3(3) − 3(2) 2 y ′ = 2 ▣ 10. Find the equation of the tangent line of y = sin x cos x at x = π 6 . [SOLUTION] y = sin x cos x y = 1 2 sin 2x y ′ = cos 2x At x = π 6 , y = 1 2 sin 2 ( π 6 ) = √3 4 y ′ = cos 2 ( π 6 ) = 1 2 The equation is y − √3 4 = 1 2 (x − π 6 ) y = 1 2 x − π 12 + √3 4