Hi. Can anybody help me with the last few questions of my HNC Mechanical Engineering homework?

Question

Task 8

A system oscillates with simple harmonic motion according to the equation y = 6 sin(5πt). Differentiate twice to find an expression for acceleration and hence find the acceleration after 0.3 seconds

Task 9

Use logrithmic differentiation to find

dy/dx for y = e^4x sin2x / (x +1)^2

Task 10

find dy/dx when y = 5tan^-1 (4x^2)

task 11

find dy/dx when y = tanh^-1 (2x^3)

Answer 1

I don't intend doing all ur homework for u but i'll help u out...

8) y = 6sin(5pit)

y' = 30picos(5pit)

y'' = -150pi^2sin(5pit)

y''(0.3) = -150pi^2sin(5pi(0.3))

work it out....

9) lny = ln(e^4xsin2x / (x+1)^2)

lny = ln(e^4xsin2x) - ln(x+1)^2

lny = lne^4x + lnsin2x - ln(x+1)^2

lny = 4x +lnsin2x - 2ln(x+1)

dy/dx(lny) = dy/dx[4x + lnsin2x - 2ln(x+1)]

1/y = 4 +2cos2x/sin2x - 2/(x+1)

1/y = 4 + 2cot2x - 2/(x+1)

y = 1 / [4 + 2cot2x - 2/(x+1)]

10) dy/dx(5arctan(4x^2)

= 40x/(1 + (4x^2)^2

= 40x / (1 + 16x^4)