Question

5) The displacement, 𝑦(m), of a body in damped oscillation is

𝒚 = 𝟐𝒆 −𝒕 𝐬𝐢𝐧 𝟑𝒕

a) Use the Product Rule to find an equation for the velocity of the object if 𝑣 = 𝑑𝑦 𝑑𝑡

6) You are given the following function which describes the readout taken from an oscilloscope

𝑦 = 3cos (2𝑥)

Your tasks are to:

a) Create a graph showing 2 full periods of the function

b) Take two gradients from the graph

c) Differentiate the function and prove your answers from part b using calculus

d) Compare your answers 

Answer 1

5)

$v(t)=y'(t)=-2e^{-t}sin3t+6e^{-t}cos3t$

6)

a)

b)

gradients from the graph:

at point (1.1, -2):

$\Delta y/\Delta x=-1.5/0.4=-3.75$

at point (5.1, -2):

$\Delta y/\Delta x=1.4/0.4=3.5$

c)

$y'=-6sin(2x)$

$y'(1.1)=-4.85$

$y'(5.1)=4.20$

d)

error at point (1.1, -2):

$\frac{4.85-3.75}{4.85}=22.7\%$

error at point (5.1, -2):

$\frac{4.2-3.5}{4.2}=16.67\%$

to compare results it was used formula for relative error:

$\frac{|gr_t-gr_g|}{gr_t}$

where gr_t is gradient from theoretical calculations,

gr_r is gradient from graph