What is the instantaneous velocity of an object moving in accordance to # f(t)= (sin2tcost,tant-sect ) # at # t=(13pi)/12 #?

2 Answers
Salvatore I.
Nov 6, 2016

#abs(v)=2,047#

Explanation:

Let's derive the vectorial function #f(t)# and evaluate it in #t=13pi/12#.
we get
#vec(v)=f'(13pi/12)=(6cos^3(13pi/12)-4cos(13pi/12),1/sin(13pi/12))#
that is
#vec(v)=(-1,54;1,35)#
and its length is #abs(v)=2,047#

Douglas K.
Nov 6, 2016

The instantaneous velocity #("speed", angle)# is #(-0.874, -0.718)#

Explanation:

Differentiate the equation, #y = tan(t) - sec(t)#, with respect to t:

#dy/dt = sec^2(t) - tan(t)sec(t)#

Use the product rule to differentiate the equation, #x = sin(2t)cos(t)#, with respect to t:

#dx/dt = 2cos(2t)cos(t) - sin(2t)sin(t)#

Solve the chain rule, #dy/dt = dy/dxdx/dt# for #dy/dx#

#dy/dx = (dy/dt)/(dx/dt)#

Substitute in the derivatives that we calculated:

#dy/dx = (sec^2(t) - tan(t)sec(t))/(2cos(2t)cos(t) - sin(2t)sin(t))#

The speed will be the above evaluated at #t = (13pi)/12#

#s = -0.874#

The angle with respect to horizontal is the inverse tangent of the speed: #theta = tan^-1(-0.874) = -0.718" radians"#

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