QCEVault

Differential equations and related rates — Question 231

Original QCE Vault practice · 9 marks

Q231 · Practice questionTechnology-freeComplex unfamiliar9 marks

QUESTION 231 (9 marks)

A bead moves on x2+xy+y2=12x^2+xy+y^2=12. At (2,2), dx/dt=3dx/dt=3 and d2x/dt2=0d^2x/dt^2=0.
The tilted ellipse x squared plus xy plus y squared equals 12, with P=(2,2) marked. Equal coordinate scales.
a)
Determine dy/dx and dy/dt at that instant.
[2 marks]
b)
Determine the second time derivative of y.
[4 marks]
c)
Treat the moving point as a bead. At the same instant, determine the rate of change of its speed and decide whether it is speeding up. Use d∣v∣dt=v⋅a∣v∣\frac{d|\mathbf v|}{dt}=\frac{\mathbf v\cdot\mathbf a}{|\mathbf v|}.
[3 marks]
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