MTH301 Midterm Past Paper 2006

¾ At critical points function can assume o maximum value. o minimum value. o both maximum and minimum value.

o zero value

¾ A plane can be perfectly determined by o one point. o one point and normal vector. o one point and parallel vector. o one normal vector.

¾ The length and width of a rectangle are measured with the errors of at most 4 % and 3%, respectively. Use differentials to approximate the maximum percentage error in the calculated area.

Solution:

Let x = length

y = width

A = area

Then So A xy

A y x

A x y

Therefore,

A A A y x y

y x x y We desire percentage change in A, which is relative change multiplied by 100 so let’s work out relative change first. This is given by A y x x y A A A

x y x y

Since

A xy

1. 04 0.04 x x and 0.03 0.04 y y

The maximum possible value for A A ∂ given the above constraints. This happens

when 0.04 x x and 0.03 y y , giving 0.07 A A . This is relative error, so the

percentage error is 7%.

Convert the Cartesian coordinates (1, 2 , 2) into cylindrical coordinates.

Solution:

We are to convert (x,y,z) int (r, θ ,z)

Given:

x =

y

z

We know that:

y x r

Put value of x and y in the above equation: