Math 1A – Chapter 4 Test II– Fall ’04       Name__________________________

Show your work for credit.  Write all responses on separate paper.

 

1.      Blanca is running around a circular track of radius r = 30 meters at a speed of 3 meters/sec.  Recall that the linear speed v is related to the angular speed ω by v = ωr. 

a.       What is Blanca’s angular velocity, ω?

b.      Recall that Blanca’s position at time t can be modeled by the parametric equations .  If Blanca starts at a bench on the east side of the track and heads north around the track, what is a parameterization of her position at time t?

c.       Write a formula for the square of Blanca’s distance from the bench at time t.

d.      Find the rate of change of Blanca’s distance from the bench when t = 1.

 

 

 

 

2.      A paper cup has the shape of a cone with height 8 cm and radius 2 cm (at the top).  I water is poured into the cup at a rate of 2 cm3/s, how fast is the water levelrising when the water is 6 cm deep?

 

 

3.      Find the point on the ellipse  which is closest to the point (3,5).

4.      An observer stands at a point P, 10 meters above a track.  Two runners start at a point S in the figure and run along the track.  One runner runs twice as fast as the other.  Find the maximum value of the observer’s angle of sight θ between the runners.

 

5.      Use Newton’s method to find all solutions to the equation  correct to twelve decimal places.

6.      Consider  

a.       Explain why g satisfies the conditions of the mean value theorem on .

b.      Set up an equation to find the point(s) in  which, by the mean value theorem are guaranteed to exist.

c.       Use Newton’s method to find a solution to the equation you set up.

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