Math 1A – Chapter 4.5 – 4.9 Test – Fall ’04       Name__________________________

Show your work for credit.  Write all responses on separate paper. Don’t abuse a calculator.

 

1.      Use L’Hôpital’s rule to evaluate the following:

a.

b.

c.

2.      In the figure shown at right, let  be the area of the sector  and let  be the area of triangle .  Recall that the area of a sector is half the product of the radian measure of the subtending angle with the square of the radius and that the area of the triangle is half  sin(θ) times the square of the radius.  Evaluate .

3.      Find the point on the parabola  which is closest to the origin.  Give the x coordinate exactly and approximate y to the nearest hundredth.

4.      Fido, a golden retriever (a dog that swims) is on one side of a circular pool of radius 10 meters.  If Fido knows (instinctively) it can swim at 1 meter per second and run at 1.2 meters per second, it may also instinctively know what angle  it should swim at to minimize the time traveling from A to C.  Note that the central angle  subtended by arc BC is , regardless of where B is. a.       Find the length of the segment  in terms of θ. b.      Find the arc length  in terms of θ.

 

c.       What is Fido’s minimum time going from A to C?

5.      Below is a graph of  .

a.       Draw tangent lines and intercepts on the graph to illustrate to iterations of Newton’s method with an initial value of x₁ = 6.

 

b.      Write the simplified formula you would iterate for Newton’s method for this function.

 

c.       Use a calculator to find the first 7 iterations of Newton’s method with x₁ = 6.  

 

d.       Explain why Newton’s method will converge regardless of the initial value of x.

 

 

6.      What constant acceleration is required to increase the speed of a car
from 5 km/sec to 10 km/sec?

7.      Suppose the graph of  is shown below.  Sketch a possible graph for f(x).