Math 1A – Chapter 3 Test – Fall ’04      Name__________________________

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

 

1.      Use the definition of the derivative to compute each of the following limits.  Follow the form of this example:  

a.

b.

c.

2.      Find equations of the lines tangent to the curve  which are parallel to the line .  Sketch a graph illustrating these tangencies.

3.      Newton’s law of gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass M is
                                                            

a.       Assume the bodies are in motion.  Find dF/dr and write a sentence of two explaining what that means.

b.      Suppose that planet Xorkon attracts an object with a force that decreases at a
rate of 3 N/km when r = 10,000 km.  How fast does the force change when r = 5000 km?

4.      Use the definition of the derivative to simplify .

5.      A curve C is defined by the parametric equations .

a.       Show that C has two tangent lines at the origin:  and find their equations.

b.      Find the points where the tangent line in the x-y plane is vertical.

6.      Find an equation for the line tangent to  at (1,1).

7.      Let . 

a.       Find the differential dy.

b.      Evaluate dy and  if x = 8 and dx =  = 0.1

c.       Estimate  using the line tangent to  at (8,2).  What is the relative error in your estimation?

8.      Find the derivative of the function  by first differentiating .