009A Sample Final 3

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This is a sample, and is meant to represent the material usually covered in Math 9A for the final. An actual test may or may not be similar.

Click on the  boxed problem numbers  to go to a solution.

 Problem 1 

Find each of the following limits if it exists. If you think the limit does not exist provide a reason.


(b)    given that  


 Problem 2 

Find the derivative of the following functions:



 Problem 3 

Find the derivative of the following function using the limit definition of the derivative:

 Problem 4 

Discuss, without graphing, if the following function is continuous at  

If you think    is not continuous at    what kind of discontinuity is it?

 Problem 5 

Calculate the equation of the tangent line to the curve defined by    at the point,  

 Problem 6 


(a) Over what  -intervals is    increasing/decreasing?

(b) Find all critical points of    and test each for local maximum and local minimum.

(c) Over what  -intervals is    concave up/down?

(d) Sketch the shape of the graph of  

 Problem 7 





 Problem 8 

Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure    and volume    satisfy the equation    where    is a constant. Suppose that at a certain instant, the volume is    the pressure is    and the pressure is increasing at a rate of    At what rate is the volume decreasing at this instant?

 Problem 9 


(a) Find all critical points of    over the  -interval  

(b) Find absolute maximum and absolute minimum of    over  

 Problem 10 


(a) Find the differential    of    at  

(b) Use differentials to find an approximate value for    Hint: