009A Sample Final A

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This is a sample final, and is meant to represent the material usually covered in Math 9A. Moreover, it contains enough questions to represent a three hour test. An actual test may or may not be similar. Click on the  boxed problem numbers  to go to a solution.


Limits

 Problem 1.   Find the following limits:
   (a)  

   (b)

   (c)  

   (d)  

   (e) 

Derivatives

 Problem 2.   Find the derivatives of the following functions:
   (a)  

   (b)  

   (c)
 

Continuity and Differentiability

 Problem 3.   (Version I) Consider the following function:  
   (a) Find a value of   which makes continuous at
   (b) With your choice of  , is differentiable at ?  Use the definition of the derivative to motivate your answer.

 Problem 3.   (Version II) Consider the following function:  
   (a) Find a value of   which makes continuous at
   (b) With your choice of  , is differentiable at ?  Use the definition of the derivative to motivate your answer.

Implicit Differentiation

 Problem 4.   Find an equation for the tangent line to the function   at the point .

Derivatives and Graphing

 Problem 5.   Consider the function  
   (a) Find the intervals where the function is increasing and decreasing.
   (b) Find the local maxima and minima.
   (c) Find the intervals on which is concave upward and concave downward.
   (d) Find all inflection points.
   (e) Use the information in the above to sketch the graph of .

Asymptotes


Problem 6. Find the vertical and horizontal asymptotes of the function  

Optimization


Problem 7. A farmer wishes to make 4 identical rectangular pens, each with 500 sq. ft. of area. What dimensions for each pen will use the least amount of total fencing?

009A SF A 7 Pens.png

Linear Approximation


Problem 8. (a) Find the linear approximation to the function at the point .
    (b) Use to estimate the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec\,(3\pi/7)} .

Related Rates


Problem 9. A bug is crawling along the -axis at a constant speed of   . How fast is the distance between the bug and the point changing when the bug is at the origin? (Note that if the distance is decreasing, then you should have a negative answer).

Two Important Theorems

Problem 10. Consider the function  
   (a) Use the Intermediate Value Theorem to show that has at least one zero.
   (b) Use Rolle's Theorem to show that has exactly one zero.