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Mathematics 141, Spring 1999, Final Exam


1. Evaluate each of the following limits or explain why it does not exist. If a limit is
state this as your answer.




2. Use the Squeeze Theorem to evaluate




3. Use the TABLE menu of your calculator to find a five decimal place estimate of



Make a table in your exam book, displaying enough values to justify your answer.


4. Find the derivative of each of the following functions:






5. Use the definition of the derivative to find f'(x) if f(x) = 1/(3x+1) .


6. Find an equation of the tangent line to the curve xy³ + x³y = 10 at the point (1,2).


7. Sketch a possible graph of a function f that satisfies all of the following conditions:

a) f'(-1) = 0 = f'(2) , f(-1) = -1 = f(2) , f(0) = 0 .

b) f'(x) = 0 for x < -3 , f'(x) < 0 on ( -3 , -1 ) and ( 0 , 2 ) and



c) f''(x) > 0 on ( -3 , 0 ) and ( 0 , 5 ) and f''(x) < 0 for x > 5 .


8. Find the point on the curve

that is closest to the point ( 5 , 0 ) .


9. Consider the function

.

a) Use f'(x) to find all intervals where f is increasing or decreasing.

b) Use f''(x) to find all intervals where f is concave up or concave down.

c) Find all points where f has a relative maximum, relative minimum or a point of inflection.

d) Use appropriate limits to find horizontal and vertical asymptotes (if any) for the graph of f(x).

e) Use the information from a) - d) to sketch the graph of

.




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