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Mathematics 152, Fall 1999, Final Exam



1. Let R be the region bounded by the parabola y = 7x - x² and the line y = 3x.

a) Find the area of R.

b) Find the volume of the solid generated by rotating R about the y-axis.

c) Set up, but do not evaluate, an integral which represents the volume of the solid generated by rotating R about the x-axis.


2. Let g(x) be the inverse of f(x) = x⁷⁷ + x¹³ . Find g'(2) .


3. Solve for the exact value of x :

ln( x + 9 ) - ln( x ) = 7 .


4. Find the derivative of




5. Evaluate the following:








6. Evaluate




7. A certain population shrinks at a rate proportional to the population at any time t. If the population halves every 140 years, how long does it take for the population to become 1/7 of its original size? (Answer may be left in terms of logarithms.)


8. Test the following for convergence or divergence:







9. For which values of x is the power series



absolutely convergent, conditionally convergent, or divergent?


10. a) Starting with



find a power series for 1 / ( 1 + x⁶ )

b) Use the series found in part a) to calculate



correct to 8 decimal places.


11. Starting with the Maclaurin series for sin x, find the first four nonzero terms in the Maclaurin series for






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