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Solutions to the math 106 final exam from december 14, 2006. The exam covers topics such as initial value problems, euler method, power series, improper integrals, partial fractions, and maclaurin series.
Typology: Exams
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Final Exam, page 1. December 14, 2006
1A. Solve the initial value problem
dy dt
et^ + 1 3 y^2 y(0) = 2
1B. Find the Euler method approximation to y(0.2) using a step-size of 4 t = 0.1. Show your work to six digits after the decimal point.
1C. Again to six digits after the decimal point, what is the exact value of y(0.2) as found using the answer to 1A?
Final Exam, page 2. December 14, 2006
2A. Use a series you know for
1 + u
to find a power series in powers of t for
1 + 2t^5
2B. What is the interval of convergence for the series
1 + 2t^5
? Write your answer with 4 decimal places.
2C. Find a series in powers of x for
∫ (^) x
0
1 + 2t^5
dt.
2D. Approximate
0
1 + 2t^5
dt using the first four non-zero terms of the series from 2C. Show your answer to five decimal places.
Final Exam, page 5. December 14, 2006
8 x^2 − 21 x − 32 (x − 2)^2 (x + 4)
dx.
arctan x dx by parts (remember your LIATE).
Final Exam, page 7. December 14, 2006
8A. Find a Maclaurin series for f(x) = cos 3x^2 using a series you know for cosine. Write the first five non-zero terms out explicitly.
8B. Express your answer to 8A in “sigma notation”, i.e., using the
-sign. Start with k = 0.
8C. Find f(16)(0).
8D. Find f(18)(0).
Final Exam, page 10. December 14, 2006
12A. What is the alternating harmonic series?
12B. What does it mean to say the alternating harmonic series is conditionally convergent? Explain fully.
12C. How many terms do you need to add of the alternating harmonic series to guarantee the sum is within 0.001 of whatever it converges to?
