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MATH118FF06N_0

# MATH118FF06N_0 - f x = e 5 x on the interval[0 4 which...

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MATH 118 FINAL EXAM SPRING 2006 1. Short Answer Problems a) What is the power series representation and radius of convergence for f ( x ) = 1 1 - x ? b) Write the formula for dy dx for a parametric curve x ( t ), y ( t ). c) Write the deﬁnition of a Taylor series for a function f ( x ) about a . d) For what values of p does the series n =1 1 n p converge? 2. Evaluate the following integrals. (If the integral is improper, state whether it converges or diverges) a) R 2 xe - x 2 dx b) R 1 ( x - 1)( x - 2) dx c) R 16 - x 2 dx d) R 3 2 x ( x 2 - 4) 3 / 2 dx 3. Consider the curve r = sin θ + cos θ . a) Draw the curve on the axis. Show all steps used to arrive at your graph. b) Find the slope of the tangent line to the curve at the point ( 2 , π 4 ) 4. Find the limit of the following sequences. a) a n = n 2 + n - n b) b 1 = 1 , b n +1 = 6 + b n 5. Determine if the following series converge or diverge. a) n =0 e - n b) n =1 ( - 1) n n ( n +1) c) n =2 ln n n 2 d) n =0 1 1+100 - n 6. Determine the interval and radius of convergence of n =1 1 n ( x + 2) n .

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Math 118 - Final Exam Page 2 of 2 7. Find a Taylor Polynomial about a = 2 to approximate the function
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Unformatted text preview: f ( x ) = e . 5 x on the interval [0 , 4] which guarentees an error of less than 0 . 1. (NOTE: e 2 < 9) 8. Use power series to approximate R 1 / 3 cos( √ x ) dx with error less than 0 . 001. 9. Use power series to ﬁnd the sum of the series ∞ ∑ n =0 n +1 n ! . 10. Solve the following diﬀerential equations. a) y dy dx = xy 2 + x b) ( x 2 + 1) dy dx + 3 xy = 6 ( x 2 +1) 2 c) 2 yy 00 = 1 + ( y ) 2 d) xy 00-y = 0, y (1) = e , y (0) = e 2 11. A 120 litre tank initially contains 90 grams of salt dissolved in 90 litres of water. Brine containing 2 grams/litre of salt ﬂows into the tank at a rate of 4 L/min. At the same time, the tank is perfectly mixed and 3 L/min of the mixture is being drained out of the tank. How much salt does the tank contain when it is full? (Note: ( 3 4 ) 3 ≈ . 4)...
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MATH118FF06N_0 - f x = e 5 x on the interval[0 4 which...

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