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Exam3Revf06

# Exam3Revf06 - MATH 1431 EXAM 3 REVIEW Let f(x = x2 4x 6...

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MATH 1431 EXAM 3 REVIEW Upper sums, lower sums, Riemann sums Let f ( x ) = x 2 - 4 x + 6 on [ - 1 , 4], and let P = {- 1 , 0 , 1 , 2 , 3 , 4 } be a partition of [ - 1 , 4]. -1 1 2 3 4 x 2 4 6 8 10 y 1. Calculate the lower sum L f ( P ) and the upper sum U f ( P ). 2. Let s 1 , s 2 , s 3 , s 4 be the midpoints o f the subintervals determined by P . Use these values to calculate the corresponding Riemann sum S ( P ). 3. Calculate 4 0 f ( x ) dx . Which of the values in (a) and (b) gives the best approximation of the definite integral? 4. Calculate the average value of f on the interval [0 , 4]. The Fundamental Theorem of Calculus; Properties of the Definite Integral 1. Show that F ( x ) = x 16 + x 2 is an antiderivative for f ( x ) = 16 (16 + x 2 ) 3 / 2 . Use this result to evaluate 3 0 16 (16 + x 2 ) 3 / 2 dx . 2. The function G is defined by: G ( x ) = cos x 1 t 1 - t 2 dt . ( i ) Determine G (2 π ) ( ii ) Determine G ( π/ 6) 3. Evaluate the definite integral 1 0 5 x (1 + x 2 ) 4 dx . 4. Assume that f is a continuous function and that 2 0 f ( x ) dx = 3 , 3 0 f ( x ) dx = 1 , 5 3 f ( x ) dx = 8 .

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Exam3Revf06 - MATH 1431 EXAM 3 REVIEW Let f(x = x2 4x 6...

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