Math124 practice questions - MATH124 CALCULUS II for Engineers Quiz 1 Lab Section(12 pts Please print names and IDs in ink Family Name INSTRUCTIONS NSID

# Math124 practice questions - MATH124 CALCULUS II for...

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MATH124 CALCULUS II for Engineers Lab Section: Quiz 1 (12 pts) January 14, 2009 Please print names and IDs in ink : NSID: Family Name: First Name: Student ID: INSTRUCTIONS: 1 . Time Limit: 30 minutes 3 . Closed book. Closed notes. No calculators. 2 . No cheating. 4 . Write clearly & legibly. 1. (2 pt) Write 1 2 2 4 + 3 8 4 16 + · · · + ( 1) n 1 n 2 n using sigma notation. = n
i =1 ( 1) i 1 · i 2 i 2. (4 pts) Consider the region A above y = x 2 1, below y = 0. (a) (2 pts) Find an equation for S n and clearly explain your answer. Do not simplify the equation. The region A is symmetric about the y -axis, so we can double the area between x = 0 and x = 1. Divide this interval into n equal subintervals of width 1 n and use the distance 0 ( x 2 1) = 1 x 2 , between y = 0 and y = x 2 1 for the heights of rectangles, to find S n = 2 n
i =1 1 n
1 i 2 n 2
= 2 n
i =1
1 n i 2 n 3
. (b) (2 pts) Given that the equation from (a) can be simplified to S n =
n n ( n + 1)(2 n + 1) 6 n 2
2 n , find the area A . A = lim n →∞ S n = lim n →∞
2 n n 2 n ( n + 1)(2 n + 1) 6 n 3
= lim n →∞
2 ( n + 1)(2 n + 1) 3 n 2
= 2 lim n →∞
2 n 2 + 3 n + 1 3 n 2
= 2 2 3 = 4 3 .
3. (2 pts) Find the closed form value for n
i =1 (3 i + i ). n
i =1 (3 i + i ) = n
i =1 3 i + n
i =1 i = n
i =1 3 i 1 · 3 + n ( n + 1) 2 = 3 · 3 n 1 3 1 + n ( n + 1) 2 = 3 n +1 3 2 + n ( n + 1) 2 = 3 n +1 3 + n 2 + n 2 . 4. (2 pts) Evaluate the lower Riemann sum of f ( x ) = x 2 on [0 , 8] with 4 equal subintervals.
x = 8 0 4 = 2 x 1 = 0, x 2 = 2, x 3 = 4, x 4 = 6 Therefore, L ( f, P 4 ) = 2(0 2 + 2 2 + 4 2 + 6 2 ) = 2(4 + 16 + 36) = 112 5. (2 pts) Evaluate the upper Riemann sum of f ( x ) = cos x on [0 , π ] with 4 equal subintervals.
x = π 0 4 = π 4 x 1 = 0, x 2 = π 4 , x 3 = π 2 , x 4 = 3 π 4 Therefore, U ( f, P 4 ) = π 4 (cos 0 + cos π 4 + cos π 2 + cos 3 π 4 ) = π 4
1 + 2 2 + 0 2 2
= π 4 Quiz Score:
MATH124 CALCULUS II for Engineers Lab Section: Quiz 2 (12 pts) January 21, 2009 Please print names and IDs in ink : NSID: Family Name: First Name: Student ID: INSTRUCTIONS: 1 . Time Limit: 30 minutes 3 . Closed book. Closed notes. No calculators. 2 . No cheating. 4 . Write clearly & legibly. 1. (1 pt) If
5 3 f ( x ) dx = 1 and
5 4 f ( x ) dx = 3, then
4 3 f ( x ) dx =
4 3 f ( x ) dx =
5 3 f ( x ) dx
5 4 f ( x ) dx = 1 ( 3) = 2 2. (3 pts) Evaluate the following integrals using the properties of the definite integral and interpreting integrals as areas. Do not use anti-di ff erentiation. (a) (1 pt)
1 1 x 5 cos x dx = 0 Observe that cos x
= 0 for any x in [ 1 , 1] and that f ( x ) is an odd function over an interval symmetric about zero. Therefore, the integral is zero. (b) (2 pts)
1 2 (3 x + 4) dx The intersection of y = 3 x + 4 with the x -axis is the point
4 3 , 0
. Let A 1 be the area of the right triangle below the x -axis, with sides 2 and 4 3 ( 2) = 2 3 . Then A 1 = 1 2
2 · 2 3
= 2 3 . Let A 2 be the area of the right triangle above the x -axis, with sides 7 and 1
4 3
= 7 3 . Then A 2 = 1 2
7 · 7 3
= 49 6 . Therefore,
1 2 (3 x + 4) dx = A 2 A 1 = 49 6 2 3 = 45 6 = 15 2 . 3. (2 pts) Find the average values of the function f ( x ) = 25 x 2 over the interval [0 , 5]. Average value = 1 5 0
5 0
25 x 2 = 1 5
1 4 π (5) 2
= 5 π 4 .
4. (1 pt) Find f
( x ) for the following function: f ( x ) = cos x +
x 0 sec( t 1) dt .