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School: JMU
Course: ELEM STATISTICS
Department of Mathematics and Statistics Math 220 (Section 21 and Section 22): Spring 2014 Instructor: Dr. Hasan Hamdan Course Website: Course is on Canvas Email: hamdanhx@jmu.edu Phone: 540.568.2844 Sections: 21 & 22 Class Location: Tuesday Roop 127 and
School: JMU
School: JMU
Course: Calculus II
H OMEWORK 5 5.5.44. To solve becomes (x2 + 4)3/2 dx we let x = 2 tan so dx = 2 sec2 d. The integral 1 1 (sec )1 d = 4 4 (2 sec )3 (2 sec2 d) = = 5.5.50. To solve becomes cos d = 1 sin + C 4 1 1 x sin(tan1 (x/2) + C = + C. 4 4 x2 + 4 x2 4 2x dx we let x =
School: JMU
Course: Calculus II
H OMEWORK 8 6.3.22. We can approximate this by 1 + (e0 e1 )2 + 1 + (e1 e0 )2 + 1 + (e2 e1 )2 + 1 + (e3 e2 )2 . 6.3.32. Since f (x) = 1, the arclength is given by 1 + (1)2 dx = 3 2. 5 2 6.3.38. Since f (x) = (1/2)(1 x2 )1/2 (2x), the arclength is given by
School: JMU
Course: Calculus II
H OMEWORK 6 5.7.10. In this situation, we can rewrite the integral as b c b f (x) dx. f (x) dx + f (x) dx = c a a Each of the integrands on the right side is monotonic, so we can use LEF T (n) or RIGHT (n) sums to evaluate each integral. On the rst integr
School: JMU
Course: Calculus II
H OMEWORK 10 7.2.34. We have k2 2 2k 2 = lim = lim = 1 2 + 2k + 2 k k k 2k + 2 k 2 by LHopitals rule. Since the sequence converges, it is bounded above and below. To see this, let = 1. Then the limit statement tells us that there exists an N > 0 such that
School: JMU
Course: Calculus II
H OMEWORK 5 5.5.44. To solve becomes (x2 + 4)3/2 dx we let x = 2 tan so dx = 2 sec2 d. The integral 1 1 (sec )1 d = 4 4 (2 sec )3 (2 sec2 d) = = 5.5.50. To solve becomes cos d = 1 sin + C 4 1 1 x sin(tan1 (x/2) + C = + C. 4 4 x2 + 4 x2 4 2x dx we let x =
School: JMU
Course: Calculus II
H OMEWORK 8 6.3.22. We can approximate this by 1 + (e0 e1 )2 + 1 + (e1 e0 )2 + 1 + (e2 e1 )2 + 1 + (e3 e2 )2 . 6.3.32. Since f (x) = 1, the arclength is given by 1 + (1)2 dx = 3 2. 5 2 6.3.38. Since f (x) = (1/2)(1 x2 )1/2 (2x), the arclength is given by
School: JMU
Course: Calculus II
H OMEWORK 6 5.7.10. In this situation, we can rewrite the integral as b c b f (x) dx. f (x) dx + f (x) dx = c a a Each of the integrands on the right side is monotonic, so we can use LEF T (n) or RIGHT (n) sums to evaluate each integral. On the rst integr
School: JMU
Course: Calculus II
H OMEWORK 10 7.2.34. We have k2 2 2k 2 = lim = lim = 1 2 + 2k + 2 k k k 2k + 2 k 2 by LHopitals rule. Since the sequence converges, it is bounded above and below. To see this, let = 1. Then the limit statement tells us that there exists an N > 0 such that
School: JMU
Course: Calculus II
H OMEWORK 7 1 b 6.2.12. Fitting 0 2xx2 dx to the general formula a (radius)(height) dx, we nd that the radius of the shell is x and the height is x2 . Hence, the region is bounded by y = x2 and the x-axis, from x = 0 to x = 1, revolved around the y -axis.
School: JMU
Course: Calculus II
Name: Q UIZ 5 Please write your solutions in complete sentences, as simply as you can, and justify each step. 1. Let h and r be positive constants. Consider the region R bounded by y = h x, y = h, r and the y -axis. Use the shell method to nd the volume o
School: JMU
Course: Calculus 1
MATH235 Calculus 1 Quiz 1 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. Solve the following inequality : 3 2 < . x1 x+1 3 2 < 0, x1 x+1 3(x + 1) 2(x 1) < 0
School: JMU
Course: Calculus 1
MATH 235 Calculus 1 Quiz 6 Solution 1. Find the following values. 1 a. sin1 ( ) = . 2 6 1 3 b. cos1 ( ) = 4 2 Note that for both part a and part b, there is one and only one answer, since if you plug in any x-value into sin1 the function value will always
School: JMU
Course: Calculus 1
MATH 235 Calculus 1 Quiz 5 1. At time t, the position of a body moving along the s-axis is s = t3 6t2 + 9t. Time is given in seconds and distance is given in meters. a. Find the bodys acceleration each time the velocity is zero. Since v (t) = s (t) = 3t2
School: JMU
Course: Calculus 1
MATH 235 Calculus 1 Quiz 4 10/04/2010 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. y = ( 4 t 2 dy ) . Find . t+1 dt dy 4 t 3 d 4 t = 2( ) ( ) dt t+1 dt t
School: JMU
Course: Calculus 1
MATH235 Calculus 1 Quiz 2 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. Find the slope of the line tangent to the curve f (x) = x2 2x 3 at the point (2, 3)
School: JMU
Course: Calculus 1
Name : MATH 235 Calculus 1 Quiz 3 09/27/2010 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. By using the precise denition of limits, prove that limx1 f (x)
School: JMU
Course: Calculus 1
Worksheet 6 MATH 235 10/21/2010 1. (The Derivative Rule for Inverses.) Let f be a continuous one-to-one function dened on an interval. Suppose f is differentiable at x = a and f (a) = 0. If f (a) = b, show that (f 1 ) (b) exists and that 1 ( f 1 ) ( b) =
School: JMU
Course: Calculus 1
Worksheet 5 MATH 235 10/07/2010 Discuss the following problems with your group and write down a complete solution. Show all work. 1. Prove the Product Rule: if f and g are differentiable, then so is their product P (x) = f (x)g (x), and P ( x) = f ( x) g
School: JMU
Course: Calculus 1
Name: Worksheet 4 MATH 235 Fall, 2010. Discuss the following problems with your group and write down a complete solution. Show all work. 1. Prove that if f is differentiable at x = c, then f is continuous at x = c. Proof. To show that f is continuous at x
School: JMU
Course: Calculus II
H OMEWORK 1 4.3.26. The integrand is the upper half circle of radius r centered at 0. Here, r is a constant and x is the variable of integration since it appears in the differential dx. Hence, r r2 x2 dx r is 1 2 of the area of a circle of radius r. This
School: JMU
Course: Calculus II
H OMEWORK 2 4.7.2 (c) We cannot anti-differentiate this function directly. However, we can use the second Fundamental Theorem of Calculus to write x d 2 2 esin t dt = esin x , dx 0 x d 2 2 esin t dt = esin x , dx 1 x d 2 2 esin t dt = esin x , dx 2 x sin2
School: JMU
Course: Calculus II
H OMEWORK 3 1 1 5.2.24. If we let u = ln x and dv = x dx then we have du = x dx and v = ln x. Integration by parts gives ln x ln x dx = (ln x)2 dx. x x If we rearrange by combining the integrals, then we get ln x dx = (ln x)2 + C x 2 so ln x (ln x)2 dx =
School: JMU
Course: ELEM STATISTICS
Department of Mathematics and Statistics Math 220 (Section 21 and Section 22): Spring 2014 Instructor: Dr. Hasan Hamdan Course Website: Course is on Canvas Email: hamdanhx@jmu.edu Phone: 540.568.2844 Sections: 21 & 22 Class Location: Tuesday Roop 127 and
School: JMU
School: JMU
Course: Stats
Math 220 Syllabus Spring 2008 1. Class Information: Section 8 : M,W 10:10 AM to 11:00 AM, Burruss 141 F 10:10 AM to 11:00 AM, Roop 127 MW 11:15 AM to 12:05 PM , Roop 141 F 11:15 AM to 12:05 PM, Roop 127 Section 11: Section 16: M,W 1:25 PM to 2:1
School: JMU
Course: ELEM STATISTICS
Department of Mathematics and Statistics Math 220 (Section 21 and Section 22): Spring 2014 Instructor: Dr. Hasan Hamdan Course Website: Course is on Canvas Email: hamdanhx@jmu.edu Phone: 540.568.2844 Sections: 21 & 22 Class Location: Tuesday Roop 127 and
School: JMU
School: JMU
Course: Calculus II
H OMEWORK 5 5.5.44. To solve becomes (x2 + 4)3/2 dx we let x = 2 tan so dx = 2 sec2 d. The integral 1 1 (sec )1 d = 4 4 (2 sec )3 (2 sec2 d) = = 5.5.50. To solve becomes cos d = 1 sin + C 4 1 1 x sin(tan1 (x/2) + C = + C. 4 4 x2 + 4 x2 4 2x dx we let x =
School: JMU
Course: Calculus II
H OMEWORK 8 6.3.22. We can approximate this by 1 + (e0 e1 )2 + 1 + (e1 e0 )2 + 1 + (e2 e1 )2 + 1 + (e3 e2 )2 . 6.3.32. Since f (x) = 1, the arclength is given by 1 + (1)2 dx = 3 2. 5 2 6.3.38. Since f (x) = (1/2)(1 x2 )1/2 (2x), the arclength is given by
School: JMU
Course: Calculus II
H OMEWORK 6 5.7.10. In this situation, we can rewrite the integral as b c b f (x) dx. f (x) dx + f (x) dx = c a a Each of the integrands on the right side is monotonic, so we can use LEF T (n) or RIGHT (n) sums to evaluate each integral. On the rst integr
School: JMU
Course: Calculus II
H OMEWORK 10 7.2.34. We have k2 2 2k 2 = lim = lim = 1 2 + 2k + 2 k k k 2k + 2 k 2 by LHopitals rule. Since the sequence converges, it is bounded above and below. To see this, let = 1. Then the limit statement tells us that there exists an N > 0 such that
School: JMU
Course: Calculus II
H OMEWORK 7 1 b 6.2.12. Fitting 0 2xx2 dx to the general formula a (radius)(height) dx, we nd that the radius of the shell is x and the height is x2 . Hence, the region is bounded by y = x2 and the x-axis, from x = 0 to x = 1, revolved around the y -axis.
School: JMU
Course: Calculus II
Name: Q UIZ 5 Please write your solutions in complete sentences, as simply as you can, and justify each step. 1. Let h and r be positive constants. Consider the region R bounded by y = h x, y = h, r and the y -axis. Use the shell method to nd the volume o
School: JMU
Course: Calculus II
H OMEWORK 9 6.5.26. We have 1 dy = sin x y dx so integrating both sides with respect to x, we obtain 1 dy dx = sin x dx y dx ln |y | = cos x + C |y | = e cos x eC y = eC e cos x . If we let A represent the constant, we obtain y (x) = Ae cos x . 6.5.34. Se
School: JMU
Course: Calculus II
H OMEWORK 4 5.3.24. The integrand is a proper rational function so we can apply partial fractions. The rational function has the form 25x2 B C A + + = . 2 (x + 3)(x 2) x + 3 x 2 (x 2)2 Clearing denominators, we obtain 25x2 = A(x 2)2 + B (x + 3)(x 2) + C (
School: JMU
Course: Calculus 1
MATH235 Calculus 1 Quiz 1 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. Solve the following inequality : 3 2 < . x1 x+1 3 2 < 0, x1 x+1 3(x + 1) 2(x 1) < 0
School: JMU
Course: Calculus 1
MATH 235 Calculus 1 Quiz 6 Solution 1. Find the following values. 1 a. sin1 ( ) = . 2 6 1 3 b. cos1 ( ) = 4 2 Note that for both part a and part b, there is one and only one answer, since if you plug in any x-value into sin1 the function value will always
School: JMU
Course: Calculus 1
MATH 235 Calculus 1 Quiz 5 1. At time t, the position of a body moving along the s-axis is s = t3 6t2 + 9t. Time is given in seconds and distance is given in meters. a. Find the bodys acceleration each time the velocity is zero. Since v (t) = s (t) = 3t2
School: JMU
Course: Calculus 1
MATH 235 Calculus 1 Quiz 4 10/04/2010 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. y = ( 4 t 2 dy ) . Find . t+1 dt dy 4 t 3 d 4 t = 2( ) ( ) dt t+1 dt t
School: JMU
Course: Calculus 1
MATH235 Calculus 1 Quiz 2 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. Find the slope of the line tangent to the curve f (x) = x2 2x 3 at the point (2, 3)
School: JMU
Course: Calculus 1
Name : MATH 235 Calculus 1 Quiz 3 09/27/2010 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. By using the precise denition of limits, prove that limx1 f (x)
School: JMU
Course: Calculus 1
MATH235 Calculus 1 Quiz 7 11/01/2010 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. Estimate the volume of material in a cylindrical shell with height 30in.
School: JMU
Course: Calculus 1
MATH235 Calculus 1 Quiz 8 11/10/2010 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. Find the absolute extreme values of f (x) = ln(cos x) on [ , ]. 43 f (x)
School: JMU
Course: Calculus 1
Worksheet 6 MATH 235 10/21/2010 1. (The Derivative Rule for Inverses.) Let f be a continuous one-to-one function dened on an interval. Suppose f is differentiable at x = a and f (a) = 0. If f (a) = b, show that (f 1 ) (b) exists and that 1 ( f 1 ) ( b) =
School: JMU
Course: Calculus 1
Worksheet 5 MATH 235 10/07/2010 Discuss the following problems with your group and write down a complete solution. Show all work. 1. Prove the Product Rule: if f and g are differentiable, then so is their product P (x) = f (x)g (x), and P ( x) = f ( x) g
School: JMU
Course: Calculus 1
Name: Worksheet 4 MATH 235 Fall, 2010. Discuss the following problems with your group and write down a complete solution. Show all work. 1. Prove that if f is differentiable at x = c, then f is continuous at x = c. Proof. To show that f is continuous at x
School: JMU
Course: Calculus 1
MATH235 Calculus 1 Proof of the Squeeze Theorem. Theorem 0.1 (The Squeeze Theorem). Suppose that g (x) f (x) h(x) for all x in some open interval containing c except possibly at c itself. If limxc g (x) = L = limxc h(x) then limxc f (x) = L. Proof. Let >
School: JMU
Course: Calculus 1
MATH235 Calculus 1 Homework problems on the Squeeze Theorem. 09/08/2010 Use the Squeeze Theorem to solve the following problems: 2 1. Find limx0 x4 cos( ). x Since the range of cosine is always between 1 and 1, 2 1 cos( ) 1. x By multiplying x4 0 to each
School: JMU
Course: Calculus II
H OMEWORK 1 4.3.26. The integrand is the upper half circle of radius r centered at 0. Here, r is a constant and x is the variable of integration since it appears in the differential dx. Hence, r r2 x2 dx r is 1 2 of the area of a circle of radius r. This
School: JMU
Course: Calculus II
H OMEWORK 2 4.7.2 (c) We cannot anti-differentiate this function directly. However, we can use the second Fundamental Theorem of Calculus to write x d 2 2 esin t dt = esin x , dx 0 x d 2 2 esin t dt = esin x , dx 1 x d 2 2 esin t dt = esin x , dx 2 x sin2
School: JMU
Course: Calculus II
H OMEWORK 3 1 1 5.2.24. If we let u = ln x and dv = x dx then we have du = x dx and v = ln x. Integration by parts gives ln x ln x dx = (ln x)2 dx. x x If we rearrange by combining the integrals, then we get ln x dx = (ln x)2 + C x 2 so ln x (ln x)2 dx =
School: JMU
Course: Calculus II
Name: Q UIZ 1 Please write your solutions in complete sentences, as simply as you can, and justify each step. 1. Evaluate 2 4 |2x| dx 2 using geometry (not the Fundamental Theorem of Calculus). 2. Find 4x3 dx. x4 5 3. Use the Fundamental Theorem of Calcul
School: JMU
Course: Calculus II
Name: D IAGNOSTIC Q UIZ 1. Factor x4 16 as much as possible. 2. Prove that x = 1 is the only real solution of x3 1 = 0. 3. Simplify 1 2x3 x2 + 4 (3x1/2 )3 x x7 p/q to show that f (x) is a function of the form Ax . f (x) = 4. List all values of x in [0, ]
School: JMU
Course: Calculus II
Name: Q UIZ 2 Please write your solutions in complete sentences, as simply as you can, and justify each step. 1. Find d dx 2. Find 0 2sin t dt. x 1 dx. 1 x2 3. It looks as if we can integrate the function 2 sin x cos x in two different ways: (a) Let u = s
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 3 DUE: Friday, Jan. 22 NAME: DIRECTIONS: Attach this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will loose points if you work is not in order. When required, do not forget the units! C
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 2 DUE: Friday, Jan. 15 NAME: DIRECTIONS: Attach this page to the front of your homework (dont forget your name!). Show all work, clearly and in order. When required, do not forget the units! Circle your nal answers. You will loose points if y
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 1 MATH 200 NAME: DIRECTIONS: Attach this page to the front of your homework (dont forget your name!). Please read article A Guide to Writing Mathematics by Dr. K. P. Lee which can be found on the class website, homework page. Note, homeworks in
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 4 DUE: Friday, Jan. 29 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will loose points if you work is not in order. When required, do not forget the units! C
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 5 DUE: Friday, Feb. 5 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will loose points if you work is not in order. When required, do not forget the units! Ci
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 10 DUE: Wed., Mar. 10 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Cir
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 9 DUE: Fri., Mar. 5 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circl
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 8 DUE: Fri., Feb. 26 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circ
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 6 DUE: Wed., Feb. 10 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circ
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 7 DUE: Fri., Feb. 19 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circ
School: JMU
Course: Mathematical Methods For The Physical Sciences II
HOMEWORK 1 DUE: Fri., Apr. 2 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circl
School: JMU
Course: Mathematical Methods For The Physical Sciences II
HOMEWORK 4 DUE: Fri., Apr. 23 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circ
School: JMU
Course: Mathematical Methods For The Physical Sciences II
HOMEWORK 3 DUE: Fri., Apr. 16 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circ
School: JMU
Course: Mathematical Methods For The Physical Sciences II
HOMEWORK 2 DUE: Fri., Apr. 9 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circl
School: JMU
Course: Mathematical Methods For The Physical Sciences II
HOMEWORK 5 DUE: Fri., Apr. 30 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circ
School: JMU
Course: Mathematical Methods For The Physical Sciences II
HOMEWORK 6 DUE: Fri., May 7 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circle
School: JMU
Course: Calculus II
H OMEWORK 5 5.5.44. To solve becomes (x2 + 4)3/2 dx we let x = 2 tan so dx = 2 sec2 d. The integral 1 1 (sec )1 d = 4 4 (2 sec )3 (2 sec2 d) = = 5.5.50. To solve becomes cos d = 1 sin + C 4 1 1 x sin(tan1 (x/2) + C = + C. 4 4 x2 + 4 x2 4 2x dx we let x =
School: JMU
Course: Calculus II
H OMEWORK 8 6.3.22. We can approximate this by 1 + (e0 e1 )2 + 1 + (e1 e0 )2 + 1 + (e2 e1 )2 + 1 + (e3 e2 )2 . 6.3.32. Since f (x) = 1, the arclength is given by 1 + (1)2 dx = 3 2. 5 2 6.3.38. Since f (x) = (1/2)(1 x2 )1/2 (2x), the arclength is given by
School: JMU
Course: Calculus II
H OMEWORK 6 5.7.10. In this situation, we can rewrite the integral as b c b f (x) dx. f (x) dx + f (x) dx = c a a Each of the integrands on the right side is monotonic, so we can use LEF T (n) or RIGHT (n) sums to evaluate each integral. On the rst integr
School: JMU
Course: Calculus II
H OMEWORK 10 7.2.34. We have k2 2 2k 2 = lim = lim = 1 2 + 2k + 2 k k k 2k + 2 k 2 by LHopitals rule. Since the sequence converges, it is bounded above and below. To see this, let = 1. Then the limit statement tells us that there exists an N > 0 such that
School: JMU
Course: Calculus II
H OMEWORK 7 1 b 6.2.12. Fitting 0 2xx2 dx to the general formula a (radius)(height) dx, we nd that the radius of the shell is x and the height is x2 . Hence, the region is bounded by y = x2 and the x-axis, from x = 0 to x = 1, revolved around the y -axis.
School: JMU
Course: Calculus II
Name: Q UIZ 5 Please write your solutions in complete sentences, as simply as you can, and justify each step. 1. Let h and r be positive constants. Consider the region R bounded by y = h x, y = h, r and the y -axis. Use the shell method to nd the volume o
School: JMU
Course: Calculus II
H OMEWORK 9 6.5.26. We have 1 dy = sin x y dx so integrating both sides with respect to x, we obtain 1 dy dx = sin x dx y dx ln |y | = cos x + C |y | = e cos x eC y = eC e cos x . If we let A represent the constant, we obtain y (x) = Ae cos x . 6.5.34. Se
School: JMU
Course: Calculus II
H OMEWORK 4 5.3.24. The integrand is a proper rational function so we can apply partial fractions. The rational function has the form 25x2 B C A + + = . 2 (x + 3)(x 2) x + 3 x 2 (x 2)2 Clearing denominators, we obtain 25x2 = A(x 2)2 + B (x + 3)(x 2) + C (
School: JMU
Course: Elementary Statistics
standard error Recall x = n . s standard error of x = n . exercise 16.1. x Recall z = /n Normal (0, 1). Now t = x s/ n has a t distribution with n 1 degrees of freedom. (James Madison University) April 1, 2013 1/9 Condence interval level C condence inte
School: JMU
Course: Elementary Statistics
Probabilities chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run. We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a la
School: JMU
Course: Elementary Statistics
Sampling distributions A parameter is a number that describes the population. A statistic is a number that can be computed from the sample data. In practice, we often use a statistic to estimate an unknown parameter. e.g., sample mean x is a statistic, a
School: JMU
Course: Elementary Statistics
General rules about probability For any event A, 0 P (A) 1. If S is the sample space, P (S ) = 1. If A an B are disjoint, P (AorB ) = P (A) + P (B ). P (notA) = 1 P (A). (James Madison University) May 22, 2012 1 / 10 multiplication rule If A and B are ind
School: JMU
Course: Elementary Statistics
Conditions for inference Where did data come from? If data do not come from a random sample or a randomized comparative experiment, the conclusions my be suspect. What is the shape of the population distribution? outliers can distort the results of infere
School: JMU
Course: Elementary Statistics
Introduction to Inference Statistical inference draws conclusions about a population from sample data. Conditions about estimating a population mean : We have an SRS from the population. The variable has a Normal distribution N (, ). We do not know . But
School: JMU
Course: Elementary Statistics
Experiments An observational study observes individuals and measures variables of interest but does not attempt to inuence the responses. An experiment deliberately imposes some treatment on individuals in order to observe their responses. The purpose of
School: JMU
Course: Elementary Statistics
Sampling The population is the entire group of individuals about which we want information. A sample is a part of the population from which we actually collect information. A sampling design describes how to choose a sample from the population. exercise:
School: JMU
Course: Elementary Statistics
individuals and variables individualsobjects described by a data set. variableany characteristic of an individual. e.g, age, height, gender. A categorical variableplaces an individual into one of several groups or categories. A quantitative variable takes
School: JMU
Course: Elementary Statistics
the mean 1 x = x1 +x2 +xn = n xi . n The mean is sensitive to extreme values. (James Madison University) August 30, 2011 1/9 the median The median M is the midpoint of a distribution. To nd the median of a distribution: Arrange all observations from small
School: JMU
Course: Elementary Statistics
density curves gure 3.1 p 52. A density curve is is always on or above the horizontal axis, and has area exactly 1 underneath it. A density curve describes the overall pattern of a distribution. (James Madison University) September 4, 2013 1 / 12 A Normal
School: JMU
Course: Elementary Statistics
regression line A regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes. A straight line has the form y = a + bx . (James Madison University) January 30, 2012 1 / 13 Least squares regressi
School: JMU
Course: Elementary Statistics
response variable and explanatory variable A response variable measures an outcome of a study. An explanatory variable may explain or inuence changes in a response variable. (James Madison University) September 12, 2011 1/5 scatterplot The most useful gra
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MATH 387 COMPUTERS & NUMERICAL ALGORITHMS Instructor: Dr. Caroline Smith Room B112 Phone x2922 E-mail: smithc@math.jmu.edu Office Hours: 10.00 11:00 Monday, Wednesday & Friday, or by appointment. Classes Meet: MWF @ 8.00 8.50 am in Burruss 34 Te
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Time Series Plot: a simple graph of data collected over time that can be invaluable in identifying trends or patterns that might be of interest. Life Expectancy over Time Year Life Expectancy over Time 1940 62.9 1950 68.2 1960 69.7 1970 70.8 1980 73.
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Inductive and deductive reasoning method of successive differences to find out the last number in a set personal finance convert bases truth tables negate a conditional statement
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Finding the Value of x Given a Proportion. Draw a picture, with the area given shaded on it. Look up the cumulative area in the middle of the z table, and look at the margins to find the zscore corresponding to that area. Use the z-score formula to s
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6.1 How Can We Summarize Possible Outcomes and Their Probabilities? Random Variable its value is a numerical outcome of a random phenomenon. We use letters such as X and Y to denote a RV. Examples: Toss a coin 10 times, record number of heads. Roll
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Quiz: A professor regularly gives multiple choice quizzes with 5 questions. Over time, he has found the distribution of the number of wrong answers on his quizzes is as follows x 0 1 2 3 4 5 x 0 1 2 3 4 5 Variance of a discrete random variable x: St
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7.3: How Can We Construct a Confidence Interval to Predict a Population Mean One-sample z Confidence Interval for : The general formula for a confidence interval for a population mean when 1. x = is the sample mean from a random sample. 2. the sampl
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8.3: Significance Tests about Means To test the hypothesis H0: = hypothesized mean, we will compute a t statistic x t s/ n The approximate P-value for this test is found using a t random variable with degrees of freedom df = n-1. H0: = hypothesize
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Confidence Intervals: Continued In general: Bigger Confidence -> Bigger Interval Bigger Sample Size -> Smaller Interval If you want to have a lot of confidence, but not a huge interval, increase the sample size. Examples: For each of the following st
School: JMU
Course: Calculus 1
MATH235 Calculus 1 Quiz 1 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. Solve the following inequality : 3 2 < . x1 x+1 3 2 < 0, x1 x+1 3(x + 1) 2(x 1) < 0
School: JMU
Course: Calculus 1
MATH 235 Calculus 1 Quiz 6 Solution 1. Find the following values. 1 a. sin1 ( ) = . 2 6 1 3 b. cos1 ( ) = 4 2 Note that for both part a and part b, there is one and only one answer, since if you plug in any x-value into sin1 the function value will always
School: JMU
Course: Calculus 1
MATH 235 Calculus 1 Quiz 5 1. At time t, the position of a body moving along the s-axis is s = t3 6t2 + 9t. Time is given in seconds and distance is given in meters. a. Find the bodys acceleration each time the velocity is zero. Since v (t) = s (t) = 3t2
School: JMU
Course: Calculus 1
MATH 235 Calculus 1 Quiz 4 10/04/2010 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. y = ( 4 t 2 dy ) . Find . t+1 dt dy 4 t 3 d 4 t = 2( ) ( ) dt t+1 dt t
School: JMU
Course: Calculus 1
MATH235 Calculus 1 Quiz 2 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. Find the slope of the line tangent to the curve f (x) = x2 2x 3 at the point (2, 3)
School: JMU
Course: Calculus 1
Name : MATH 235 Calculus 1 Quiz 3 09/27/2010 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. By using the precise denition of limits, prove that limx1 f (x)
School: JMU
Course: Calculus 1
MATH235 Calculus 1 Quiz 7 11/01/2010 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. Estimate the volume of material in a cylindrical shell with height 30in.
School: JMU
Course: Calculus 1
MATH235 Calculus 1 Quiz 8 11/10/2010 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. Find the absolute extreme values of f (x) = ln(cos x) on [ , ]. 43 f (x)
School: JMU
Course: Calculus II
Name: Q UIZ 1 Please write your solutions in complete sentences, as simply as you can, and justify each step. 1. Evaluate 2 4 |2x| dx 2 using geometry (not the Fundamental Theorem of Calculus). 2. Find 4x3 dx. x4 5 3. Use the Fundamental Theorem of Calcul
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Course: Calculus II
Name: D IAGNOSTIC Q UIZ 1. Factor x4 16 as much as possible. 2. Prove that x = 1 is the only real solution of x3 1 = 0. 3. Simplify 1 2x3 x2 + 4 (3x1/2 )3 x x7 p/q to show that f (x) is a function of the form Ax . f (x) = 4. List all values of x in [0, ]
School: JMU
Course: Calculus II
Name: Q UIZ 2 Please write your solutions in complete sentences, as simply as you can, and justify each step. 1. Find d dx 2. Find 0 2sin t dt. x 1 dx. 1 x2 3. It looks as if we can integrate the function 2 sin x cos x in two different ways: (a) Let u = s
School: JMU
Course: Introduction To Real Analysis And Linear Algebra
QUIZ 2 WED., APR. 4 NAME: DIRECTIONS: No papers, phones, calculators, or gadgets are permitted to be out during the quiz. Show all work, clearly and in order You will lose points if any of these instructions are not followed. Questions Points 1 1.5 2 1.
School: JMU
Course: Introduction To Real Analysis And Linear Algebra
QUIZ 1 WED., MAR. 28 NAME: DIRECTIONS: No papers, phones, calculators, or gadgets are permitted to be out during the quiz. Show all work, clearly and in order You will lose points if any of these instructions are not followed. Questions Points 1 1 2 2 3
School: JMU
Course: Introduction To Real Analysis And Linear Algebra
QUIZ 3 SOLUTIONS WED., APR. 11 NAME: DIRECTIONS: No papers, phones, calculators, or gadgets are permitted to be out during the quiz. Show all work, clearly and in order You will lose points if any of these instructions are not followed. Questions Points
School: JMU
Course: Introduction To Real Analysis And Linear Algebra
QUIZ 4 WED., APR. 18 NAME: DIRECTIONS: No papers, phones, calculators, or gadgets are permitted to be out during the quiz. Show all work, clearly and in order You will lose points if any of these instructions are not followed. Questions Points 1 1 2 2 3
School: JMU
Course: Introduction To Real Analysis And Linear Algebra
QUIZ 9 WED., MAY 30 NAME: DIRECTIONS: No papers, phones, calculators, or gadgets are permitted to be out during the quiz. Show all work, clearly and in order You will lose points if any of these instructions are not followed. Questions Points 1 1 2 2 2
School: JMU
Course: Introduction To Real Analysis And Linear Algebra
QUIZ 8 WED., MAY. 23 NAME: DIRECTIONS: No papers, phones, calculators, or gadgets are permitted to be out during the quiz. Show all work, clearly and in order You will lose points if any of these instructions are not followed. Questions Points 1 1 2 2 2
School: JMU
Course: Introduction To Real Analysis And Linear Algebra
QUIZ 7 Wed., May 16 NAME: DIRECTIONS: No papers, phones, calculators, or gadgets are permitted to be out during the quiz. Show all work, clearly and in order You will lose points if any of these instructions are not followed. Questions Points 1 1 2 2 2
School: JMU
Course: Introduction To Real Analysis And Linear Algebra
QUIZ 5 WED., OCT. 28 NAME: DIRECTIONS: No papers, phones, calculators, or gadgets are permitted to be out during the quiz. Show all work, clearly and in order You will lose points if any of these instructions are not followed. Questions Points 1 1 2 2 3
School: JMU
Course: Introduction To Real Analysis And Linear Algebra
QUIZ 6 WED., May. 9 NAME: DIRECTIONS: No papers, phones, calculators, or gadgets are permitted to be out during the quiz. Show all work, clearly and in order You will lose points if any of these instructions are not followed. Questions Points 1 1 2 2 3
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School: JMU
Haveagoodlongweekend AnswerstoAPDynamicsPretest2007 1. AnetforceFacceleratesamassmwithanaccelerationa.Ifthesamenetforceisappliedtomassm/2,thenthe accelerationwillbe a) b) c) d) 2. (F/m) velocityincreased. velocitydecreased. speedremainedthesame,butit'stur
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Exam 3 Chapters 13 & 14 Chapter 13 1. Be able to determine Bronsted-Lowry acid/base 2. Memorize strong acid/base 3. Calculate pH, pOH, [H+], [OH-] 4. Calculate weak acids & bases using ICE table and equilibrium concept. Also, percent ionization 5. Calcula
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NaturalandPositiveLaw Law:Abodyofenforceablerulesgoverningrelationshipsamongindividualsandbetween individualsandtheirsociety o NaturalLaw:principlesinherentinhumannature o PositiveLaw:lawofaparticularsociety Jurisprudence(science/philosophyoflaw) Thest
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Terms: Helix = a spiral Antiparallel = Pertaining to molecular orientation in which a molecule or parts of a molecule have opposing directions Conservative = the two old strands come back together after replication; one molecule is completely old strand a
School: JMU
School: JMU
School: JMU
School: JMU
School: JMU
School: JMU
Course: Calculus II
MATH 236 MATH 236 SPECIMEN TEST ONE Name: Social Security / Student ID Number: Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 TOTAL (/75) TOTAL (%) Instructions: Answer all questions
School: JMU
Course: Calculus 1
Worksheet 6 MATH 235 10/21/2010 1. (The Derivative Rule for Inverses.) Let f be a continuous one-to-one function dened on an interval. Suppose f is differentiable at x = a and f (a) = 0. If f (a) = b, show that (f 1 ) (b) exists and that 1 ( f 1 ) ( b) =
School: JMU
Course: Calculus 1
Worksheet 5 MATH 235 10/07/2010 Discuss the following problems with your group and write down a complete solution. Show all work. 1. Prove the Product Rule: if f and g are differentiable, then so is their product P (x) = f (x)g (x), and P ( x) = f ( x) g
School: JMU
Course: Calculus 1
Name: Worksheet 4 MATH 235 Fall, 2010. Discuss the following problems with your group and write down a complete solution. Show all work. 1. Prove that if f is differentiable at x = c, then f is continuous at x = c. Proof. To show that f is continuous at x
School: JMU
Course: Calculus II
H OMEWORK 1 4.3.26. The integrand is the upper half circle of radius r centered at 0. Here, r is a constant and x is the variable of integration since it appears in the differential dx. Hence, r r2 x2 dx r is 1 2 of the area of a circle of radius r. This
School: JMU
Course: Calculus II
H OMEWORK 2 4.7.2 (c) We cannot anti-differentiate this function directly. However, we can use the second Fundamental Theorem of Calculus to write x d 2 2 esin t dt = esin x , dx 0 x d 2 2 esin t dt = esin x , dx 1 x d 2 2 esin t dt = esin x , dx 2 x sin2
School: JMU
Course: Calculus II
H OMEWORK 3 1 1 5.2.24. If we let u = ln x and dv = x dx then we have du = x dx and v = ln x. Integration by parts gives ln x ln x dx = (ln x)2 dx. x x If we rearrange by combining the integrals, then we get ln x dx = (ln x)2 + C x 2 so ln x (ln x)2 dx =
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 3 DUE: Friday, Jan. 22 NAME: DIRECTIONS: Attach this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will loose points if you work is not in order. When required, do not forget the units! C
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 2 DUE: Friday, Jan. 15 NAME: DIRECTIONS: Attach this page to the front of your homework (dont forget your name!). Show all work, clearly and in order. When required, do not forget the units! Circle your nal answers. You will loose points if y
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 1 MATH 200 NAME: DIRECTIONS: Attach this page to the front of your homework (dont forget your name!). Please read article A Guide to Writing Mathematics by Dr. K. P. Lee which can be found on the class website, homework page. Note, homeworks in
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 4 DUE: Friday, Jan. 29 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will loose points if you work is not in order. When required, do not forget the units! C
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 5 DUE: Friday, Feb. 5 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will loose points if you work is not in order. When required, do not forget the units! Ci
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 10 DUE: Wed., Mar. 10 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Cir
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 9 DUE: Fri., Mar. 5 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circl
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 8 DUE: Fri., Feb. 26 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circ
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 6 DUE: Wed., Feb. 10 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circ
School: JMU
Course: Mathematical Methods For The Physical Sciences I
HOMEWORK 7 DUE: Fri., Feb. 19 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circ
School: JMU
Course: Mathematical Methods For The Physical Sciences II
HOMEWORK 1 DUE: Fri., Apr. 2 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circl
School: JMU
Course: Mathematical Methods For The Physical Sciences II
HOMEWORK 4 DUE: Fri., Apr. 23 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circ
School: JMU
Course: Mathematical Methods For The Physical Sciences II
HOMEWORK 3 DUE: Fri., Apr. 16 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circ
School: JMU
Course: Mathematical Methods For The Physical Sciences II
HOMEWORK 2 DUE: Fri., Apr. 9 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circl
School: JMU
Course: Mathematical Methods For The Physical Sciences II
HOMEWORK 5 DUE: Fri., Apr. 30 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circ
School: JMU
Course: Mathematical Methods For The Physical Sciences II
HOMEWORK 6 DUE: Fri., May 7 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circle
School: JMU
Course: Mathematical Methods For The Physical Sciences II
HOMEWORK 10 DUE: Wed., June 2 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circ
School: JMU
Course: Mathematical Methods For The Physical Sciences II
HOMEWORK 8 DUE: Fri., May 21 NAME: DIRECTIONS: STAPLE this page to the front of your homework (dont forget your name!). Show all work, clearly and in order You will lose points if you work is not in order. When required, do not forget the units! Circl
School: JMU
Course: Introduction To Real Analysis And Linear Algebra
HOMEWORK 1 NAME: DIRECTIONS: Turn in your homework as SINGLE-SIDED typed or handwritten pages. STAPLE your homework together. Do not use paper clips, folds, etc. STAPLE this page to the front of your homework. Be sure to write your name on your homewo
School: JMU
Course: Introduction To Real Analysis And Linear Algebra
HOMEWORK 4 DUE: Fri., APR. 20 NAME: DIRECTIONS: Turn in your homework as SINGLE-SIDED typed or handwritten pages. STAPLE your homework together. Do not use paper clips, folds, etc. STAPLE this page to the front of your homework. Be sure to write your
School: JMU
Course: Introduction To Real Analysis And Linear Algebra
HOMEWORK 2 DUE: Fri., Apr. 6 NAME: DIRECTIONS: Turn in your homework as SINGLE-SIDED typed or handwritten pages. STAPLE your homework together. Do not use paper clips, folds, etc. STAPLE this page to the front of your homework. Be sure to write your n
School: JMU
Course: Introduction To Real Analysis And Linear Algebra
HOMEWORK 3 DUE: Fri., Apr. 13 NAME: DIRECTIONS: Turn in your homework as SINGLE-SIDED typed or handwritten pages. STAPLE your homework together. Do not use paper clips, folds, etc. STAPLE this page to the front of your homework. Be sure to write your
School: JMU
Course: Non-linear Dynamics And Chaos
EXERCISE SET 1 1. Determine whether the equation s = s0 + v0t 0.5gt2 is dimensionally compatible, where s is the position (measured vertically from a fixed reference point) of a body at time t, s0 is the position at t = 0, v0 is the initial velocity
School: JMU
SPSS Assignment 2 Due October 9, 2007 1) Open the document labeled "classdata". 2) Find the mean, median, and standard deviation for the high school GPAs (HSGPA). 3) Find the mean, median, and standard deviation for the college GPAs (GPA). 4) Find th
School: JMU
SPSS Assignment 1 Due Thursday September 20, 2007 1) Open the document labeled "classdata". 2) For the variable ,politics, construct a bar chart. (Frequency) 3) Using the variable ,gender, construct a pie chart. (Percent) 4) Using the variable ,heig
School: JMU
Course: Stats
Math 220 HW #4 Due Friday, March 21 Name _ Section _ Honor Pledge: I have neither given nor received help from any person on this homework. Signed _ 1. Assume that 30% of students at a university wear contact lenses. Consider the sample proportion
School: JMU
Course: ELEM STATISTICS
Department of Mathematics and Statistics Math 220 (Section 21 and Section 22): Spring 2014 Instructor: Dr. Hasan Hamdan Course Website: Course is on Canvas Email: hamdanhx@jmu.edu Phone: 540.568.2844 Sections: 21 & 22 Class Location: Tuesday Roop 127 and
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School: JMU
Course: Stats
Math 220 Syllabus Spring 2008 1. Class Information: Section 8 : M,W 10:10 AM to 11:00 AM, Burruss 141 F 10:10 AM to 11:00 AM, Roop 127 MW 11:15 AM to 12:05 PM , Roop 141 F 11:15 AM to 12:05 PM, Roop 127 Section 11: Section 16: M,W 1:25 PM to 2:1