Department of Mathematical Sciences, UAEU Spring 2008 Midterm Exam MATH 1120 Calculus II for Engineers U A E University, College of Science Department of Mathematical Sciences Midterm Exam Spring 2008 MATH 1120 CALCULUS II FOR ENGINEERS Student’s Name Student’s I.D. Section # Check the Name of Your Instructor Dr. Nabila Azam - Section 51 Dr. Moh. El Bachraoui - Section 01 Dr. Nabila Azam - Section 52 Dr. Adama Diene -Section 02 Dr. John Abraham - Section 53 Dr. Jae Lee - Section 03 Dr. John Abraham - Section 54 Dr. Adama Diene - Section 04 Allowed time is 1 hours. You can use the back of the sheets. NO BOOKS. NO NOTES. NO PROGRAMING CALCULATORS Section I Problem # Points Section II Problem # Section III Points 1-6 Problem # Points Total Points Department of Mathematical Sciences, UAEU Spring 2008 Midterm Exam MATH 1120 Calculus II for Engineers Section I: Multiple choice problems [20 Points, 5 each] (No Partial Credits for this Section) G G K G G G G 1. The area of the triangle with two sides represented by u = i + 3 j − k and v = i − k , is A) 12 B) 2 18 C) 1 18 2 D) 18 2. The symmetric equations of the line passing through the points P (1, 2, −1) and Q ( 5, −3, 4 ) A) C) x –1 = –4 x −1 = −4 y+2 = –5 y −2 = −5 3. Evaluate the given integral. z +3 7 z +1 5 ∫ B) D) 3 ln t + 9t , t 2 + 8t + c 2 B) C) 3 ln t + 9t , t 2 + 8t + c 2 D) z +1 5 z +1 5 x , then f y is y x B) f y = e xy + 2 y x D) f y = xe xy − 2 y 1 3 − 2 + 9t , t 2 + 8t + c 2 t 3 ln t + 9, t 2 + 8 + c 2 4. Let f ( x, y ) = e xy + C) y −2 = −5 y+2 = −5 1 + 9,3t + 8 dt t A) A) x −1 = 4 x +1 = 4 x f y = ye xy + 2 y 2x f y = xe xy − 2 y 2-6 Department of Mathematical Sciences, UAEU Spring 2008 Midterm Exam MATH 1120 Calculus II for Engineers Section II: Multiple-Step problems [65 Points, 14 each+9 for last] 1. The thrust of an airplane’s engine produces a speed of 500 mph in still air. The wind velocity is given by 20, 80 . In what direction should the plane head to fly due to east. G G 2. Let a = 2 ,1 and b = 3 ,4 . Find G G G a) 3a − 2b b) CompbG a 3-6 Department of Mathematical Sciences, UAEU G Spring 2008 Midterm Exam G MATH 1120 Calculus II for Engineers G 3. Consider the circle r (t ) = a cos t i + a sin t j with radius a. a) Find the unit tangent b) Show that the curvature is the reciprocal of the radius. 4. A cone has the height h and the circular base of radius r. The height and radius are changing with time. At the instant in which r = 2 cm, h = 3 cm, it was found that dr dh = 0.02 cm / sec and = −0.03 cm / sec . At what rate is the cone's volume changing dt dt 1 at that instant? (Hint: you may use the formula V = πr 2 h ) 3 4-6 Department of Mathematical Sciences, UAEU G G G Spring 2008 Midterm Exam G G MATH 1120 Calculus II for Engineers G G 5. Let c = b − a where a = 1 and b = 1 . If θ is the angle between a and c and β is G G the angle between b and c , then find the relation between cos θ and cos β , hence find the relation between the angles θ and β . 5-6 Department of Mathematical Sciences, UAEU Spring 2008 Midterm Exam MATH 1120 Calculus II for Engineers Section III: Concept problems [15 Points, 5 each] 1. Decide which of the following statements is true or false: (1 point each) G a. If u G G G and v are large then, u ⋅ v is also large. b. For a given vector, there is one unit vector parallel to it. c. The cross product of two unit vectors is a unit vector. d. The minimum number of points required to determine an equation of a plane in 3-space is two points. K dr (t ) d K K e. If r (t ) is a differentiable vector function, then r (t ) = dt dt [ ] [ [ ] ] [ ] [ ] 2- Decide if each of the following quantities is a vector, a scalar, or undefined (write your answer over the dots). Give Reasons for your answer. (1 point each) a. G G G (u ⋅ v ) w .………………….. b. G G G u (v ⋅ w) ……………………. c. G G G u ⋅v + w ……………………. d. GG G G (v .w) × (u × w) …………………….. G 3 w e. 2v − G ……………………. 3. a) What does the equation x + y = 3 represent in 3-space? (2 Points) b) What it is the object in 3-space that can be represented by the vector equation G G G K r (t ) = t 2 i + 2t 2 j + 3t 2 k ? Explain your answer. (3 Points) 6-6