Final Exam Review - Math 2412 Fall 2013 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the term indicated in the expansion. 1) (x + 3y)11; 4th term 1) ID: PCALC5Z 10.5.3-7 Write the first three terms in the binomial expansion, expressing the result in simplified form. 2) (2 x + 3 y) 8 2) ID: PCALC5Z 10.5.3-4 Use the Binomial Theorem to expand the binomial and express the result in simplified form. 3) (5x + 4)3 3) ID: PCALC5Z 10.5.2-2 Evaluate the given binomial coefficient. 11 4) 4 4) ID: PCALC5Z 10.5.1-1 Find the sum of the infinite geometric series, if it exists. 5) 18 + 6 + 2 + 2 3 +... 5) ID: PCALC5Z 10.3.6-3 Solve the problem. Round to the nearest dollar if needed. 6) Looking ahead to retirement, you sign up for automatic savings in a fixed-income 401K plan that pays 6% per year compounded annually. You plan to invest $3000 at the end of each year for the next 25 years. How much will your account have in it at the end of 25 years? ID: PCALC5Z 10.3.5-2 6) Use the formula for the sum of the first n terms of a geometric sequence to solve. 7) Find the sum of the first 10 terms of the geometric sequence: 4, 12, 36, 108, 324,.... 7) ID: PCALC5Z 10.3.4-2 Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence with the given first term, a1, and common ratio, r. 8) Find a12 when a1 = 2, r = -2. 8) ID: PCALC5Z 10.3.3-5 1
Write out the first three terms and the last term of the arithmetic sequence. 9) 40-3i 9) i=1 ID: PCALC5Z 10.2.4-10 Convert the equation to the standard form for an ellipse by completing the square on x and y. 10) 36x 2 + 9y 2-72x + 54y - 207 = 0 10) ID: PCALC5Z 9.1.3-10 Solve the problem. 11) The arch beneath a bridge is semi-elliptical, a one-way roadway passes under the arch. The width of the roadway is 34 feet and the height of the arch over the center of the roadway is 10 feet. Two trucks plan to use this road. They are both 10 feet wide. Truck 1 has an overall height of 9 feet and Truck 2 has an overall height of 8 feet. Draw a rough sketch of the situation and determine which of the trucks can pass under the bridge. 11) ID: PCALC5Z 9.1.4-1 Write the equation in terms of a rotated x'y'-system using, the angle of rotation. Write the equation involving x' and y' in standard form. 12) xy +16 = 0; = 45 12) ID: PCALC5Z 9.4.2-3 Parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. 13) x = 4 + 5 cos t, y = 6 + 4 sin t; t = 2 13) ID: PCALC5Z 9.5.1-2 Use point plotting to graph the plane curve described by the given parametric equations. 14) x = 2t - 1, y = t 2 + 1; -4 t 4 14) ID: PCALC5Z 9.5.1-5 Identify the equation without completing the square. 15) 2x 2 + 4y 2-4x - 3y = 0 15) ID: PCALC5Z 9.4.1-3 2
Use synthetic division and the Remainder Theorem to find the indicated function value. 16) f(x) = 3x 3-3x 2-3x + 3; f(-2) 16) ID: PCALC5Z 2.4.3-2 Find the slant asymptote, if any, of the graph of the rational function. 17) f(x) = x2 + 9x - 9 x - 3 17) ID: PCALC5Z 2.6.7-2 Use a vertical shift to graph the function. 18) y = -3 sin 1 2 x - 2 18) ID: PCALC5Z 4.5.5-5 Use a sketch to find the exact value of the expression. 19) cos tan -1 5 6 19) ID: PCALC5Z 4.7.5-9 Use a right triangle to write the expression as an algebraic expression. Assume that x is positive and in the domain of the given inverse trigonometric function. 20) sin(tan -1 x 2 ) 20) ID: PCALC5Z 4.7.5-16 Complete the identity. 21) 1 - sin2 x 1 + cos x =? 21) ID: PCALC5Z 5.1.1-19 Solve the equation on the interval [0, 2 ). 3 22) cos 2x = 2 22) ID: PCALC5Z 5.5.2-3 3
Solve the problem. 23) Two tracking stations are on the equator 170 miles apart. A weather balloon is located on a bearing of N36 E from the western station and on a bearing of N15 W from the eastern station. How far is the balloon from the western station? Round to the nearest mile. ID: PCALC5Z 6.1.4-2 24) Two airplanes leave an airport at the same time, one going northwest (bearing 135 ) at 407 mph and the other going east at 337 mph. How far apart are the planes after 2 hours (to the nearest mile)? ID: PCALC5Z 6.2.2-2 25) A painter needs to cover a triangular region 63 meters by 69 meters by 72 meters. A can of paint covers 70 square meters. How many cans will be needed? ID: PCALC5Z 6.2.2-6 23) 24) 25) The rectangular coordinates of a point are given. Find polar coordinates of the point. Express in radians. 26) (3, -3) 26) ID: PCALC5Z 6.3.4-7 Find another representation, (r, ), for the point under the given conditions. 27) 7, 6, r < 0 and 2 < < 4 27) ID: PCALC5Z 6.3.2-4 Convert the rectangular equation to a polar equation that expresses r in terms of. 28) 6x - 3y + 10 = 0 28) ID: PCALC5Z 6.3.5-4 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of a polar equation is given. Select the polar equation for the graph. 29) 29) A) r = 6 cos B) r = 3 + sin C) r = 6 sin D) r = 3 + cos ID: PCALC5Z 6.4.1-5 4
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find all the complex roots. Write the answer in the indicated form. 30) The complex cube roots of 343(cos 33 + i sin 33 ) (polar form) 30) ID: PCALC5Z 6.5.8-2 Solve the problem. 31) Two forces, F1 and F2, of magnitude 60 and 70 pounds, respectively, act on an object. The direction of F1 is N40 E and the direction of F2 is N40 W. Find the magnitude and the direction angle of the resultant force. Express the direction angle to the nearest tenth of a degree. ID: PCALC5Z 6.6.7-3 31) 32) A force of 5 pounds acts in the direction of 5 to the horizontal. The force moves an object along a straight line from the point (5, 6) to the point (18, 19), with distance measured in feet. Find the work done by the force. Round the answer to one decimal place, if necessary. ID: PCALC5Z 6.7.6-4 32) Write the partial fraction decomposition of the rational expression. 33) 7x2 - x - 18 x 3 - x 33) ID: PCALC5Z 7.3.1-7 34) 7x 2 + 21x - 9 x 3 + 6x 2 + 3x + 18 ID: PCALC5Z 7.3.3-4 34) 35) y = -3 sin 3x + 3-2 35) ID: PCALC5Z 4.5.5-7 36) sin 2 x + tan 2 x + cos 2 x =? 36) ID: PCALC5Z 5.1.1-8 5
37) A guy wire to a tower makes a 70 angle with level ground. At a point 39 ft farther from the tower than the wire but on the same side as the base of the wire, the angle of elevation to the top of the tower is 35. Find the length of the wire (to the nearest foot). ID: PCALC5Z 6.1.4-4 37) 38) 4x 2 + 16y 2-16x - 96y + 96 = 0 38) ID: PCALC5Z 9.1.3-9 39) Find the work done by a force of 2 pounds acting in the direction of 41 to the horizontal in moving an object 6 feet from (0, 0) to (6, 0). ID: PCALC5Z 6.7.6-2 40) The magnitude and direction of two forces acting on an object are 35 pounds, N45 E, and 55 pounds, S30 E, respectively. Find the magnitude, to the nearest hundredth of a pound, and the direction angle, to the nearest tenth of a degree, of the resultant force. ID: PCALC5Z 6.6.7-2 39) 40) 41) The complex cube roots of 8i (rectangular form) 41) ID: PCALC5Z 6.5.8-6 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 42) 42) A) r = 4 cos(4 ) B) r = 4 + cos(4 ) C) r = 4 sin(4 ) D) r = 4 ID: PCALC5Z 6.4.1-6 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Identify the equation without applying a rotation of axes. 43) x 2 + 4xy + 4y 2 + 3x - 3y - 8 = 0 43) ID: PCALC5Z 9.4.4-1 6
44) x = t, y = 2t + 4; 0 t 4 44) ID: PCALC5Z 9.5.1-10 45) A new exhibit is scheduled to open at the local museum. Museum officials expect that 10,000 people will visit the exhibit in its first week, and that the number of visitors will drop by 40 people per week after the first week during the first 6 months. Find the total number of visitors expected in the exhibit's first 7 weeks. ID: PCALC5Z 10.2.4-19 45) 46) (3x + 5)4 46) ID: PCALC5Z 10.5.2-6 47) (x 2 + y 4 ) 9 ; 6th term 47) ID: PCALC5Z 10.5.3-9 7
Answer Key Testname: FALL 2013 - FINAL REVIEW 1) 4455x8y3 2) 256x 8 + 3072 x 7 y + 16,128 x 6 y2 3) 125x3 + 300x2 + 240x + 64 4) 330 5) 27 6) $164,594 7) 118,096 8) -4096 9) -3-6 - 9 -... - 120 10) (x - 1)2 9 + (y + 3)2 36 = 1 11) Both Truck 1 and Truck 2 can pass under the bridge. 12) y'2 32 - x'2 32 = 1 13) (4, 10) 14) 20) x x2 + 2 x 2 + 2 21) cos x 22) 12, 11 12, 13 12, 23 12 23) 211 miles 24) 1376 miles 25) 29 cans 26) (-3 2, 135 ) 27) -7, 19 6 28) r = -10 (6 cos - 3 sin ) 29) D 30) 7(cos 11 + i sin 11 ), 7(cos131 + i sin 131 ), 7(cos 251 + i sin 251 ) 31) F = 99.37; = 93.7 32) 91.6 ft-lb 33) 18 x + -5 x + 1 + -6 x - 1 34) 35) 3 x + 6 + 4x - 3 x 2 + 3 15) ellipse 16) -27 17) y = x + 12 18) 19) 6 61 61 36) sec 2 x 37) 39 feet 38) (x - 2)2 16 + (y - 3)2 4 = 1 39) 9.1 ft-lb 40) F = 57.04; = -23.6 41) -2i, 3 + i, - 3 + i 42) A 43) parabola 8
Answer Key Testname: FALL 2013 - FINAL REVIEW 44) 45) 69,160 visitors 46) 81x4 + 540x3 + 1350x2 + 1500x + 625 47) 126x 8 y 20 9