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mathematics
calculus
Questions and Answers of
Calculus
Two tugboats are towing a loaded barge and the magnitude of the resultant is 6000 pounds directed along the axis of the barge (see figure). Find the tension (in pounds) in the tow lines when they
To carry a 100-pound cylindrical weight, two people lift on the ends of short ropes that are tied to an eyelet on the top center of the cylinder. Each rope makes a 20° angle with the vertical. Draw
A force of F pounds is required to pull an object weighing W pounds up a ramp inclined at degrees from the horizontal. 1. Find F when W = 100 pounds and = 12°. 2. Find W when F = 600 pounds and
Determine whether u and v are equivalent. Explain.1.2.
An airplane travels in the direction of 148° with an airspeed of 875 kilometers per hour. Due to the wind, its groundspeed and direction are 800 kilometers per hour and 140°, respectively
A commercial jet travels from Miami to Seattle. Thejet's velocity with respect to the air is 580 miles per hour, and its bearing is 332°. The jet encounters a wind with a velocity of 60 miles per
Determine whether the statement is true or false. Justify your answer. 1. If u and v have the same magnitude and direction, then u and v are equivalent. 2. If u is a unit vector in the direction of
1. Describe the error in finding the component form of the vector u that has initial point (3, 4) and terminal point (6, 1). The components are u1 = 3
1. The ________ ________ of two vectors yields a scalar, rather than a vector. 2. The dot product of u = (u1, u2) and v = (v1, v2) is u ∙ v = ________. 3. If is the angle between two nonzero
Use the vectors u = (3, 3), v = (-4, 2), and w = (3, 1) to find the quantity. State whether the result is a vector or a scalar. 1. u ∙ u 2. 3u ∙ v 3. (u ∙ v)v
Use the dot product to find the magnitude of u. 1. u = (-8, 15) 2. (4, -6) 3. 20i + 25j 4. u = 12i - 16j 5. u = 6j 6. u = -21i
Find the angle (in radians) between the vectors. 1. u = (1, 0) v = (0, -2) 2. u = (3, 2) v = (4, 0) 3. u = 3i + 4j v = -2j
Find the angle (in degrees) between the vectors. 1. u = 3i + 4j v = −7i + 5j 2. u = 6i − 3j v = −4i − 4j
Use vectors to find the interior angles of the triangle with the given vertices. 1. (1, 2), (3, 4), (2, 5) 2. (−3, −4), (1, 7), (8, 2)
Find u ∙ v, where is the angle between u and v. 1. ∥u∥ = 4, ∥v∥ = 10, θ = 2π/3 2. ∥u∥ = 4, ∥v∥ = 12, θ = π/3
Determine whether u and v are orthogonal. 1. u = (3, 15), v = (-1, 5) 2. u = (30, 12), v = (1/2, -5/4) 3. u = 2i - 2j, v = -i - j
Find the projection of u onto v. Then write u as the sum of two orthogonal vectors, one of which is projvu. 1. u = (2, 2), v = (6, 1) 2. u = (0, 3), v = (2, 15)
Use the graph to find the projection of u onto v. (The terminal points of the vectors in standard position are given.) Use the formula for the projection of u onto v to verify your result.1.2. 3. 4.
Find two vectors in opposite directions that are orthogonal to the vector u. 1. u = (3, 5) 2. u = (-8, 3) 3. u = ½ i - 2/3 j 4. -5/2 i - 3j
Determine the work done in moving a particle from P to Q when the magnitude and direction of the force are given by v. 1. P (0, 0), Q (4, 7), v = 1, 4 2. P (1, 3), Q (−3, 5), v = −2i + 3j
Find u ∙ v. 1. u = (7, 1) v = (−3, 2) 2. u = (6, 10) v = (−2, 3) 3. u = (−6, 2) v = (1, 3) 4. u = −2, 5 v = (−1, −8) 5. u = (4i − 2j) v = (i - j) 6. u = (i − 2j) v = (−2i -
1. The vector u = 1225, 2445 gives the numbers of hours worked by employees of a temporary work agency at two pay levels. The vector v = 12.20, 8.50 gives the hourly wage (in dollars) paid at
A truck with a gross weight of 30,000 pounds is parked on a slope of d° (see figure). Assume that the only force to overcome is the force of gravity.(a) Find the force required to keep the truck
A sport utility vehicle with a gross weight of 5400 pounds is parked on a slope of 10°. Assume that the only force to overcome is the force of gravity. Find the force required to keep the vehicle
1. Determine the work done by a person lifting a 245-newton bag of sugar 3 meters. 2. Work Determine the work done by a crane lifting a2400-pound car 5 feet. 3. Work a constant force of 45 pounds,
1. One of the events in a strength competition is to pull a cement block 100 feet. One competitor pulls the block by exerting a constant force of 250 pounds on a rope attached to the block at an
Determine whether the statement is true or false. Justify your answer. 1. The work W done by a constant force F acting along the line of motion of an object is represented by a vector. 2. A sliding
Describe the error in finding the quantity when u = (2, 1) and v = (-3, 5)2. u 2v = (2, -1) (-6, 10) = 2 (-6) - (-1) (10) = -12 + 10
Find the value of k such that vectors u and v are orthogonal. 1. u = 8i + 4j v = 2i − kj 2. u = −3ki + 5j v = 2i − 4j
1. Let u be a unit vector. What is the value of u u? Explain.2. What is known about , the angle between two nonzero vectors u and v, under each condition (see figure)?(a) u
1. What can be said about the vectors u and v under each condition? (a) The projection of u onto v equals u. (b) The projection of u onto v equals 0. 2. Proof Use vectors to prove that the diagonals
Prove that ∥u - v∥2 = ∥u∥2 + ∥v∥2 - 2u ∙ v.
Plot the complex number and find its absolute value. 1. - 7i 2. - 7 3. - 6 + 8i 4. 5 - 12i
Find the sum of the complex numbers in the complex plane.1. (3 + i) + (2 + 5i)2. (5 + 2i) + (3 + 4i)3. (8 2i) + (2 + 6i)4. (3 i) + (1 + 2i)5.6.
Find the difference of the complex numbers in the complex plane. 1. (4 + 2i) − (6 + 4i) 2. (−3 + i) − (3 + i) 3. (5 − i) − (−5 + 2i) 4. (2 − 3i) − (3 + 2i) 5. 2 − (2 + 6i) 6. −3
Plot the complex number and its complex conjugate. Write the conjugate as a complex number. 1. 2 + 3i 2. 5 − 4i 3. −1 − 2i 4. −7 + 3i
Find the distance between the complex numbers in the complex plane. 1. 1 + 2i, −1 + 4i 2. −5 + i, −2 + 5i 3. 6i, 3 − 4i
Find the midpoint of the line segment joining the points corresponding to the complex numbers in the complex plane. 1. 2 + i, 6 + 5i 2. −3 + 4i, 1 − 2i 3. 7i, 9 − 10i
1.Ship A is 3 miles east and 4 miles north of port. Ship B is 5 miles west and 2 miles north of port (see figure).(a) Using the positive imaginary axis as north and the positive real axis as east,
Determine whether the statement is true or false. Justify your answer. 1. The modulus of a complex number can be real or imaginary. 2. The distance between two points in the complex plane is always
What does the set of all points with the same modulus represent in the complex plane? Explain.
1. Determine which graph represents each expression. (a) (a + bi) + (a - bi) (b) (a + bi) - (a - bi) 2. The points corresponding to a complex number and its complex conjugate are plotted in the
Match the complex number with its representation in the complex plane. [The representations are labeled (a)-(h).]a.b. c. d. e. f. g. h. 1. 2 2. 3i 3. 1 + 2i 4. 2 + i 5. 3 - i 6. - 3 + i 7. - 2 -
Show that the negative of z = r (cos θ + i sin θ) is - z = r [cos (θ + π) + i sin (θ + π)].
Show that z̅ = r [cos(−θ) + i sin(−θ)] is the complex conjugate of z = r (cos θ + i sin θ ) Then find (a) zz̅ and (b) z / z̅, z̅ ≠ 0.
Write the standard form of the complex number. Then plot the complex number. 1. 2 (cos 60° + i sin 60°) 2. 5 (cos 135° + i sin 135°) 3. √48 [cos (−30°) + i sin (−30°)]
Use a graphing utility to write the complex number in standard form. 1. 5 (cos π / 9 + i sin π / 9) 2.10 (cos 2π / 5 + i sin 2π / 5) 3. 2(cos 155o + i sin 155o) 4. 9 (cos 58o + i sin 58o)
Find the quotient. Leave the result in trigonometric form. 1. 3(cos 50° + i sin 50°) / 9(cos 20° + i sin 20°) 2. cos 120° + i sin 120° / 2(cos 40° + i sin 40°)
(a) Write the trigonometric forms of the complex numbers, (b) Perform the operation using the trigonometric forms, and (c) Perform the operation using the standard forms, and check your result with
Plot the complex number. Then write the trigonometric form of the complex number. 1. 1 + i 2. 5 - 5i 3. 1 - √3i
Find the product in the complex plane.1.2.
Use DeMoivre's Theorem to find the power of the complex number. Write the result in standard form. 1. [5(cos 20° + i sin 20°)]3 2. [3(cos 60° + i sin 60°)]4 3. (cos π / 4 + i sin π / 4)12
Represent the powers z, z2, z3, and z4 graphically. Describe the pattern. z = √2 / 2 (1 + i)
(a) Use the formula on page 610 to find the roots of the complex number, (b) Write each of the roots in standard form, and (c) Represent each of the roots graphically. Square roots of 5(cos 120° +
Use the formula on page 610 to find all solutions of the equation and represent the solutions graphically. x4 + i = 0
Ohm's law for alternating current circuits is E = IZ, where E is the voltage in volts, I is the current in amperes, and Z is the impedance in ohms. Each variable is a complex number.(a) Write E in
The figure shows onw of the fourth roots of a complex number z.(a) How many roots are not shown? (b) Describe the other roots.
Determine whether the statement is true or false. Justify your answer. 1. Geometrically, the nth roots of any complex number z are all equally spaced around the unit circle. 2. The product of two
Given two complex numbers z1 = r1 (cos θ1 + i sin θ1) and z2 = r2 (cos θ2 + i sin θ2), z2 ≠ 0, show that z1 / z2 = r1 / r2 [cos (θ1 θ2) + i sin (θ1 θ2)].
Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. 1. A = 38°, B = 70°, a = 8 2. A = 22°, B = 121°, a = 19 3. B
Find the sum of the complex numbers in the complex plane. 1. (2 +3i) + (1 - 2i) 2. (-4 + 2i) + (2 +i) Find the difference of the complex numbers in the complex plane. 3. (1 + 2i) − (3 + i) 4. (−2
Plot the complex number and its complex conjugate. Write the conjugate as a complex number. 1. 3 + i 2. 2 − 5i
Find the distance between the complex numbers in the complex plane. 1. 3 + 2i, 2 - i 2. 1 + 5i, −1 + 3i Find the midpoint of the line segment joining the points corresponding to the complex numbers
Plot the complex number. Then write the trigonometric form of the complex number. 1. 4i 2. −7 3. 7 − 7i
Find the product. Leave the result in trigonometric form. 1. [2 (cos π/ 4 + i sin π/4)] [2 (cos π/3 + i sin π/3)] 2. [4 (cos π/3 + i sin π/3)] [3 (cos 5π / 6 + i sin 5π / 6)]
Find the quotient. Leave the result in trigonometric form. 1. 2(cos 60° + i sin 60°) / 3(cos 15° + i sin 15°) 2. cos 150° + i sin 150° / 2(cos 50° + i sin 50°)
Use DeMoivre's Theorem to find the power of the complex number. Write the result in standard form. 1. [5 (cos π / 12 + i sin π / 12)]4 2. [2 (cos 4π / 15 + i sin 4π / 15)]5
(a) find the roots of the complex number, (b) write each of the roots in standard form (c) represent each of the roots graphically. 1. Sixth roots of −729i 2. Fourth roots of 256i
Find all solutions of the equation and represent the solutions graphically. x4 + 81 = 0
Find the area of the triangle. Round your answers to one decimal place. 1. A = 33°, b = 7, c = 10 2. B = 80°, a = 4, c = 8 3. C = 119°, a = 18, b = 6 4. A = 11°, b = 22, c = 21
Determine whether the statement is true or false. Justify your answer. 1. The Law of Sines is true when one of the angles in the triangle is a right angle. 2. When the Law of Sines is used, the
From a certain distance, the angle of elevation to the top of a building is 17°. At a point 50 meters closer to the building, the angle of elevation is 31°. Find the height of the building.
A surveyor finds that a tree on the opposite bank of a river flowing due east has a bearing of N 22° 30 E from a certain point and a bearing of N 15° W from a point 400 feet downstream. Find the
Use the Law of Cosines to solve the triangle. Round your answers to two decimal places. 1. a = 6, b = 9, c = 14 2. a = 75, b = 50, c = 110 3. a = 2.5, b = 5.0, c = 4.5 4. a = 16.4, b = 8.8, c = 12.2
Determine whether the Law of Cosines is needed to solve the triangle. Then solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. 27. C = 64°,
The lengths of the diagonals of a parallelogram are 10 feet and 16 feet. Find the lengths of the sides of the parallelogram when the diagonals intersect at an angle of 28°.
Two planes leave an airport at approximately the same time. One flies 425 miles per hour at a bearing of 355°, and the other flies 530 miles per hour at a bearing of 67°. Draw a diagram that gives
Use Heron's Area Formula to find the area of the triangle. 1. a = 3, b = 6, c = 8 2. a = 15, b = 8, c = 10 3. a = 12.3, b = 15.8, c = 3.7 4. a = 4/5, b = 3 / 4, c = 5 / 8
Determine whether u and v are equivalent. Explain.1.2.
Find the component form and magnitude of the vector v. 1. Initial point: (0, 10) Terminal point: (7, 3) 2. Initial point: (1, 5) Terminal point: (15, 9)
Find (a) u + v, (b) u -v, (c) 4u, and (d) 3v + 5u. Then sketch each resultant vector. 1. u = 〈-1 ,-3〉, v = 〈-3 , 6〉 2. u = 〈4 , 5〉, v = 〈0 , - 1〉
The initial and terminal points of a vector are given. Write the vector as a linear combination of the standard unit vectors i and j. Initial Point ..................................................
Find the component form of w and sketch the specified vector operations geometrically, where u = 6i - 5j and v = 10i +3j. 1. w = 3v 2. w = 1 / 2v 3. w = 2u + v
Find the magnitude and direction angle of the vector v. 1. v = 5i + 4j 2. v = −4i + 7j 3. v = −3i − 3j 4. v = 8i - j
Find the component form of v given its magnitude and the angle it makes with the positive x-axis. Then sketch v. Magnitude ................................................. Angle 1. ∥v∥ = 8
Forces with magnitudes of 85pounds and 50pounds act on a single point at angles of 45° and 60°, respectively, with the positive x-axis. Find the direction and magnitude of the resultant of these
Two ropes support a 180-pound weight, as shown in the figure. Find the tension in each rope.
Find u · v 1. u = 〈6, 7〉 v = 〈-3, 9〉 2. u = 〈-7, 12〉 v = 〈-4, -14〉 3. u = 3i + 7j v = 11i - 5j 4. u = -7i + 2j v = 16i - 12j
Use the vectors u = 〈-4, 2〉 and v = 〈5, 1〉 to find the quantity. State whether the result is a vector or a scalar. 1. 2u · u 2. 3u · v 3. 4 - ∥u∥ 4. ∥v∥2
Find the angle θ (in degrees) between the vectors. 1. u = 〈2 √2, -4〉, v = - √2, 1〉 2. u = 〈3, √3〉, v = 〈4, 3 √3〉 3. u = cos 7π / 4i + sin 7π / 4j, v = cos 5π / 6i + sin 5π
Determine whether u and v are orthogonal. 1. u = 〈-3, 8〉 v = 〈8, 3〉 2. u = 〈1/4 , 1/2〉 v = 〈-2, 4〉 3. u = -i v = i + 2j 4. u = -2i + j v = 3i +6j
Find the projection of u onto v. Then write u as the sum of two orthogonal vectors, one of which is projv u. 1. u = 〈-4, 3〉, v = 〈-8, -2〉 2. u = 〈5, 6〉, v = 〈10, 0〉
Determine the work done in moving a particle from P to Q when the magnitude and direction of the force are given by v. 1. P (5, 3), Q (8, 9), v = 〈2, 7〉 2. P (-2, -9), Q (-12, 8), v = 3i - 6j
1. Determine the work done by a crane lifting an 18,000-pound truck 4 feet. 2. A constant force of 25 pounds, exerted at an angle of 20° with the horizontal, is required to slide a crate across a
Plot the complex number and find its absolute value. 1. 7i 2. −6i 3. 5 + 3i 4. −10 − 4i
Solve the system by the method of substitution.1.2.
The given amount of annual interest is earned from a total of $12,000 invested in two funds paying simple interest. Write and solve a system of equations to find the amount invested at each given
Solve the system by the method of substitution.1.2.
Solve the system graphically.1.2. 3. 4. 5.
Use a graphing utility to solve the system of equations graphically. Round your solution(s) to two decimal places, if necessary1.2. 3. 4.
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