Vector Calculus Questions and Answers

Let be an angle in standard position where 0 ≤ 0 ≤ 2л with: cos(0) = -and cot(0) < 0 Answer the following questions. Express your answer using exact values. (a) In what quadrant does the angle reside? 3 (b) tan(0) = (c) csc(0) =

A surveyor measures the sides of a triangular lot to be a = 145 feet, b = 70 feet, and c = 91 feet. a. Draw a diagram accurately labeling the information. b. Approximate the angles of A, B, and C to the nearest degree.

Solve the initial value problein x Axt) foct 0, with x(0) 2.1) Classify the nature of the origin as an attractor, repeller, or saddie point of the dynamical system described try Ax Finst the dre greatest attraction and/or repulsion A 0 -6 15 -21 Solve the initial value problem A Classify the nature of the origin as an attractor, repeter, or saddle point Choose the correct answer below Repoller Saddle Point Attractor CECH hoose the correct graph below that represents the direction(s) of greatest attraction and/or repulsion OB GO Oc x OD

3) Given the following information for an ellipse; foci at (1, 2), and (-3, 2); vertex at (-4,2) 5 pts State: a) the center point and the values for a, b, c, a², b², c². b) the equation for the ellipse

Calculate the Taylor polynomial T4 centered at x = 1 for the function f(x) = √x.

3 The surface area S (in square meters) of a hot-air balloon is given by S(r) = 4x² where r is the radius of the balloon (in meters). If the radius r is increasing with time t (in seconds) according to the formula r(t) = - t20, find the surface area S of the balloon as a function of the time t. CLS S(t) = (Type an expression using t as the variable. Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression. Simplify your answer.)

Write the expression as a single logarithm. 8 2 log√6x-1-log 6 6 (3 X 8 2 log6√6x-1-log 6 0 ( 1 ) X (Simplify your answer.) + log 68 + log 68=

The domain of a one-to-one function f is [7,00), and its range is [-7,00). State the domain and the range of f1. What is the domain of f¹? 1 The domain of f is (Type your answer in interval notation.) What is the range of f-¹? The range of f¹ is. (Type your answer in interval notation.)

1. Unlike savings accounts, banks expect people to make frequent withdrawals and deposits to checking accounts. (2 pts.) True False

A traveler pulls a suitcase, as shown in the figure. LLLL 0 The force acting on the handle is 30 N at an angle of 8 = 40°. How much work is done if the traveler moves 120 meters? Round to the nearest whole number.

2. Banks and other financial institutions offer incentives for people to keep their money in a savings account. (2 pts.) True False

7. (a) Determine such a < 0 so that the forward Euler method with step h = 0.1 is stabl when applied to solve (3+7 10 marks) (b) Determine for which values of step h the forward Euler method is stable when appliec to solve the following first-order equations dy dr dx/dt = ay(x). 1 [4]-[32][6] dy/dt y(t)

Complete the table. y = f(g(x)) 2x 9 y = sin U= u = g(x) y = f(u) y = sin u

Indicate whether the following table specifies a function. Domain 4 8 12 Range 235 Choose the correct answer below. OA. The table specifies a function. OB. The table does not specify a function.

Banks and credit unions are the only types of financial institutions. O True O False

Checks can be used to: (4 pts.) O pay bills Opay rent Omake purchases all of the above

C9 Consider the plane in R³ with vector equation 1+S2+t1 s,teR (a) Find parametric equations for the plane. (b) Use the parametric equations you found in (a) to find a scalar equation for the plane.

A railroad crew can lay 5 miles of track each day. They need to lay 135 miles of track. The length, L (in miles), that is left to lay after d days is given by the following function. L (d) = 135-5d Answer the following questions. (a) How many days will it take the crew to lay all the track? days (b) How many miles of track does the crew have left to lay after 18 days? miles S

Convert the Cartesian coordinate (4,5) to polar coordinates, 0 ≤ 0 < 2π r = Enter exact value. 0 =

. (4 pts) The volume of a sound, in decibels, is given by D = 10 log where I is the intensity 10-1 in watts per square meter. What is the volume, in decibels, of a sound whose intensity is 1.7783 * 10-9 watts per meter?

Express the vector t as a linear combination of the vectors u, v, and w. 1 2 THHH - 20. u= -4 V= 0 W = -3 5 1 t= t= 0 27 Qu+v+C W 1 5

Aldo spends $16 each time he travels the toll roads. He started the month with $240 in his toll road account. The amount, A (in dollars), that he has left in the account after t trips on the toll roads is given by the following function. A (t)=240-16t Answer the following questions. (a) How much money does Aldo have left in the account after 11 trips on the toll roads? s (b) How many trips on the toll roads can he take until his account is empty? trips

(d) For any two vectors u and v in R³, u xv||≤|u||||v||. True False Justification:

Which of the following represents vector w = -45i + 28j in trigonometric form? Round 0 to the nearest degree. Ow= 53cos 32°i + 53sin 32°j Ow= 53cos 148°i + 53sin 148°j w = 35sin 32°i + 35cos 32°j w = 35sin 148°i + 35cos 148°j

Solve the following system of equations. = 2 X+ y + z 2x + 5y + 2z -x+6y-3z = - 10 8 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. OA. The solution is (.1.1). (Type integers or simplified fractions.) B. There are infinitely many solutions. C. There is no solution.

Write the vector, parametric, and symmetric equations of the line passing through A(-1, 4, 1) and B(-1, 7, -2).

Use your calculator to evaluate cos -¹(0.53) to at least 3 decimal places. Give the answer in radians.

The reference angle of 239 degrees is __________ degrees. The reference angle of 342 degrees is __________ degrees. The reference angle of -136 degrees is _________ degrees.

A rectangular pool table centered at the origin has a width of 4 ft. and a length of 8 ft., with pockets at each of the four corners of the table. A ball sitting at the origin is struck at an angle of - 63.4° below the x-axis toward the bottom right pocket of the pool table, at a constant velocity of 2 ft. per second. a) Create a pair of parametric equations modeling the location of the ball t seconds after it is struck. b) After how many seconds will the ball reach the bottom right pocket of the pool table?

The cost of a can of Coca Cola in 1960 was $0.10. The function that models the cost of a Coca Cola can by year is C(t) = 0.10e⁰∙⁰⁵⁷⁶ᵗ , where t is the number of years since 1960. In what year was it expected that a can of Coca Cola will cost $1.00?

What is the range of the data? What percentage of data is below upper quartile? What percentage of data is located between the lower quartile and the median? What percentage of data is above the median?

Find sin(2x), cos(2x), and tan(2x) from the given information sin(x)=8/17 , x in quadrant I A)sin(2x) B)cos(2x) = C)tan(2x)

The driver of a car travelling at 45 km/h applies the brakes 50 m from a crosswalk. Assuming the driver applies a constant braking deceleration; find the deceleration required to bring the vehicle to a full stop at the crosswalk.

You take a group of 20 people to the movies, Kids' tickets cost $ 4.- each, and adults tickets cost $ 6.- each. Your total cost is $ 96.- . The number of kids in your group is Hint: Set up and solve a linear system.

Dylan invested $700 in an account paying an interest rate of 1(3/4) % compounded quarterly. Sebastian invested $700 in an account paying an interest rate of 1(1/4)% compounded daily. After 12 years, how much more money would Dylan have in his account than Sebastian, to the nearest dollar?

Suppose sin 2x=-(3^-2)/2 Find all solutions 0 ≤x ≤ 2π. Give exact values in radians, and type "pi" for π. x=2π/3, 5π/6, 5π/3,11π/6 (If there is more than one answer, enter the answers separated by commas)

What are all the solutions to the equation sin 2x = 2cos x in the interval [0, 2π)? {n} {0, π} {π/2} {π/2, 3π/2}

A formula of order 4 for approximating the first derivative of a function f gives: f'(0) ≈ 4.50557 for h = 1 f'(0) ≈ 2.09702 for h = 0.5 By using Richardson's extrapolation on the above values, a better approximation of f'(0) is: A) 0.17095 B) 1.00177 C) 0.97318 D) 1.93645

Vector ü has a magnitude of 3 units and it creates an angle of 160° with the positive x-axis. Find the magnitude and direction of 5ų. Give the direction as an angle measure between 0 and 360°. ||50|| = units 50 creates an angle of with the positive -axis.

Given vector a = (4, -2, 3) and b= (0,3,-5) find: 1. lal 2. a * b 3. the angle between a and b 4. la x bl Blank #1 Blank #2 Blank #3 Blank #4

Solve the inhomogeneous equation ∇²u = -f in an infinite cylindrical region for zero boundary conditions (of first or second kind) and construct the source function.

Given the complex number z₁ = 30( cos 23π/12 + i sin 23π/12) and z₂ = 5( cos 19π/12 + i sin 19π/12), express the result of z₁/z₂ in rectangular form with fully simplified fractions and radicals.

Let 0.02x+1.02 if x< -1 f(x)= -1.4x-4.4 if x> -1 4 if x=-1 The graph of f is given below. Determine the following function value. Enter DNE if the function is undefined at x = – 1. f(-1) =_______

Vector v has a magnitude of 3 units and it creates an angle of 160° with the positive x-axis. Find the magnitude and direction of 5v. Give the direction as an angle measure between 0° and 360°. ||5v|| = ________units. 5v creates an angle of __________° with the positive x-axis.

Determine the equation of the plane that contains the point A(-2, 2, 3) and the line r = m(2, -1,7), m ∈ R

Determine a vector equation for the plane containing the points O(0,0,0), A(-1,2,5), and C(3, -1, 7). Determine a vector equation for the plane containing the points P(-2, 2, 3),Q(-3, 4, 8), and R(1, 1, 10).

Determine parameters corresponding to the point P(5,3, 2), where P is a point on the plane with equation π:r= (2,0,1) + s(4, 2, -1) + t(-1, 1, 2), s, t∈R. b. Show that the point A(0, 5, -4) does not lie on π.

(1) Determine two pairs of direction vectors that can be used to represent the xy-plane in R³ . (2) Write a vector and parametric equations for the xy-plane in R³ .

A plane passes through the points P(-2, 3,1), Q(-2,3, 2), and R(1, 0, 1). (a) Using PQ and PR as direction vectors, write a vector equation for this plane. (b) Using QR and one other direction vector, write a second vector equation for this plane.

A plane has 7 = (-3, 5, 6) + s(-1, 1, 2) + v(2, 1, -3), s, v∈R as its equation. a. Give the equations of two intersecting lines that lie on this plane. b. What point do these two lines have in common?