Vector Calculus Questions and Answers

What are the magnitude and direction of w = (-5, -14)? Round your answer to the thousandths place.

Change from rectangular to spherical coordinates. (Lèt p ≥ 0, 0 ≤ θ ≤ 2π, and 0 ≤ φ≤ π.) (a) (0, 3, -3) (p, θ, φ) = (b) (-7,7,7√6) (p, θ, φ) =

Solve the system of equations. x-3y - z=-9 -x+8y-4z=24 2x-15y+7z=-45 Select the correct choice below and fill in any answer boxes within your choice. A. There is one solution. The solution is ( , , ) (Type exact answers in simplified form.) B. There are infinitely many solutions. The solutions are ( , ,z), where z is any real number.(Type exact answers in simplified form.) C. There is no solution.

Vector v has an initial point at (8, 8) and a terminal point at (16, 11). Which of the following represents the linear form of v? v=-3i-8j v=3i+ 8j v=-8i-3j v = 8i + 3j

Given u = -3i+5j and v= -24i + 40j, are u and v parallel or orthogonal? Explain. The vectors are orthogonal because cos θ = 1. The vectors are orthogonal because u • v = 0. The vectors are parallel because cos θ = 1. The vectors are parallel because u • v = 0.

Determine the measure of angle in standard position, and the diagram: Show all your work to receive full marks. a) Sin = -0.7502, if is in the 3rd quadrant b) Sin = 0.9122, if is in the 1st quadrant

Determine the solution set for the equation. (x-6)²+(y+9)² = -9 The solution set is

An object is suspended by two cables attached at a single point. The force applied on one cable has a magnitude of 125 pounds and acts at an angle of 37°. The force on the other cable is 75 pounds at an angle of 150°. Part A: Write each vector in component form. Show all necessary work. (4 points) Part B: Find the dot product of the vectors. Show all necessary calculations (3 points) Part C: Use the dot product to find the angle between the cables. Round the answer to the nearest degree. Show all necessary calculations. (3 points)

Consider the system of equations dx dt = x(1-x 2-y) dy dt = y(1-y 3-x), taking (x, y) > 0. (a) Write an equation for the (non-zero) vertical (x-)nullcline of this system: (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: (Enter your equation, e.g., y=x.) (Note that there are also nullclines lying along the axes.) (b) What are the equilibrium points for the system? Equilibria = (Enter the points as comma-separated (x,y) pairs, e.g., (1,2), (3,4).) (c) Use your nullclines to estimate trajectories in the phase plane, completing the following sentence: If we start at the initial position (1 2,3 2). trajectories ? the point (Enter the point as an (x,y) pair, e.g., (1,2).)

The Coffee Counter charges $9 per pound for Kenyan French Roast coffee and $8 per pound for Sumatran coffee. How much of each type should be used to make a 20 pound blend that sells for $8.50 per pound? The Coffee Counter should mix pounds of Kenyan Roast coffee and pounds of Sumatran coffee to make 20 pounds of a blend that sells for $8.50 per pound.

The functions f, g, and h are defined as follows. f(x) = −11+|6x| g(x) = x 1 +x2 h(x) = √-2x+15 Find f(-3 2), g (-2), and h (3). Simplify your answers as much as possible.

Given u = 4i +2j and v = −7+ 5j, find the dot product u . v.

The point A(-6, 6) is reflected over the point (0,7) and its image is point B. What are the coordinates of point B?

Use the log properties to rewrite the expression. Simplify as much as possible. Enter the final result without any blank spaces. In (²²)

To verify the identity, start with the more complicated side and transform it to look like the other side. Choose the correct transformations and transform the expression at each step. csc θ - sin θ

Create each of the following during your work: Det A, Det Ax, and Det Ay. 4x + y = 5 x - ky = 2 Complete the ordered pair: (x,y) where

On our moon, the acceleration of gravity is 1.6 m/sec². If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 45 seconds later?

Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph whe function has a local maximum or local minimum value. Then graph the function in a region large enough to show these points simultaneously. Add to your picture the graphs of the function's first and second derivatives. y=x5-4x4-200

Explain why the following four statements ask for the same information. (i) Find the roots of f(x) = 3x3 - 3x - 1 (ii) Find the x-coordinates of the intersections of the curve y = x3 with the line y = 3x + 1. (iii) Find the x-coordinates of the points where the curve y = x³ - 3x crosses the horizontal line y = 1. (iv) Find the values of x where the derivative of g(x) = 3 4x4 - 3 2x² - x + 5 equals zero.

At noon, ship A was 12 nautical miles due north of ship B. Ship A was sailing south at 12 knots (nautical miles per hour; a nautical mile is 2000 yards) and continued to do so all day. Ship B was sailing east at 9 knots and continued to do so all day. The visibility was 5 nautical miles. Did the ships ever sight each other?

One 8-oz serving each of coffee, energy drink, and soda contains 201 mg of caffeine. One serving of coffee has 7 mg more caffeine than two servings of soda. One serving of energy drink contains 35 mg less caffeine than one serving each of coffee and soda. Find the amount of caffeine in one serving of each beverage.

For the real-valued functions g(x)=x² + 1 and h(x)=√x-4, find the composition goh and specify its domain using interval notation.

Use mathematical induction to prove that the alternating sum of n numbers is for odd n and for even n. 1-2+3+.........+n=n+1/2 1-2+3-..........-n= -n/2

Tutorial Exercise Solve for x accurate to three decimal places. In(√x + 6) = 3 Step 1 The given function is In (√x + 6) = 3. Take the natural antilog of each side of the equation. Step 2 Simplify the above expression. x + 6 = e___ x = e__

To measure the height of the Eiffel Tower in Paris, a person stands away from the base and measures the angle of elevation to the top of the tower to be 60°. Moving 210 feet closer, the angle of elevation to the top of the tower is 70°. How tall is the Eiffel Tower? Give your answer as a decimal rounded to 2 places after the decimal.

(1 point) Consider the two points (3,-1) and (6, 10). The distance between them is: The x co-ordinate of the midpoint of the line segment that joins them is: The y co-ordinate of the midpoint of the line segment that joins them is:

What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve y = 5 x at some point?

Draw a vector representing the course of a ship that travels 6 miles on a course with bearing 225

The figure shows a 800 foot tower on the side of a hill that forms a 8° angle with the horizontal. Find the length of each of the two guy wires that are anchored 50 feet uphill and downhill from the tower's base and extend to the top of the tower. Part (a) What is the length of the uphill guy wire? feet (Round to the nearest tenth as needed.)

Use a calculator to perform the indicated operation. Give the answer in rectangular form. (4 cis 2π 9)2 (Type your answer in the form a + bi. Use integers or decimals for any numbers in the expression. Round to four decimal places as needed.)

Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of f(x) = x+6/x-6 Find any vertical asymptotes of f(x). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one vertical asymptote, (Type an equation.) B. The function has two vertical asymptotes. The leftmost asymptote is and the rightmost asymptote is (Type equations.) C. There are no vertical asymptotes. Find the intervals where f(x) is increasing or decreasing. Select the correct choice below and fill in the answer box(es) to complete your choice. (Type your answer in interval notation. Use a comma to separate answers as needed.) A. The function is increasing on It is never decreasing. B. The function is increasing on It is decreasing on C. The function is decreasing on It is never increasing. Find the location of any local extrema of f(x). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. There is a local minimum at x = (Type an integer or a decimal. Use B. There is a local maximum at x = (Type an integer or a decimal. Use C. There is a local maximum at x = There is no local maximum. a comma to separate answers as needed.) There is no local minimum. a comma to separate answers as needed.) and there is a local minimum at x = (Type integers or decimals. Use a comma to separate answers as needed.) D. There are no local extrema. Find the intervals where f(x) is concave upward or downward. Select the correct choice below and fill in the answer box(es) to complete your choice. (Type your answer in interval notation. Use a comma to separate answers as needed.) It is never concave downward. A. The function is concave upward on___It is never concave download B. The function is concave downward on__It is never concave upward. C. The function is concave upward on___It is concave downward on

Use elimination to solve the system of equations. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. 5x+y = -3 7x-3y=-35 Select the correct choice below and fill in any answer boxes in your choice. A. The solution of the system is (Simplify your answer. Type an ordered pair.) B. There are infinitely many solutions. C. There is no solution. Choose the correct statement. A. The system is consistent and the equations are independent. B. The system is consistent and the equations are dependent. C. The system is inconsistent and the equations are independent. D. The equations are dependent.

(1 point) A ferris wheel is 28 meters in diameter and makes one revolution every 8 minutes. Suppose you board the ferris wheel at ground level when time is t = 0. For how many minutes of one revolution will your seat be above 21 meters? For minutes of any one revolution you will be above 21 meters.

Suppose that triangle ABC is a right triangle with a right angle at C and hypotenuse c. Also note that a is the length of the side opposite angle A and b is the length of the side opposite angle B. Given that a = 1 and A = 17°, determine the values indicated below. Round to four decimal places when needed. b = C = B =

Find the volume of the given solid. enclosed by the paraboloid z = x² + y2 + 1 and the planes x = 0, y = 0, z = 0, and x + y = 7

A lemonade recipe calls for 1/4 cup of lemon juice for every 1 cup of water. How many cups of water would be needed if you used 1 cup of lemon juice? 1/4 cups 1 cup 2 cups 4 cups

Find the average value of f over the given rectangle f(x, y) = 4x²y, R has vertices (-1,0), (-1, 4), (1, 4), (1, 0) favg=

A company markets exercise DVDs that sell for $14.95, including shipping and handling. The monthly fixed costs (advertising, rent, etc.) are $14,630 and the variable costs (materials, shipping, etc.) are $5.45 per DVD. (A) Find the cost equation and the revenue equation. (B) How many DVDs must be sold each month for the company to break even? (C) Graph the cost and revenue equations in the same coordinate system and show the break-even point. Interpret the regions between the lines to the left and to the right of the break-even point. (A) What is the cost equation?

Rewrite each equation as requested. (a) Rewrite as an exponential equation. log3 1/81 = -4 (b) Rewrite as a logarithmic equation. 8² = 64

Find the interest earned when $22825 is invested at 6% compounded continuously for 5 years. Round the answer to the nearest cent. A. $14,804.52 B. $15,885.06 C. $1,566.87 D. $7,985.53

The amount of money, A(t), in a savings account that pays 3% interest, compounded quarterly for t years, with an initial investment of P dollars, is given by At =P[1+ 0.03/4)^4t If $5000 is invested at 3%, compounded quarterly, how much will the investment be worth after 6 years? How much money will there be in the account at the end of 6 years?

Solve for t to two decimal places. 4 = e^0.08t

In a frame of three-dimensional space, the coordinates of the points O, A, B and C as well as vector ū are: 0= (0,0,0) A= (1,2,3) B= (8,6,7) C= (-2,1, 8) ū = (-3.4, -1) a) Determine the angle formed between the vectors AB and AC. b) Determine the coordinates of point P located 3 units from point A on segment AB. c) Find the coordinates of a point Q such that the vector AQ is parallel to the vector ū. d) Find the centroid of A, B and C. e) Find the volume of the tetrahedron OABC. f) Determine the normal equation of the plane containing the points A, B and C.

Given a RIGHT triangle with one LEG measured 5m in length, and its LONGEST side is 6m; find the measure, in degrees, of the SMALLEST of the 3 angles in that triangle and round it off to the nearest tenth. A. 50.2 B. 39.8 C. 56.4 D. 33.6 E. NO correct choices

Of all points (x,y,z) that satisfy x + 4y + z = 24, find the one that minimizes (x-1)² + (y-1)²+(z-1)² The given function has a minimum at which satisfies x + 4y + z = 24.

Solve x³- 2x² - 4x + 8 = 0 by factoring by grouping. Smaller x = Larger x =

Use long division to find the quotient and remainder. (-x² +11) ÷ (x + 2) What is the quotient? What is the remainder?

Find the extreme values of f subject to both constraints. (If an answer does not exist, enter DNE.) f(x, y, z) = yz + xy; xy = 1, y² + z² = 49 maximum minimum

Find the maximum volume (in cubic units) of a rectangular box that is inscribed in a sphere of radius r units. cubic units

Find any points on the hyperboloid x² - y² - z² = 1 where the tangent plane is parallel to the plane z = 4x + 4y. (If an answer does not exist, enter DNE.) (x, y, z) =