Vector Calculus Questions and Answers

This problem has two parts. Part 1: Here is the graph of 3cos(3x +(1/4)π ) the interval [- π, π]. Give the coordinates of point A, the y-intercept:_______(To enter, type "pi" or select in the math editor.) Give the coordinates of point B, the first x-intercept along the positive x-axis:__________ Give the coordinates of point C, the first peak of the graph with positive value:___________

Determine if the equation specifies a function with independent variable x. If it does, find the domain. If it does not, find a value of x to which there corresponds more than one value of y. x² + y² = 49 Does the equation specify a function with independent value x? (a) No (c) Yes

Find fxx, fxy, fyx, and fyy for the following function. (Remember, fyx means to differentiate with respect to y and then with respect to x.) f(x, y) = e^(9xy) fxx = 0 fxy = 0 fyx = 0 fyy = 0

Write the equation of the line through the indicated point with the indicated slope. Write the final answer in the form y = mx + b m =7/2; (2, 2) The equation of the line is_____ (Type your answer in slope-intercept form. Use integers or fractions for any numbers in the equation.)

For each of the following, show that the given line and plane do not intersect. Explain how the equation that results implies there is no intersection. a). L: r = (-1, 1, 0) + s(-1, 2, 2), s∈ R; π: 2x - 2y + 3z − 1 = 0

For each of the following, show that the line lies on the plane with the given equation. Explain how the equation that results implies this conclusion. a. L: x = -2 + t, y = 1 - t, z = 2 + 3t, t∈ R; π : x + 4y + z-4 = 0

The CN tower is 553m tall. A person is south of the tower, in a boat on Lake Ontario at a position that makes an angle of elevation of 10° to the top pf the tower. A second person is sitting on a park bench, on a bearing of 60° relative to the tower, observing the top of the tower at an angle of elevation of 9°. Determine the displacement between the two people.

For the following, determine points of intersection between the given line and plane, if any exist: a. L: r = (-1, 3, 4) + p(6, 1, −2), p∈ R; π : x + 2y - z + 29 = 0

An airplane is flying at 650 km/h on N60 °W. The wind is blowing at 40km/h from N75°E. Calculate the magnitude and direction of the resultant force. Draw a diagram of this situation

Find the inverse of by hand. Use matrix multiplication to prove they are inverses

The line has an equation of y=2x intersects the curve y² = 8x, Determine the area between the curve and the line. A) 17/3 B) 11/3 C) 4/3 D) 7/3

For the equation 2x + 4y = 11, determine a). the number of solutions b). a generalized parametric solution c). an explanation as to why it will not have any integer solutions

Let u =(5,-2), v = (-1,0), and w = (-8,6). Calculate u + 3w - 5v.

Find two unit vectors perpendicular to (2,-2,-3) and (0,2,1). Use the dot product to verify the result is perpendicular to the two original vectors.

Find an equation of the circle that has center (-1, 6) and passes through (1, -3).

Express as a complex number in simplest a+bi form: -3 + 21i/9 - 3i

Marshal invests $4500 into an account bearing 6% interest, compounded semi-annually. How much interest will be added into this account after five years? Round to the nearest cent.

Find an equation of the circle that has center (6, -4) and passes through (-2, 0).

Find the absolute maximum and the absolute minimum of f(x, y) = 2x³ + y² on the closed and bounded set {(x, y): x² + y² ≤ 1}.

Determine the Cartesian equation of the plane. F = (0,0,3) + S(1,1,4) + t(-2,1,5), (S, t) ∈ R

A sinusoidal function has an amplitude of 4 units, a period of 180°, and a maximum point at (0, 9). a) Represent the function with an equation using the cosine function. b) Create a graph of this function over two cycles. (Make sure to label the y-axis with the correct numbers)

Determine the angle between the following lines: R1= (1,2,3) + € (0, 1,-3), t{R R2= (-5,2,0) +5(1,-1,4), SER

Determine the vector and parametric equations of the plane: 3x - 2y +z - 5 = 0

A town has a population of 6000 and grows at 3.5% every year. To the nearest year, how long will it be until the population will reach 9400?

An audio cable is 14.5 dm long. How long is the cable in meters? Be sure to include the correct unit in your answer.

Determine the angle between the following lines: P₁ = (1,2,3) + t(0, 1,-3), t ∈ R P₁ = (-5,2,0) +S(1,-1,4), S ∈R

Determine the angle between the following lines: F₁ = (1,2,3) + t( 0, 1,-3), t ∈ R F₂ = (-5,2,0) +S(1,-1,4), S ∈ R

Find the exact value. sin(17π/12) Write the given angle17π/12 as a sum of two special angles: Enter the combination with the smallest angle first. Enter the smaller angle on each possible combination. 17π/12 = ______ OR 17π/12 = _____ OR 17π/12 = _____ b) Use one of the combination to Sum/Difference Formula to evaluate exactly. sin(17π/12) = _______

Determine the coordinates of the point where the plane with the below equation crosses the z-axis: (2, -1, 4)+ s(3, 1, -2) + t(-1, -1, 2)

At the movie theatre, child admission is $5.80 and adult admission is $8.90. On Tuesday, 163 tickets were sold for a total sales of $1246.10. How many adult tickets were sold that day?

Zachary wants to build a rectangular pen for his animals. One side of the pen will be against the barn; the other three sides will be enclosed with wire fencing. If Zachary has 400 feet of fencing, what dimensions would maximize the area of the pen? a) Let w be the length of the pen perpendicular to the barn. Write an equation to model the area of the pen in terms of w Area = b) What width w would maximize the area? W=>

Find a plane through the points (-3, 7 ,-4), (-3, -1, 8), (3, 1, -3).

A person standing close to the edge on top of a 28-foot building throws a ball vertically upward. The quadratic function h = - 16t² + 108t + 28 models the ball's height about the ground, h, in feet, t seconds after it was thrown. a) What is the maximum height of the ball? feet b) How many seconds does it take until the ball hits the ground? seconds

Use the Pythagorean identity to find cos x. sin x = 9/13 cos x=

Find all solutions of the system of equations. (If there is no solution, enter NO SOLUTION.) xy = 24 { 2x2. (x, y) = (x, y) = - y² + 4 = 0 (smaller x-value) (larger x-value)

In Exercises 7-10, find the slope of the line passing through the pair of points. Then use a graphing utility to plot the points and use the draw feature to graph the line segment connecting the two points. (Use a square setting.) 7. (0, -10), (-4, 0) 9. (-6, -1), (-6, 4) 8. (2, 4), (4, -4) 10. (-3,-2), (1,6)

In a solar system, two comets pass near the sun, which is located at the origin. Comet E is modeled by (Y+162 (y+13) x2 1, where all measurements are in astronomical units. Part A: What are the vertices for the path of Comet E? Show your work. (4 points) Part B: Which comet travels closer to the sun? Give evidence to support your answer. (5 points) Part C: What key feature does the sun represent for both comets? Give evidence to support your answer. (6 points) = 1 and Comet H is modeled by

In how many ways can the expression a b c be rewritten as a product without exponents?

Assume A and B are independent events with P(A) = 0.9 and P(B) = 0.6. Find (a) P(A n B), (b) P(AUB), (c) P (A' B'), (d) P (A'NB), and (e) P (ANB). (a) P(A n B)= (b) P(A U B)= (c) P (A'n B') = (d) P (A' n B) = (e) P (ANB) =

Use the Test for Independence to determine if events A and B are independent. P(A)=0.5, P(B) = 0.8, P(An B) = 0.3 Choose the correct answer below. A. Events A and B are not independent because P(A) P(B) #P(An B). They are dependent. B. Events A and B are independent because P(A) P(B) #P(An B). C. Events A and B are not independent because P(A) P(B) = P(An B). They are dependent. D. Events A and B are independent because P(A) P(B) = P(An B). E. It is not possible to determine based on the given information.

Find r and a₁ for the geometric sequence with a₃= 2 and a₈= 1/16 r= ____ (Simplify your answer.)

Find all solutions of the equation 2sin²x - cos x =1 in the interval [0, 2π) with x1 < x2 < x3. The answers are X1 = ______ x2 = ______ x3 = ______

Use the tangent identity to find tan x. sin x = 2√10/7 COS X = 3/7 tan x =

The plane containing the point (7,-7,6) and parallel to the plane 2x+8y-5z=3 has equation Ax+By+Cz=D. If A=2 find the following: B = __________________ C = __________________ D = __________________

For what values of a and b will make the two complex numbers equal? 5-2i = 10a + (3 + b)i

Calculate the Fourier Transform in Continuous Domain (CFDF) of the following signs: 15 f(t) = e‾ᵗtu(t)cos 3t =

Find t for the following terminal points, 0 < t < 2π (a) P(0, 1), t = (b) P ( - V2/2 , V2/2),t= (c) P(-1,0), t = (d) P ( - V2/2, - V2/2), t= (e) P(0, - 1), t = (f) P(-V2/2, - V2/2).,t=

Given p with magnitude of 40 and a direction of 25°, q with magnitude of 75 and a direction of 270°, and r with magnitude of 30 and a direction of 220°, what is the magnitude of p + q + r? Round to the thousandths place.

If vector r = <4, 6> and s= <8, -3> then which is r-s?

Solve 2x³-3x²-11x <-6. Include an Interval Chart/Number Line.