Vector Calculus Questions and Answers

Let f(3) = √|x - 1|. Determine the following function value. Enter DNE if the function is undefined at x = 1. f(1) = ______

The function given by f(x) = cos x is (A) Decreasing in (0, 2π) (B) Increasing in (0, 2π) (C) Neither increasing nor decreasing in (0, 2π) (D) Increasing in (0, π )

Run a regression analysis on the following data set with y as the response What is the predicted value of the response variable when x = 74.5? y=Round to 1 decimal place.

Given the ellipse (x - 3)/25 + (y - 4)/36 = 1 Find the center point: List the vertices (separated by a comma):

Suppose sin Φ = 8/17, cos Φ = -15/17 sin(2Φ) = ________ cos(2Φ) = _________

The center of the conic section with the equation x² +(y+1)²/=1 is 64 (A) (-1,0) (B) (0, -1) (C) (0, 1) (D) (1,8) (E) (8,1)

If point P has rectangular coordinates (-8, -6), one possible pair of polar coordinates for P is: (A) (10,37°) (B) (10, 53°) (C) (10, 217°) (D) (10, 233°) (E) (10, -307°)

The center of the conic section with the equation x + (y+1)²/64 =1 is; (A) (-1,0) (B) (0, -1) (C) (0, 1) (D) (1,8) (E) (8,1)

The line with vector equation 1 = (2,0, 1) + s(-4,5,5), S∈R, lies on the plane T, as does the point P(1, 3, 0). Determine the Cartesian equation of T.

A toy manufacturer builds three holiday-themed toys, A, B, and C, and keeps track of the sales of these three toys over the first three weeks of December. The matrix X below shows the number of each toy sold during each week. The matrix Y shows the price of each toy. A B C Price(S) Week 1 [25 50 15] A [20.00] Week 2 [30 55 10] = X B [5.00] = Y Week 3 [25 60 5] C [40.00] a. Calculate the product matrix XY. b. How would you label each row and column of the product matrix XY? Explain your reasoning. c. The store owner wants to add one more row to the bottom of matrix X showing the number of each toy sold in Week 4 of December. If the owner adds one more row to matrix X with whole number values, will the product matrix XY be defined or undefined? Explain your reasoning.

A plane contains the point A(2, 2, -1) and the line 1 = (1, 1,5) + s(2, 1, 3), S∈R. Determine the Cartesian equation of this plane. no containing the point (-1. 1.0)

For sin2x + cosx = 0, use a double-angle or half-angle formula to simplify the equation and then find all solutions of the equation in the interval [0, 2π). The answers are X1 =___________ X2 =___________ X3 =___________ and X4 =___________ with x1 < x2 < x3<x4

Algebra 1 x x 1 Calculate (a) Spectral decomposition of the matrix (b) Rank-1 approximation of the matrix

Which of the following transformation creates a figure that is similar (but not congruent) to the original figure? (A) Translation (B) Rotation (C) Dilation

Determine the Cartesian equation of the plane containing the point (-1, 1, 0) and perpendicular to the line joining the points (1, 2, 1) and (3, -2, 0). Evoloin the process you would use to determine the angle formed between

Determine the angle between the planes x + 2y – 3z – 4 = 0 and x + 2y – 1 = 0. b. Determine the Cartesian equation of the plane that passes through the x - 3 y + 1 point P(1, 2, 1) and is perpendicular to the line x-3/-2 = y+1/3 = z+4/1

What is the value of k that makes the planes 4x + ky – 2z + 1 = 0 and 2x + 4y – z + 4 = 0 parallel? b. What is the value of k that makes these two planes perpendicular?

The function f(x) is defined by the rule "add 3, multiply by 2, and square root". How is the inverse, f¹(x), defined? a) Square, divide by 2, and subtract 3. Ob) Square root, multiply by 2, and add 3. c) Square, divide by 3, and subtract 2. d) Subtract 3, divide by 2, and square.

The three non collinear points P(-1, 2, 1), 2(3, 1, 4), and R(-2, 3, 5) lie on a plane. a. Using PQ and QR as direction vectors and the point R(-2, 3, 5), determine the Cartesian equation of this plane. b. Using QP and PR as direction vectors and the point P(-1, 2, 1), determine the Cartesian equation of this plane.

The table provides various values, including all minimums and maximums, of a cosine function f(x) on the interval [0, π). Angle 0 π/4 7π/12 3π/4 11π/12 π f(x) 1 3 0 -1 0 1 What is the period of the function? A) π/3. B) π/2. C) π. D) 2π.

Point P lies on the unit circle in the first quadrant at angle 8 shown below, and has coordinates (a, b). a. Kiara says that b = √1-a². Do you agree or disagree with Kiara? Explain your reasoning. b. Assume a = 4/5 and b = 3/5. i. Find sin(-θ) ii. Find sin(2θ) iii. Find θ, rounding your answer to the nearest tenth of a degree.

If Matrix A is given by A = B could be 4-6 (A) -2 4 (B) (C) -2 4 3] 46 24 and the product Matrix AB is given by AB = -64 (D) 1 (E) Matrix B cannot be determined by the information given. 10 then Matrix

What is the magnitude of the vector 8i - 15j? (A) - 7 (B)√23 (C) 7 (D) 17 (E) 23

10. The asymptotes of the hyperbola with the equation (x^2)/4-(y^2)/25=1 (A) y =±(4/25)x (B) y =±(25/4)x (C) y = ±(2/5)x (D) y =±(5/2)x (E) y =±x

In ΔABC, side a=25, side b=30, and side c=45. The measure of ∠C to the nearest degree is: (A) 132° (B) 109° (C) 71° (D) 39° (E) 32°

The center of the sphere given by the equation x² + y² + z² = 20 (A) (4, 0, -10) (B) (2, 0, -5) (C) (0, 0, 0) (D) (-2, 0, 5) (E) (-4, 0, 10)

Which of the following points is NOT the same point as the polar coordinate point (8, π/6)? (A) Rectangular coordinates: (4√3, 4) (B) Polar coordinates: (8, 30%) (C) Polar coordinates: (8, 390) (D) Polar coordinates: (-8, 210°) (E) Polar coordinates: (8, -30%)

Let X and Y be independent and Cauchy distributed. Show that (X+Y)/2 also has a Cauchy distribution.

-> -> -> -> -> -> The sum of v = 4 j - 6 k and w= -i - 2 k is: (A) < 3, -8 > (B) < 5, 4> (C) < -1, 4, -8> (D) < 4, -7, -2> (E) < 4,-1, -8>

Find the lateral area and surface area of the cone below. 515.8 527.8 639.8 703.8 728.8 lateral area = surface area =

For y=10 sin 2x, its amplitude is □ its period is □

State the x-, y-, and z-intercepts for each of the following three planes: i. π₁: 2x + 3y = 18 ii. π₂: 3x - 4y + 5z 120 iii. π₃: 13y-z = 39 State two direction vectors for each plane.

For each given equation, sketch the corresponding plane. (a)2x + 2y + z-4 = 0

For each of the following equations, sketch the corresponding plane: (a) π₁: 4x - y = 0 (b). π3:2 = 4

Which of the following is the focus of the parabola with the equation (y + 4)² = 12x? (A) (-3,-4) (B) (0, -1) (C) (0, -7) (D) (3, 0) (E) (3,-4)

Name the three planes that the equation xyz = 0 represents in R³.

Find the distance between the pair of points. (0,0) and (9,- 12) The distance between the points is _________ units.

Seawater has density 1025 kg/m³ and flows in a velocity field v= yi+xj, where x, y, and z are measured in meters and the components of V in meters per second. Find the rate of flow outward through the hemisphere x² + y² + z² = 4, z ≥ 0. ________ kg/s

The equation of the parabola with focus (2, 1) and vertex (2, 4) is: La ecuación de la parábola con foco (2, 1) y vértice (2, 4) es: (a) (x-2)² = 12(y-4) (b) (x-2)² = -12(y-4) (c)(y-4)² = -12(x-2) (d )(-4)² = 12(x-2)

Let g(x) = 2x - 4x² +9 Determine the value(s) of x, if any, for which g has a horizontal tangent line. (A) g has a horizontal tangent line at x = ______ (B) g has no horizontal tangent lines.

Si u=(-1,5); v = (5.-6), entonces la magnitud de 3u-2v es: (a)√178 (b) 4√35 (c) √898 (d) 4√10

At a tournament in San Diego, a bowman shoots a arrow. The arrow moves along a parabolic trajectory. The highest point is reached by the arrow is 320 feet in the air. The arrow lands 360 feet away from the bowman. Write a function h(x) giving the height of the arrow as a function of the horizontal distance from the bowman. h(x) =__________feet What is the height of the arrow when it its horizontal distance from the bowman is 71 feet?_________feet. What will be the horizontal distance to the arrow when its height is 156 feet?___________ feet(Round your answer(s) to three decimal places. If there is more than one horizontal distance, enter both, separated by commas)

The minimum SOP form of the following function F = xy +z is (a)xy+xz+x'y'z' (b)xyz'+xy'z+x'yz+x'y'z' (c)xyz+xy'z'+x'yz'+x'y'z (d)xz+yz+x'y'z'

Suppose sin (2x) = 3/5. Then cos(x) = __________ (If there is more than one possible value, enter the values separated by commas)

Find the intersection of the line l and the plane π. l:r =(4,-1,4)+t(5,-2,3) π: 2x+5y+z+2=0

what is the net signed area bounded by g(x) = 3x and the x-axis on [-1,1]? Use a graph and a simple geometric formula to figure this net signed area.

A small retail business has determined that the correlation coefficient between monthly expenses and profits for the past year, as measured at the end of each month, is r= 0.56. Assuming that both expenses and profits are approximately normally distributed, test at the 5% (0.05) level of significance the null hypothesis that there is no correlation between them.

(05.03, 05.04 HC) Triangle △ABC has side lengths of a = 16, b= 16√3, and c = 32 inches. Part A: Determine the m∠A. Part B: Show how to use the unit circle to find tan A. Part C: Calculate the area of ∠ABC.

In the summer, humidity interacts with the outdoor temperature, making a person feel hotter due to a reduced heat loss from the skin caused by higher humidity. The temperature-humidity index, Th, is what the temperature would have to be with no humidity in order to give the same heat effect. One index often used is given byTₕ= 1.98T 1.09(1-H)(T-58)-56.9, where T is the air temperature, in degrees Fahrenheit, and H is the relative humidity, expressed as a decimal. Find the temperature-humidity index if T = 82° and H= 55%. The temperature-humidity index at 82° and 55% humidity is □° (Round to one decimal place as needed.)

Write the equation of the conic section shown below.