Vector Calculus Questions and Answers

f(x, y) =x/y , P(8, 1), u = 3/5i +4/5 j (a) Find the gradient of f. Vf(x, y) = (b) Evaluate the gradient at the point P. Vf(8, 1) = (c) Find the rate of change of f at P in the direction of the vector u. Duf(8, 1) =

A water sprinkler sends water out in a circular pattern. Determine how large an area is watered. The radius of watering is 24 ft. Use 3.14 for π . The area of the circle is ft² (Type a whole number or decimal rounded to the nearest hundredth as needed.)

Given a triangle with a leg of 8 km and hypotenuse 22 km, find the missing side. The length of the missing side is km. (Round to the nearest thousandth.)

Solve the following equation. x⁴ - 4x² - 32 = 0

Use the distributive property to multiply a. -6(4x² - 2x - 11) b. x(x⁴ + 8x² - 11) c. 2x² (2x²-x-3) d. x² (3x³-8x² - 6x)

Factor the following expression completely: 6z² +z-1=

Solve logarithmic equation graphically log₅ (x + 1) - log₂ (x - 2) = 1

Solve the following equation. x⁴ - 81 = 0

A value of sinh x or cosh x is given. Use the definitions and the identity cosh² x - sinh² x = 1 to find the value of the other indicated hyperbolic function. sinh x = 5/12 , cosh x = sinh x = -4/3, tanh x= sinh x = 3/4, coth x= cosh x = 5/3, x>0, sinh x= cosh x= 17/15, x < 0, csch x =

Rewrite the expression as an equivalent expression functions greater than 1. 8 cos2x

Use a half-angle formula to find the exact value of the expression. COS 5π/12

Consider a ferris wheel that measures 400ft in diameter and spins at a constant speed completing a rotation once every 30 minutes. Assume you board the ferris wheel at the bottom at time zero. This is the same set up as the very first ferris wheel we considered in the video. The difference is that you need to determine the function x= g(t) which gives your horizontal position as a function of time. Assume where you board the ferris wheel at the bottom has a horizontal position of zero. Provide a plot for x = g(t) that shows at least 2 rotations around the ferris wheel. Hint: the horizontal position will take on both positive and negative values.

Let . a,β Z[i] with prime. Prove that if π|a, then 7 and a are relatively prime. Let , a. β € Z[i] with prime. Prove that if π| aß, then πa or π | β

Find the position and velocity vectors of a particle that has the given acceleration and the given initial velocity and position: a(t) = tj + tk, v(1) = 5j, and r(1) = 0.

Find the equation of the plane that passes through the points (2, 2, 1) and (-1, 1, -1) and is perpendicular to the plane 2x - 3y + z = 3.

Find an equation of the normal plane to a = sin(2t), y = cos(2t), z = 4t at the point (0, 1, 2π).

Sketch the plane curve r(t) = ti+t³j and the vectors T(t) and N(t) when t = 1. (Do NOT try to find T(t) and N(t)). Find the scalar tangential component of acceleration at of a particle with position vector r(t) =5ti + (t² + 5)j.

(a) Find the point at which the given lines intersect. r = (3, 2, 0) + t(2, -2, 3) r=(5, 0, 3) + s(-2, 2, 0) (x, y, z) = (b) Find an equation of the plane that contains these lines.

Find two unit vectors that are orthogonal to both (5, 3, 1) and (-1, 1,0).

Evaluate the Riemann sum for f(x) = ln(x) - 0.7 over the interval [1, 5] using eight subintervals, taking the sample points to be midpoints.

Show all work and upload a file to complete the following: Given that f = 3/1+x and g = 3x - 4, a) Find the composite function, f.g b) Determine the domain of the composite function.

Find the scalar equation of the plane that contains the point (2,-1,5) as well as the line (x,y,z) = (1,2,3) + t(3,4,-3)

What is the radius of a circle in which an angle of 3 radians cuts off an arc length of 30 cm?

Evaluate log3 54/log162 3 - log3 486/log18 3

Prove by definition that {(x,y,z)| x² +2²-6z=0} is a C^∞ surface.

From Times Square in New York City, you can see Central Park and the Empire State Building. The approximate distance from Times Square to Central Park is 2 miles, Central Park to the Empire State Building is 1.3 miles, and Empire State Building to Times Square is 0.9 miles. What is the approximate angle to the other two landmarks from Times Square?

Find a second-degree polynomial P such that P(3) = 17, P'(3) = 14, and P"(3) = 6. P(x) =

Graph the curve with parametric equations x = sin(t), y = 3 sin(2t), z = sin(3t).

At what point on the curve x = t³, y = 6t, z = t4 is the normal plane parallel to the plane 6x +12y - 8z = 3?

Suppose a = 11 and b = 6. Find an exact value or give least two decimal places: tan (A) = sec (A) = csc (A) = cot (A) =

Find the curvature of r(t) = (7t, t², t³) at the point (7, 1, 1).

Find the length of the curve correct to four decimal places. (Use your calculator to approximate the integral.) r(t) = (sin(t), cos(t), tan(t)), 0≤t≤π/4.

Find the equation of the plane through (3, -1, 2), (4, 0, 2), and (5, 3, 1).

Solve the following quadratic inequality: 3x² + 5x − 2 ≤ 0 Write your answer in interval notation.

Evaluate the following expressions. The answer must be given as a fraction, NO DECIMALS. If the answer involves a square root it should be entered as sqrt. For instance, the square root of 2 should be written as sqrt(2). If tan (θ) = -9/7 and sin(θ)<0, then find (a) sin(θ) (b) cos(θ) (c) sec (θ) (d) csc(θ) (e) cot(θ)

Solve: |4x-8| > 1 Give your answer as an interval. Note: Type oo for the infinity symbol ∞ and U for the union symbol U.

Use the hal zeros theorem to list all possible rational zeros of the following. f(x)= 10x³ + 6x² + 8x + 4 Be sure that no value in your list appears more than once.

The y-coordinate of a particle in curvilinear motion is given by y-8.71³-13.5t, where y is in inches and t is in seconds. Also, the particle has an acceleration in the x-direction given by ax = 3.4t in./sec². If the velocity of the particle in the x-direction is 12.7 in./sec when t=0, calculate the magnitudes of the velocity v and acceleration a of the particle when t = 3.3 sec. Construct vand a in your solution. Answers: When t - 3.3 sec. V- a- i in./sec in./sec²

Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify your answer. T(X₁X2 X3 X4) = (0,X3 +X4X3 X4 X3 X4) a. Is the linear transformation one-to-one? 1000 OA. T is not one-to-one because the standard matrix A has a free variable. OB. T is not one-to-one because the columns of the standard matrix A are linearly independent. OC. T is one-to-one because the column vectors are not scalar multiples of each other. OD. T is one-to-one because T(x)=0 has only the trivial solution. b. Is the linear transformation onto? OA. T is not onto because the first row of the standard matrix A is all zeros. OB. T is onto because the standard matrix A does not have a pivot position for every row. OC. T is not onto because the columns of the standard matrix A span Rª OD. T is onto because the columns of the standard matrix A span R¹

Find all y-intercepts and x-intercepts of the graph of the function. f(x) = -x² - 2x+48 If there is more than one answer, separate them with commas. Click on "None" if applicable. y-intercept(s): x-intercept(s): None 0/0 X 0/6 0,0,... 5 3/6 ?

Find a vector equation and parametric equations for the line. (Use the parameter t.) the line through the point (8, -5, 6) and parallel to the vector (1, 3, -2/3) r(t) = (x(t), y(t), z(t)) =

Let P be a point not on the line L that passes through the points Q and R. The distance d from the point P to the line L is d=la x b| / lal where a = QR and b = QP. Use the above formula to find the distance from the point to the given line. (5, 2, -1); x = 2 + t, y = 2 - 2t, z = 4 - 3t

Determine whether the lines L₁ and L₂ are parallel, skew, or intersecting. L₁: x = 9 + 6t, y = 12 - 3t, z = 3 + 9t L₂: x = 3 + 12s, y = 9 - 6s, z = 12 + 15s parallel skew intersecting

Find parametric equations for the line. (Use the parameter t.) the line through (3, 3, 0) and perpendicular to both i + j and j + k (x(t), y(t), z(t)) = Find the symmetric equations. x-3=y- 3 = -z x+ 3 = -(y + 3) = z x3 = -(y - 3) = z -(x- 3) = y - 3 = z x+ 3 = -(y + 3), z = 0

. If * x (x z) = (xy) xz, then which of the following option is correct ? (a) x and y are collinear. (b) and z are collinear. (c) y and z are collinear. (d) None

(5 points) Let V be a vector space over some field F a) (3 points) Let W be a subspace of a vector space V. Suppose x, y € V are not in W. Show that if re span (W,y) then y span (W, x) b) (2 point) Give an example of a vector space V, a subspace W CV, and vectors x, y € V such that are span (W,y) but y & span (W, x)

A Ferris wheel rotates 2852° per ride. How far would a rider seated 8 feet from the center of the Ferris wheel travel during the ride? Round your answer to the nearest thousandth.

Use the ∈ - δ definition of limit to show that lim z→1+i (2z + 3i) = 2 + 5i.

A jet airliner, flying initially 300 mi/hr due east, suddenly enters a region where the wind is blowing 100 mi/hr, pushing the aircraft in the direction 30° north of east. What is the resultant speed of the aircraft?

Graph the parabola. y=x² + 4x+7 Plot five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex. Then click on the graph-a-function button.