Vector Calculus Questions and Answers

The magnitude and direction of vectors t, u, and v are shown in the table. Vector Magnitude Direction t 7 240⁰ u 10 30⁰ v 15 310⁰ What is the direction of t + u + v? Round to the nearest degree.

Vector u has a magnitude of 30 and a direction of 70°. Vector v has a magnitude of 40 and a direction of 220°. What is the direction of u + v? Round to the nearest degree.

Rewrite the expression without the log of a product, power or root. log₃(x¹⁴ ∛y¹⁸) =_______________

Match the words with their definition. 1. simple interest 2. principal 3. compound interest a. When you deposit money into a savings account b. interest paid only on the principal c. the bank pays on the principal and the interest already earned

Complete the square for the following in order to rewrite the equation of the conic in standard form. State any applicable characteristics, including center, vertex (vertices), focus (foci), directrix and/or equations of asymptotes. Provide your calculations. (8pts each) i. 9x² - y² - 14y - 58 = 0 ii. 16y2 - 56y - 16x +81 = 0

a) Explain the difference between a positive angle and a negative angle. b) Consider the time 11:30 where the initial side is the hour hand and terminal side is the minute hand. Draw the angle between the two hands in standard position. State the angle in positive degrees and then restate the angle as a negative angle. c) Draw a picture depicting the definition of a radian and in your own words, write a definition. d) Draw a circle. Show approximate radian and degree values for every 1/8 of the circle. Show approximate values for 1, 2, 3, 4, 5, and 6 radians.

Algebraically find the inverse of f(x) = 3 cos (2x), showing your steps. State the domain and range of both f(x) and f-¹(x).

A club with nine members is to choose three officers: president, vice-president, and secretary-treasurer. If each office is to be held by one person and no person can hold more than one office, in how many ways can those offices be filled?

A 5-digit phone number cannot start with 0. Assume that there are no restrictions on the remaining 4 numbers. How many telephone numbers are possible in which all 5 digits are different? The possible number of 5-digit telephone numbers where all the digits are different is ___.

Assume A and B are independent events with P(A) = 0.8 and P(B)=0.5. Find (a) P(A ∩ B), (b) P(AUB), (c) P (A' ∩ B'), (d) P (A'∩B), and (e) P (A ∩ B'). (a) P(A ∩ B)= (b) P(A U B) = (c) P (A' ∩B') = (d) P (A' ∩ B) = (e) P (A ∩ B') =

Let x be a continuous random variable with a standard normal distribution. Using the accompanying standard normal distribution table, find P(0 ≤x≤2.23).

The population of a colony of rabbits grows exponentially. The colony begins with 10 rabbits; 5 years later there are 340 rabbits. Using your solution from question 19, find and interpret R−¹(y).

For the probability density function f defined on the random variable x, find (a) the mean of x, (b) the standard deviation of x, and (c) the probability that the random variable x is within one standard deviation of the mear 1 f(x)=2x. (5.7] a) Find the mean. p= (Round to three decimal places as needed.) b) Find the standard deviation. G= (Round to three decimal places as needed.) c) Find the probability that the random variable x is within one standard deviation of the mean. The probability is (Round to three decimal places as needed.)

Your parents have a credit card with a balance of $2,748.56. The interest rate is 10.5% APR. A late payment fee of $40.00 is added to the principal if they pay after 6/1. They make a payment for $300.00 on 6/1. How much interest did they pay? $24.05 $288.60 $24.40 $297.00

Solve AABC subject to the given conditions, if possible. Round the lengths of the sides and measures of the angles (in degrees) to 1 decimal place if necessary. Round intermediate steps to at least four decimal places. b = 149.1, c = 130, A = 110.1° The triangle with these conditions does not exist. The triangle with these conditions does exist. α≈ B= C

Suppose f(x) = 4* and g(x) = -3x - 2. Find a simplified formula for the function: h(x) = (g(f(x)))²

Solve the equation. 8x+16=27x 64/3 19/6 O No solution O {12}

Write the linear function whose graph passes through the points (4, 5) and (2, 13). Write the answer in the form f(x) = mx + b. Do not enter any spaces in the answer. f(x) =__________A

A test has many multiple-choice items, each with seven answer choices. A student receives 1 point for every correct answer, and each incorrect answer brings a penalty (loss) of a point. If Chris guesses on every question, what score can he expect on the test

Solve the equation for the unknown variable. 3+(n-1)7=17

Find the length and width of a rectangle that has perimeter 48 meters and a maximum area. 12 m; 12 m. 16 m; 9 m. 1m; 23 m. 13 m; 11 m. 6 m; 18 m.

Which of the following identities is equal to sec²θ?

A student was asked to prove the trigonometric identity tan1/2x + cot1/2x = 2cscx. Which of the following could be the first step in proving the identity? I. 1-cosx/sinx + 1-cosx/sinx = 2cscx II. 1+cosx/sinx + sinx/1-cosx = 2cscx III. sinx/ 1+cosx + 1+cosx/sinx = 2cscx

Find the directional derivative of the function f(x, y) = x²e^-y at the point P(-2,0) in the direction v = (2, -3).

Find the area of the surface S, where S is the part of the paraboloidz 1 -x²-y² that lies above the plane z = -2.

Given f(x, y, z) = xy + yz + xz, P(1, 2, 4) I. Find Vf. II. Evaluate Vf at a point P. III. Find the maximum rate of change of f at the point P.

Find v*v for the given vector. v=-4i +2j + 4k Select the correct choice below and fill in the answer box(es) within your choice. A. The answer is a vector. v*v = ai + bj + ck where a = (Type integers or simplified fractions.) B. The answer is a scalar. V*V= (Type an integer or a simplified fraction.) ,b= and c =

Write the given rule as an equation using function notation. Find the output when the input is 1, 2, 3, and 4. Rule: raise 2 to the power of the input. Use b for the function and n for the input. [Note: Do not use any space in your answer.] Function: b(1) = b(2) = b(3) = b(4) =

Determine whether the following relation is a function. Input Output The first 5 positive integers It is not a function. It is a function. Twice the input

Solve the linear programming problem. Maximize P = 40x+ 50y Subject to 2x+y ≤ 16 x+y ≤ 9 x+2y = 14 x, y ≥ 0 What is the maximum value of P? Select the correct choice below and fill in any answer boxes present in your choice. A. P= (Type an integer or a fraction.) B. There is no maximum value of P.

Give the first four terms of the sequence. an = 3n+2 n³

Give the first four terms of the sequence. an = -(−7)n-1 a1 =

Find the common difference for the arithmetic sequence. {4, 12, 20, 28, 36,...}

The first term of a geometric sequence is 5 and the common ratio is 3. Find the 6th term.

The first four terms of a geometric sequence are given. Find a12. an = {3, -6, 12, -24,...}

Find the first five terms of the arithmetic sequence given the first term and common difference. a₁ = −35, d = -9

Find the first term a₁ of the arithmetic sequence in which a9 = 23 and a17 = 95.

The figure below is translated left 2 units. What are the coordinates of the image of point A after this transformation?

What is the end behavior of the function f(x)=-1 4x²? As a →∞, f(x) →-∞ As →-∞, f(x) →-∞ As x→∞, f(x) → ∞ As →-∞. f(x)→-∞ As x→∞, f(x) → ∞ As x→-∞, f(x) → ∞

Find the vertex, focus, and directrix of the parabola. Then sketch the parabola. (x + 5) + (y-2)² = 0 vertex (x, y) = ( focus (x, y) = ( directrix

Figure N is the result of a transformation on Figure M. Which transformation would accomplish this? A reflection over the y -axis A reflection over the x-axis A translation 3 units up A translation 3 units down

Solve for x. If there is no solution, enter DNE. x 5 - 14x - 72 x + 9 x + 3 x² + 12x + 27

Suppose f(x) = √2x and g(x) = x2 2. Show that f(x) and g(x) are inverse functions using algebra.

Suppose f(x) = x² + 4 and g(x) = x² + 2. Calculate f(2)g(-3).

The vector < 18,5> has initial point (-10,-4). The terminal point of the vector is:

The unit circle can alternatively be parametrized by the equations: x(t) = -t² + 1 / t² + 1 y(t)= t² + 1 / 2t Everyone point on the unit circle except (-1,0) can be described using the above parametrization. Find a possible way to obtain the missing point (-1,0) from this parametrization.

Where do the graphs of f(x) = cos(x/2) and g(x) = √3-cos(x/2) intersect on the interval [0, 360°)?

Given the following information about two spherical tanks that are connected, calculate the work required to fill both tanks from the ground. The first tank: Radius = 1m The second tank: Radius = 0.5m The first tank is on the ground and the second tank is 2m above the top of the first tank. The tanks are connected by a tube that is of negligible width. The tube goes through the top of the first tank and the bottom of the second tank. If p > 0 is the density (in kg/m^3) of the liquid in both tanks, calculate the work required to deliver the liquid to fill both tanks from the ground.

An archaeological site is to be made accessible for viewing by the public. To do this, archaeologists built two straight paths from point A to point B and from point B to point C as shown in the following diagram. The length of path AB is 185 m, the length of path BC is 250 m, and angle ABC is 125°. 7a. Find the distance from A to C. The archaeologists plan to build two more straight paths, AD and DC. For the paths to go around the site, angle BÂD is to be made equal to 85° and angle BCD is to be made equal to 70° as shown in the following diagram. 7b. Find the size of angle BÂC. 7c. Find the size of angle CÂD. 7d. Find the size of angle ACD.

For the following exercises, plot the points. 45. (-2, π/3) 46. (−1, -π/2)