a) Find an equation of the plane tangent to the graph of f at the given point P. Write your answer in the form ax + by +cz= d, where a, b, c, and d are integers with no common factor, and a is greater or equal to 0. f(x,y)= 2x³y + 4x-y, P(1,3,7)

b) Use a multivariable chain rule to find a formula for the given derivative or partial derivative. w= f(x,y), x=g(u,v), y=h(u,v);  ∂w/ ∂v

List all of the values of the sine function that you know. Remember that values of sin(x) repeat every 2π radians, so your answer should include infinitely many values.

A plane flying horizontally at an altitude of 5 mi and a speed of 470 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 7 mi away from the station. For full credit I expect to see a well-labeled picture. Show all work. Round your final answer to the nearest whole number. Do not round intermediate values. Include units of measure in your answer.

A fence is to be built to enclose a rectangular area. The fence along three sides is to be made of material that costs $5 per foot. The material for the fourth side costs $15 per foot.If $3,000 is available for the fencing, find the dimensions of the rectangle that will enclose the most area.

You want to build a rectangular box in such a way that the sum of the length, width and height is 24 cm.
a) Define the equations so that the dimensions of their volume are maximum
b) Which of the equations proposed would be the restriction and which function? Explain
c) Using the technique you want to calculate maximums and minimums, what are these values? What volume will the box have?

Suppose that we have a rectangle ABCD with AB = 4 inches and BC = 16 inches.

Find the perimeter of a square that has the same area is ABCD.

Find the length and width of a rectangle that has the same area as ABCD and twice the perimeter of ABCD.

Two ships leave the same port at noon. Ship A sails north at 14 mph, and ship B sails east at 17 mph. How fast is the distance between them changing at 1 p.m.? (Round your answer to one decimal place.)

Determine which of the following vector fields F in the plane is the gradient of a scalar function f. If such an f exists, find it. (If an answer does not exist, enter DNE.)

F(x, y) = 3xi + 3yj

f(x, y) =

F(x, y) = 6xyi + 6xyj

f(x, y) =  

F(x, y) = (4x² + 4y²)i + 8xyj

f(x, y) =

find a particular solution to

y'' + 5y' + 6y = 18te^3t

Yp=?

Evaluate the integral using partial fractions. ∫▒( x+3)/((x-9)^2 (x+2))

1. A tank starts with 100 litres of water and 1,000 bacteria in it. For now we assume the bacteria do not reproduce. Let B(t) be the number of bacteria in the tank as a function of time, where t is in hours. For each of the situations below, write down a first order differential equation satisfied by B(t), of the form dB dt = f(t, B). You DO NOT need to solve it.

(a) A little goblin is pouring bacteria into the tank at a rate of 2020 bacteria per hour.

(b) Like part (a), but we are also draining the tank at a rate of 3 litres per hour.

(c) Like part (b), but now the bacteria are reproducing. Suppose that the bacteria will double the present population in every hour. A gentle reminder: make sure that you write down the meaning of each term.

Find the equation of the osculating circle at the local minimum of

f(x)=4x^3−9x^2+(15/4)x−7.

Two boats are leaving port at the same time. Ship A is sailing due east at 40 km/h, and Ship B due south at 30 km/h. How fast are they moving away from each other after 2 hours?

(a) Determine the point of intersection of the line given by (x, y, z) = (4+t, −1+8t, 3+2t), t ∈ R with the plane given by 2x − y + 3z = 15 or show that they do not intersect.

(b) Given the line L : X = (9, 13, −3) + t(1, 4, −2), t ∈ R, the point A with position vector 4i+ 16j −3k and that the point P lies on L such that AP is perpendicular to L, find the exact coordinates of P.

ind the intervals of increase or decrease, the local maximum and minimum values, the intervals of concavity, and the inflection points for each of the following:

?(?)=2?^3―3?^2―12?

?(?)= ? (Square root) ?+3

?(?)=ln(?^4+27)