Find the equation of the tangent plane to the surface,

(4x^2)(y^3) + (5yz) + (2xz^3) = 7

at the point P(-1,1,1). Also nd the parametric equation of the normal

line to that surface at that point . Sketch a picture that illustrates what this

is all about.

1. A 10 ft chain weighs 25 lb and hangs from a ceiling with a 5 lb weight attached to the end. Find the wok done lifting the lower end of the chain and the weight to the ceiling so that they are level with the upper end.

2. Use the method of cylindrical shells to find the volume formula for a sphere with radius r. (in our example we used the disk method. You formula should be the same, but the integral you use to get there should be different.

( NO HAND WRITING PLEASE )

Q1: Suppose that a and b are integers, a ≡ 11 (mod 19), and

b ≡ 3 (mod 19). Find the integer c with 0 ≤ c ≤ 18 such

that

a) c ≡ 13a (mod 19).

b) c ≡ 8b (mod 19).

c) c ≡ a − b (mod 19).

d) c ≡ 7a + 3b (mod 19).

e) c ≡ 2a² + 3b² (mod 19).

Q2:

List all the steps used to search for 10 in the sequence 1,3, 4, 5, 6, 8, 9, 11 using

a) A linear search.

b) A binary search.

Utilize identities to calculate the exact value of the five remaining trigonometric functions if:

csc(α)= √3, π/2, < α < π

Calculate the equivalent algebraic expression for:

sec(arcsin(x - 1)

x' = x − y

y' = 4y − x^2*y

Linearize the system about the point (2, 2). Classify the type and stability of the critical point at (2, 2) by examining the linearized system.

a) A salt-water tank has pure water flowing into it at 5 L/min. The contents of the tank are kept thoroughly mixed, and the contents flow out at 5 L/min. Initially, the tank contains 1 kg of salt in 10L of water. How much salt will be in the tank after 20 minutes? Let ?? represent the amount of salt in the tank at time t and let ?? represent the volume of saltwater in the tank at time t.

b) Now assume that salt is added into the tank at a rate of 0.1 kg/min with pure water still flowing into it at 5 L/min. The contents of the tank are kept thoroughly mixed, and the contents flow out at 5 L/min. Initially, the tank contains 1 kg of salt in 10L of water. How much salt will be in the tank after 20 minutes?

Given f(x) = (x⁴ - 2)(x⁵ - 10x + 1)³

Find the definite integral of f(x) on the closed interval [0, 1].

1) 0.05

2) 204.75

3) 4095

4) None

Use exponential regression to find an exponential function that best fits this data.

f(x) =     

Use linear regression to find an linear function that best fits this data.

g(x) =    

given DE has a regular singular point at x=0 determine two solutions for x>0

x^2y''+3xy'+(1+x)y=0

Let F(x,y,z) = < z tan^-1(y²), z³ ln(x² + 1), z >. Find the flux of F across S, the top part of the paraboloid x² + y² + z = 2 that lies above the plane z = 1 and is oriented upward. Note that S is not a closed surface.

Question : y''=4y'+8y=0 , (y''-4y'+13y)^2=0 , (y''+2y'+2y)^2=0 ,

y''-6y'+13y=0,y(0)=3 , y'(0)=13 , 2y''-6y'+17y=0,y(0)=2, y'(0)=13

Approximently 72% of freshmen entering public high schools in the United States in 2005 graduated with their class in 2009. A random sample of 136 freshmen is chosen. around your answers to four decimal places as needed.

Part 1 of 6

Find the mean u ^p is

Find ??, ?? and ?? of F(x, y, z) = tan(x+y) + tan(y+z) – 1

Consider the function f(x)= 1 + 1/x - 1/x²

^{Find the domain, the vertical and horizontal asymptotes, the intervals of increase or decrease, the local minimum and maximum values, the intervals of concavity and the inflection points.}