Question 1: Show that x = h + r cos t and y = k + r sin t represents the equation of a circle.

Question 4: Find the area above the polar x-axis and enclosed by r = 2−cos(θ).

Question 5: If r = f(θ) is a polar curve, find the slope of the tangent line at a point (r₀,θ₀).

second order linear non-homogeneous

solve the following equations

d²y/dx²+ y=5e^x-4x²

la place transform of

1. y´´+4y´+6y_=0, y(0)=1, y´(0)=-4

2. y´+y=t^2 , y(0)=0

Find T(t), N(t), aT, and aN at the given time t for the space curve r(t). [Hint: Find a(t), T(t), aT, and aN. Solve for N in the equation a(t)=aTT+aNN. (If an answer is undefined, enter UNDEFINED.)

Function    Time

r(t)=9ti-tj+(t^2)k t=-1

T(-1)=

N(-1)=

aT=

aN=

(6) Consider the function f(x, y) = 9 − x^2 − y^2 restricted to the domain x^2/9 + y^2 ≤ 1. This function has a single critical point at (0, 0)

(a) Using an appropriate parameterization of the boundary of the domain, find the critical points of f(x, y) restricted to the boundary.

(b) Using the method of Lagrange Multipliers, find the critical points of f(x, y) restricted to the boundary.

(c) Assuming that the critical points you found were (±3, 0) and (0, ±1, find the absolute maximum and minimum of f(x, y) restricted to this domain.

What is the absolute max / min value for the function f(x) = x sqrt 1 - x on the interval [ 1 , 1 ]

Prove the transitive property of similarity: if A~B and B~C, then A~C.

Set up (Do Not Evaluate) a triple integral that yields the volume of the solid that is below

       the sphere x^2+y^2+z^2=8 and above the cone z^2=1/3(x^2+y^2)

1. Rectangular coordinates

       b) Cylindrical coordinates

       c)   Spherical coordinates

A company manufactures two types of electric hedge trimmers, one of which is cordless. The cord-type trimmer requires 2 hours to make, and the cordless model requires 4 hours. The company has only 800 work hours to use in manufacturing each day, and the packaging department can package only 300 trimmers per day. If the company profits for the cord-type model for $28.50 and the cordless model for $57.00, how many of each type should it produce per day to maximize profits? Scenario #1 (with the smaller number of cord-type trimmers): corded models cordless models Scenario #2 (with the larger number of cord-type trimmers): corded models cordless models

Use the position function ?(?) = −16?^ 2 + ?0? + ?0 for the following vertical motion of objects on earth.

5) An object is launched vertically from a 75ft building with an initial velocity of 240 ft/sec.

a) Find the velocity of the object 2 second after launch. Is it travelling up or down?

b) When will the object reaches its maximum height?

c) How long will the object travel in the air? (Hint: how long does it take to hit the ground?)

d) Determine the velocity when the subject hits the ground.

6) An object is fired downward with a speed of 12ft/sec from the top of a 80ft building.

a) Write the position function ?(?) that represents the height of the object, in feet, from the ground.

b) Determine the time for the object to hit the ground.

c) Determine the velocity at impact

a.)Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves y^2=x and x = 2y about the y-axis

b.) Find the volume of the solid that results when the region bounded by x=y^2 and x=2y+15 is revolved about the y-axis

c.) Find the length of the curve y=ln(x) ,1≤x≤sqrt(3)

d.)Consider the curve defined by the equation xy=5. Set up an integral to find the length of curve from x=a to x=b

A chemist needs 10 liters of a 25% acid solution. The solution is to be mixed from three solutions whose concentrations are 10%, 20% and 50%. How many liters of each solution will satisfy each condition?

a) Use 2 liters of the 50% solution.

b) Use as little as possible of the 50% solution.

c) Use as much as possible of the 50% solution.