1. State the dimension theorem. Explain how it proved.

Find vectors of the Frenet frame of the curve at any point of the curve:

x = a(t-sin(t)), y = a(1-cos(t)), z = 4acos(t), where a is a positive constant.

Please show all steps and relevant formulae!

Jennifer took clothes to the cleaners three times last month.​ First, she brought 3 shirts and 1 pair of slacks and paid ​$9.96. Then she brought 6 ​shirts, 4 pairs of​ slacks, and 2 sports coats and paid ​$40.88. ​Finally, she brought 4 shirts and 2 sports coats and paid ​$20.94. How much was she charged for each​ shirt, each pair of​ slacks, and each sports​ coat? Jennifer was charged ​$ nothing for each​ shirt, ​$ nothing for each pair of​ slacks, and ​$ nothing for each sports coat.

A mass that weight 6lb stretches a spring 4in. The system is acted on by an external force 2sin(4sqrt(6)t)lb.If the mass is pulled down 3in and then released, determine the position of the mass at any time t. Use 32ft/s^2 as the acceleration due to gravity. Pay close attention to the units. Answer must be in inches

Consider the orthogonal coordinate system x = uv cosθ, y = uv sinθ, z = (u2 −v2)/2, with u,v ≥ 0 and 0 ≤ θ < 2π.
Express the volume element dV in terms of u, v, θ.

A building has a 9’ deep basement (foundation) wall that is 160’ long. The wall has no windows. On average, 8’ of the wall is below grade (underground). Assume an outside air temperature of 10°F, an inside temperature of 70°F, and an average ground temperature of 45°F.

a. Determined the rate of heat loss through the wall below grade if it is uninsulated.

b. Determine the rate of heat loss through the wall below grade if it is insulated with an R-value of about 12 hr*°F*ft2/Btu.

A commercial building is built on a slab on grade foundation that is a 120’ by 48’ rectangular shape.
Assume the outside design temperature is -5°F. Determine the rate of heat loss through the exterior
edge of a concrete slab.

a. For an uninsulated slab edge perimeter.

b. For a slab edge perimeter insulated with 1” of perimeter insulation (an R-value of about 2.5).

c. For a slab edge perimeter insulated with 2” of perimeter insulation (an R-value of about 5).

What are the conditions on the parameters for linear multistep methods so that the methods are regular? Would appreciate a detailed explanation.

A mass weighing 19lb stretches a spring 8in. The mass is attached to a viscous damper with damping constant 4lb⋅s/ft. The mass is pushed upward, contracting the spring a distance of 2in, and then set into motion with a downward velocity of 7in/s. Determine the position u of the mass at any time t. Use 32ft/s2 as the acceleration due to gravity. Pay close attention to the units. Answer must be in inches

Suppose that f(t) is the unique solution to the IVP y' = t + y^2 , y(0) = 5 and g(t) is the unique solution to the IVP y' = 1/(y + t^2) , y(5) = 0.

a. Determine an IVP that the function y = f(g(t)) solves. [Hint: You differential equation part will contain the functions t, g(t), and y in its expression.

b. (2 points) Show that the function y = t also solves this IVP.

c. (2 points) Use the uniqueness part of the Existence & Uniqueness Theorem to conclude that f and g are inverses.

1. Methylergonovine maleate is available in 0.2 mg tablets. The prescription is for 0.4 mg every 8 hours after delivery for 48 hours. How many tablets are needed for one day (24 hours)? Round your final answer to one decimal place. Type your final answer and label in the answer. Thank you!

2. An infant was born at 30 weeks weighing 2 pounds 9 oz. (1.162 kg). The baby is being fed Special Care Formula (24 calories per ounce). Based on a premature infant's cardiac needs of 130 calories/kg/day.

a. How many calories per day does this baby require?

b. How many ml of formula is required every 24 hours to provide the needed calories determined in the previous question?

c. Using the conversion factor of 1 ounce=30 ml, how many ml should be given at each feeding in order to provide the needed calories, if the infant is feed every three hours? Round your final answer to the nearest whole calorie. Type you final answer and label. Thank you!

3. The client is experiencing a postpartum hemorrhage. The physician prescribes 30 IU Oxytocin added to a 500 ml bag of Lactated Ringers to run at 150 ml/hr. Oxytocin is available 10 IU per ml.

a. How many ml of Oxytocin will be added to the bag of Lactated Ringers? Round your final answer to one decimal place with label.

b. After one hour at the prescribed rate of 150 ml/hr, how many units of Oxytocin has the client received?

c. The mother is still hemorrhaging. The primary care provider prescribes Carboprost tromethamine 350 mcg intramuscularly. The drug is available in 250 mcg per 1 ml. how much volume in ml will the nurse administer? Please round answer to one decimal place. Thank you!

4. The nurse is preparing to administer 1 mg Phytonadione Intramuscularly to a newborn. The medication on hand is 2 mg/1 ml. What is the correct volume of the injection in ml? Round your final answer to two decimal places.

5. The nurse is to give a new mother Ibuprofen 800 mg by mouth for severe uterine cramping after delivery. The pharmacy stocks Ibuprofen as 200 mg tablets. How many tablets will the nurse give? Round your final answer to one decimal place. Thank you!

Compute the Fourier series for the triangle and sawtoothwaves, i.e. the following functions, periodically extended to R:   
  
f(x) = |x|, −1 < x ≤ 1; g(x) = x−π ,−π ≤ x < π . Plot each function and its Fourier polynomials of degrees up to 4.