A particle of mass 1 is attached to a spring dashpot mechanism. The stiffness constant of the spring is 3 N/m and the drag force exerted on the particle by the dashpot mechanism is 4 times its velocity. At time t=0, the particle is stretched 1/4 m from its equilibrium position. At time t=3 seconds, an impulsive force of very short duration is applied to the system. This force imparts an impulse of 2 N*s to the particle. Find the displacement of the particle from its equilibrium position.

This question is from the dirac delta function chapter in my diff. equations book. Thanks!!

Q1-Find all possible time domain signals corresponding to the following z-transform:

X(z) = (z³ + z² + 3/2 z + 1/2 ) / (z³ + 3/2 z² + 1/2 z)

Q2-A digital linear time invariant filter has the following transfer function:

H(z) = (5 + 5z^-1) / (1 - 3/8 z^-1 + 1/16 z^-2)

a) Find the impulse response if the filter is causal.

​​​​​​Variable Identification: how do i know if they have 1 or 2 variables?

1. A study was conducted to investigate the possible link between vaccination in children and autism.

Individual: a single child

Variable 1:

QN or CL :   

Variable 2 :

QN or CL:

2. An employee at a supermarket samples shoppers to determine if a majority of them are wearing face masks.

Individual: a single shopper

Variable 1:

QN or CL :   

Variable 2 :

QN or CL:

3. Does the average movie gross more than $10,000,000?

Individual: a single movie

Variable 1:

QN or CL :

Variable 2 :

QN or CL:

4. You want to conduct a study to compare how quickly pain relief is felt for 2 different brands of aspirin.

Individual: a person with pain

Variable 1:

QN or CL :

Variable 2 :

QN or CL:

5. A realtor wants to determine if property values in a particular area have decreased since last year.

Individual: a piece of property

Variable 1:  

QN or CL :   

Variable 2 :

QN or CL:

The price of a new car is $16,000. Assume that an individual makes a down payment of 25% toward the purchase of the car and secures financing for the balance at the rate of 7%/year compounded monthly. (Round your answers to the nearest cent.)

(a) What monthly payment will she be required to make if the car is financed over a period of 36 months? Over a period of 72 months?

(b) What will the interest charges be if she elects the 36-month plan? The 72-month plan?

• Sketch a drawing that matches the values given
• Determine how to begin (Law of Sines/Law of Cosines)
• Solve for the missing values of the triangle (if possible)

1. a=4.2 in, b=2.5 in, c=5.8 in

2. b=3.9 cm, β=94.7°, α=30.6°

3. α=39.6°, a=3.7 m, c=18.4 m

4. α=41.2°, a=8.1 ft, b=10.6 ft

5. Sketch a triangle that fits the description below. Then find the area.

1. a=3 in, b=5 in, c=7 in

2. a=12.9 m, b= 6.4 m, ?=13.7°

1. Explain the process, in general terms, that the ancient Chinese mathematician Liu Hui used to approximate pi, and how was this the same or different from what Archimedes had done?

Demand for propane is given by D(x) = 6.5 − 0.25x, and supply is given by S(x) = 2.1 + 0.15x, where x is in gallons per month customer and D(x) and S(x) are dollars per gallon. Find the followings:   

(c) Assume a price ceiling of $3 per gallon of propane is imposed. Find the point (Xc, Pc)   

(d) Find the new producer surplus and the new consumer surplus at (Xc, Pc).

(e) Find the deadweight loss.

a. Prove the Archimedes result that the area of a section of a parabola equals times the area of the inscribed triangle of the same height. (You may use calculus here.)

b. In what sense can one consider Archimedes to be the inventor of calculus?

c. Find at least five axioms equivalent to the parallel lines axiom. Give the dates when they were proposed and the person that proposed them.

What interest will be earned if $6600 is invested for 8 years at 12% compounded monthly? (Round your answer to the nearest cent.) $

(a) Show that the diagonal entries of a positive definite matrix are positive numbers.

(b) Show that if B is a nonsingular square matrix, then B^TB is an SPD matrix.(Hint. you simply need to show the positive definiteness, which does requires the nonsingularity of B.)