# Calculus Review on Integrals, 8 real dynamic PROBLEMS,

1. Evaluate the following indefinite integrals:

1. 1/x^2 dx

2. e^(-x) dx

3. sin(t) cos(t)dt

4. sqrt(s) ds

2. On a dark night in 1915, a German zeppelin bomber drifts menacingly over London. The men on the ground train a spotlight on the airship, which is traveling at 90 km/hour, and at a constant altitude of 1 km. The beam of the

spotlight makes an angle with the ground.

1. Draw a diagram of the situation.

2. When the airship is 3 kilometers from the spotlight, how fast is changing?

3. An electrical circuit consists of two parallel resistors, with resistances R1 and R2 respectively. The total resistance R of the circuit (measured in Ohms ) is specied by 1/R = 1/R1 + 1/R2

The resistors are heating up, so their resistances are increasing over time. Suppose that R1 is increasing at a rate of 0:3

ohm/s and R2 is increasing at a rate of 0:2 ohm/s. When R1 = 80 and R2 = 100, how fast is the total resistance increasing?

4. The ideal gas law states that the pressure P, volume V , and temperature T of a gas are related by PV = NkT

where, N is the number of molecules of gas, and k is Bolzmann's constant, about 1:38*10^23J=K where J is Joules and K is Kelvins (look it up on Wikipedia if you're curious). Say that I have 1024 molecules of gas. The gas begins at a pressure of 200 kPa, inside a 100 cm3 container, and at a temperature of 400K.

1. Say that I hold the temperature xed at 400K and begin to decrease the volume of the container at a rate of 10

cm^3/s. At what rate is the pressure changing?

2. What if instead the pressure is kept xed at 200 kPa, and the gas is cooled at a rate of -2K=s. At what rate is

the volume changing?

5. Use the fundamental theorem of calculus to evaluate sin(x)dx. You must show your work.

6. What is d/dx(int 0 to x, exp(sin(s) + ss)ds? (Hint: use the fundamental theorem of calculus.)

7. Approximate exp(-t^2)dt by partitioning the interval [0; 1] into 5 parts of equal length and sampling exp(-t^2) at the midpoint of each part. Round to three decimal places.

8. The graph of 2^x crosses the graph of x62 at x = 2 and x = 4; see the diagram below. Given that 2^x dx = (1= ln(2))2^x + C, and the area of the region A enclosed by the two curves. (See diagram in attachment)

(See all questions in attachment)

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#### Solution Summary

All eight problems are solved step-by-step with consideration of actual situation for the real life-dynamic problems. The solution contains guidance for wise derivatives from the basic formulas. See all questions and solutions in the attachment for the best view of formulas and diagrams. For solutions consider attached file named "538993_Revised Sol.pdf".

Operations Management and Process Analysis

The bathtub theory of operations management is being promoted as the next breakthrough for global competitiveness. The factory is a bathtub with 50 gallons of capacity. The drain is the outlet to the market and can output 3 gallons per hour wide open. The faucet is the raw material input and can let material in at a rate of 4 gallons per hour. (Assume the bathtub is empty)

a) Draw a diagram of the factory and determine the maximum rate at which the market can be served if all valves are set to maximum. What happens to the system over time?

b) Suppose that instead of a faucet, a 5 gallon container is used for filling the bathtub (assume its full and next to the tub to begin with), it takes 2 hours to refill the container and return it to the bathtub. What happens to the system over time?

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