Finding Derivatives:

Find y’ for

(A) y = 3e^x + 5 ln x

(B) y = x^4 – ln x^4

Please help me solve these two questions.

Thank you (:

# Category: Calculus homework help

Please research at least one source of information on engineering applications of integration. Important: The purpose of this assignment is for you to share your engineering expertise and teach us how integral calculus is applied in your field. Under no circumstances should you copy and paste any content from a web reference. Instead, explain these applications in your own words.

In your original post, answer the following:

Create a summary of what you found (in your own words!) and describe an example application. Keep your post clear and concise, under 500 words.

You can use any source you like, including (but not limited to) the Internet, ECPI Library Resources, and your Electric Circuits textbook. Be sure to include the citation of your source in APA format.

Find a resource with a tutorial/information that will help you create a specific, original, math problem related to your topic. Remember, you are NOT teaching the whole topic, but just how to solve ONE PROBLEM You’ve created. You must cite your source in APA format. (Even if you know how to create and solve a problem related to your topic, you still must include a source that your classmates can reference to read more about what you are teaching.)

Tell us the problem you’ve created. Remember, your problem must be ORIGINAL: made up by YOU. DO NOT use a problem that has a full solution from a resource.

Tell us how to solve your problem step by step. Your solution should include all mathematical steps as well as explanations in your own words. You can type this with the equation editor right here in the reply box, you can use an image file to show hand-written work, or you can make a video using the media tool to provide your explanations! I encourage you to get creative with how you explain your solution!

Using a simulation of the manufacturing process Mr. Gürkan found a method to find the optimal location to place buffers to help machines run smoother and longer without breaking down. The simulation ran a maximum of 50 machines and took data on cycle times and downtime to set up the production process. Then using the data collected and forming a directional derivative could be used to infer the approximate location where the process became too loud or uneven. These points were isolated and selected for the optimal location for the buffers. Once added into the simulation the productivity of the process was found to increase as was the time between break down on the production line.

To further test the simulation additional production line simulations were produced. These additional simulations were made with fewer machines used in the process. These simulations were tested utilizing the directional derivative and more traditional stochastic testing methods and the level of error found between the two on the location of buffers was found to be negligible.

References

Gürkan, G. (2000). Simulation optimization of buffer allocations in production lines with unreliable machines. Annals of Operations Research, 93(1–4), 117–216. https://libproxy.ecpi.edu:2111/10.1023/a:1018900729338

There are many types of derivatives used in real life engineering situations but, the one I am going to talk about is tangents used in daily life. I specifically want to explain an example of a secant tangent line used in daily life for engineers. A secant line is a line that connects to two points on a circle. Astronauts use this to find things like the distance from the moon while it is orbiting to many different locations on earth. The formula used for this real life equation is y-b=[(d-b)/(c-a)](x-a). For example, if you are using this equation and are given the values (5-3)/(-2-1). Your equation of the secant line should look like so y-3=-2/3(x-1) then y=(-2/3)x+2/3+3 with the solution being y=(-2/3)x+11/3.

Reference:

Application of tangents and normal in real life. Unacademy. (2022, April 19). Retrieved October 19, 2022, from https://unacademy.com/content/upsc/study-material/mathematics/application-of-tangents-and-normal-in-real- life/#:~:text=of%20the%20tangent.-,For%20example%2C%20when%20a%20cycle%20travels%20down%20a%20road%2C%20that,and%20turned%20from%20another%20endLinks to an external site..

For my discussion, I choose to talk about how derivatives are used for servo motor controls. First, we need to discuss what a servo motor is and what it does. A servo motor is a current and voltage-controlled electrical motor. This motor works on a closed-loop system through the commands of a servo controller which uses a feedback device to control the velocity and position of the servo. A great example of this would be the cruise control in a car. The servo controller would be the driver setting cruise control to a set speed which sends a voltage signal and varying current to the servo which controls the throttle until you get to a certain speed, then maintains that speed. The feedback device would be your tachometer telling the servo to lower the current being sent if you go over the set speed and to raise the current if you go under the set speed. The servo is using Proportional Integral Derivative or PID which changes the motor’s output based on the set speed and what the tachometer reads. The PID algorithm uses Proportional feedback which tells the servo that it needs to go faster to reach the set speed increasing current sent to the servo. Derivative feedback tells the servo that we are over the set speed, and it doesn’t need to run anymore decreasing current sent to the servo. Integral feedback holds the current at its set amps and holds the position of the servo to keep the set speed without any outside interactions that would call for the need of the other two. Below is an image of how the PID algorithm works.

(Collins, 2022)

When put into an equation it will look something like this:

Apmonitor.com (n.d.)

Thank you for reading.

References:

Collins, D. (2022, October 17). FAQ: What are servo feedback gains, overshoot limits, and position error limits? Motion Control Tips. https://www.motioncontroltips.com/faq-what-are-servo-feedback-gains-overshoot-limits-position-error-limits/Links to an external site.

Proportional Integral Derivative (PID). (n.d.). https://apmonitor.com/pdc/index.php/Main/ProportionalIntegralDerivativeLinks to an external site.

Let’s go beyond the mechanics. Differential calculus is a tool, invented to solve the most challenging problems in science and engineering. For instance, Newton used differential calculus to express the equations of planetary motion around the Sun, as well as the rate at which a warm object cools off in a colder environment. This week, you’ll become familiar with more of those applications by presenting them to your classmates.

Please research at least one source of information on engineering applications of derivatives.

Create a summary of what you found (in your own words!) and describe an example application. Keep your post clear and concise, under 500 words.

You can use any source you like. Be sure to include the citation of your source in APA format.

1. Given the demand function D(p)=100−3p^2,

Find the Elasticity of Demand at a price of $4

At this price, we would say the demand is:

Inelastic

Elastic

Unitary

Based on this, to increase revenue we should:

Raise Prices

Keep Prices Unchanged

Lower Prices

2. Given that f'(x)=−5(x−5)(x+4),

The graph of f(x)f(x) at x=3 is Select an answer increasing concave up increasing concave down decreasing concave up decreasing concave down

3. Find ∫4e^xdx

+ C

4. Find ∫(7x^6+6x^7)dx

+ C

5. Find ∫(7/x^4+4x+5)dx

+ C

6. The traffic flow rate (cars per hour) across an intersection is r(t)=200+600t−90t^2, where t is in hours, and t=0 is 6am. How many cars pass through the intersection between 6 am and 10 am?

cars

7. A company’s marginal cost function is 5/√x where x is the number of units.

Find the total cost of the first 81 units (of increasing production from x=0 to x=81)

Total cost: $

8. Evaluate the integral

∫x^3(x^4−3)^48dx

by making the substitution u=x^4−3.

+ C

NOTE: Your answer should be in terms of x and not u.

9. Evaluate the indefinite integral.

∫x^3(8+x^4)^1/2dx

+ C

10. A cell culture contains 2 thousand cells, and is growing at a rate of r(t)=10e^0.23t thousand cells per hour.

Find the total cell count after 4 hours. Give your answer accurate to at least 2 decimal places.

_thousand cells .

11. ∫4xe^6xdx = + C

12. Find ∫6x/7x+5dx

+ C

13. Sketch the region enclosed by y=4x and y=5x^2. Find the area of the region.

14. Determine the volume of the solid generated by rotating function f(x)=(36−x^2)^1/2 about the x-axis on [4,6].

Volume =

15. Suppose you deposit $1000 at 4% interest compounded continuously. Find the average value of your account during the first 2 years.

$

16. Given: (x is number of items)

Demand function: d(x)=3362√x

Supply function: s(x)=2√x

Find the equilibrium quantity: items

Find the consumers surplus at the equilibrium quantity: $

17. Given: (x is number of items)

Demand function: d(x)=3072/√x

Supply function: s(x)=3√x

Find the equilibrium quantity: items

Find the producer surplus at the equilibrium quantity: $

18. Find the accumulated present value of an investment over a 9 year period if there is a continuous money flow of $9,000 per year and the interest rate is 1% compounded continuously.

$

19. A company manufactures 2 models of MP3 players. Let x represent the number (in millions) of the first model made, and let y represent the number (in millions) of the second model made.

The company’s revenue can be modeled by the equation

R(x,y)=110x+170y−4x^2−2y^2−xy

Find the marginal revenue equations

Rx(x,y) =

Ry(x,y) =

We can achieve maximum revenue when both partial derivatives are equal to zero. Set Rx=0and Ry=0 and solve as a system of equations to the find the production levels that will maximize revenue.

Revenue will be maximized when:

x =

y =

20. An open-top rectangular box is being constructed to hold a volume of 200 in^3. The base of the box is made from a material costing 5 cents/in^2. The front of the box must be decorated, and will cost 11 cents/in^2. The remainder of the sides will cost 2 cents/in^2.

Find the dimensions that will minimize the cost of constructing this box.

Front width: in.

Depth: in.

Height: in.

1. Farmer Jones, and his wife, Dr. Jones, decide to build a fence in their field, to keep the sheep safe. Since Dr. Jones is a mathematician, she suggests building fences described by y=2x^2 and y=x^2+8. Farmer Jones thinks this would be much harder than just building an enclosure with straight sides, but he wants to please his wife. What is the area of the enclosed region?

2. Sketch the region enclosed by y=e^4x, y=e^9x, and x=1. Find the area of the region.

3. Find the volume of the solid obtained by rotating the region bounded by y=6x^2, x=1, x=4 and y=0, about the xx-axis.

4. Find the volume of the solid formed by rotating the region enclosed by

y=e^3x+3, y=0, x=0, x=0.7

about the x-axis.

5. Suppose you deposit $3000 at 3% interest compounded continuously. Find the average value of your account during the first 2 years.

6. If a cup of coffee has temperature 98°C in a room where the ambient air temperature is 20°C, then, according to Newton’s Law of Cooling, the temperature of the coffee after t minutes is T(t)=20+78e−t/50. What is the average temperature of the coffee during the first 28 minutes?

7. Given: (x is number of items)

Demand function: d(x)=3920√x

Supply function: s(x)=5√x

Find the equilibrium quantity: items

Find the consumers surplus at the equilibrium quantity: $

8. Given: (x is number of items)

Demand function: d(x)=4205√x

Supply function: s(x)=5√x

Find the equilibrium quantity: items

Find the producer surplus at the equilibrium quantity: $

9. Given: (x is number of items)

Demand function: d(x)=300−0.3x

Supply function: s(x)=0.5x

Find the equilibrium quantity:

Find the consumers surplus at the equilibrium quantity:

10. Given: (x is number of items)

Demand function: d(x)=200−0.6x

Supply function: s(x)=0.2x

Find the equilibrium quantity:

Find the producers surplus at the equilibrium quantity:

11. Given: (x is number of items)

Demand function: d(x)=784−0.4x^2

Supply function: s(x)=0.6x^2

Find the equilibrium quantity:

Find the consumers surplus at the equilibrium quantity:

12. Given: (x is number of items)

Demand function: d(x)=588.7−0.3x^2

Supply function: s(x)=0.4x^2

Find the equilibrium quantity:

Find the producers surplus at the equilibrium quantity:

13. Suppose the demand function for a product is given by the function:

D(q)=−0.016q+54.4

Find the Consumer’s Surplus corresponding to q=650 units.

(Do no rounding of results until the very end of your calculations. At that point, round to the nearest tenth, if necessary. It may help you to sketch the demand curve, which crosses the horizontal at q=3,400)

Answer: dollars

14. Find the accumulated present value of an investment over a 10 year period if there is a continuous money flow of $10,000 per year and the interest rate is 1.3% compounded continuously.

15. A company is considering expanding their production capabilities with a new machine that costs $52,000 and has a projected lifespan of 7 years. They estimate the increased production will provide a constant $8,000 per year of additional income. Money can earn 1.1% per year, compounded continuously. Should the company buy the machine?

a. Select an answer Yes, the present value of the machine is greater than the cost by No, the present value of the machine is less than the cost by

b. $ over the life of the machine

16. Find the present value of a continuous income stream F(t)=20+7t, where t is in years and F is in thousands of dollars per year, for 30 years, if money can earn 2.5% annual interest, compounded continuously.

Present value = thousand dollars.

17. Given f(x,y)=−2x^2+4xy^6+3y^5, find

fxx(x,y) =

fxy(x,y) =

18. Find the critical point of the function f(x,y)=−4−7x−5x^2+6y−2y^2

19. Suppose that f(x,y)=x^4+y^4−xy

Then the minimum is

20. Find and classify the critical points of z=(x^2−8x)(y^2−6y)

Local maximums:

Local minimums:

Saddle points:

For each classification, enter a list of ordered pairs (x, y) where the max/min/saddle occurs. If there are no points for a classification, enter DNE.

21. A company manufactures 2 models of MP3 players. Let x represent the number (in millions) of the first model made, and let y represent the number (in millions) of the second model made.

The company’s revenue can be modeled by the equation

R(x,y)=50x+110y−4x^2−2y^2−xy

Find the marginal revenue equations

Rx(x,y)=

Ry(x,y)=

We can achieve maximum revenue when both partial derivatives are equal to zero. Set Rx=0and Ry=0 and solve as a system of equations to the find the production levels that will maximize revenue.

Revenue will be maximized when:

x =

y =

21. A chemical manufacturing plant can produce z units of chemical Z given p units of chemical P and r units of chemical R, where:

z=70p^0.9r^0.1

Chemical P costs $100 a unit and chemical R costs $700 a unit. The company wants to produce as many units of chemical Z as possible with a total budget of $140,000.

A) How many units each chemical (P and R) should be “purchased” to maximize production of chemical Z subject to the budgetary constraint?

Units of chemical P, p=

Units of chemical R, r =

B) What is the maximum number of units of chemical Z under the given budgetary conditions? (Round your answer to the nearest whole unit.)

Max production, z= units

22. An open-top rectangular box is being constructed to hold a volume of 300 in^3. The base of the box is made from a material costing 8 cents/in^2. The front of the box must be decorated, and will cost 9 cents/in^2. The remainder of the sides will cost 2 cents/in^2.

Find the dimensions that will minimize the cost of constructing this box.

Front width: in.

Depth: in.

Height: in.

1. Evaluate the integral

∫x5(x^6−10)^47dx

by making the substitution u=x^6−10.

=_+C

2. Evaluate the indefinite integral.

∫7dx/xln(8x)

=_ + C

3. Evaluate the indefinite integral.

∫x^8e^x^9dx

=_+C

4. Evaluate the indefinite integral.

∫x4(15+x^5)^1/2dx

=_+C

5. Evaluate the indefinite integral.

∫4/(t+5)^8 dt

=_+C

6. Evaluate the indefinite integral:

∫x/x^2+4 dx

=_+C

7. A cell culture contains 4 thousand cells, and is growing at a rate of r(t)=11e^0.24t thousand cells per hour.

Find the total cell count after 4 hours. Give your answer accurate to at least 2 decimal places.

_ thousand cells

8. Use integration by parts to evaluate the integral:

∫2te^tdt =

9. ∫4xe^7xdx = + C

10. Use integration by parts to evaluate the integral:

∫ln(7x−1)dx

11. Use integration by parts to evaluate the integral:

∫ln(z)√z^13dz

12. Find ∫4x/3x+2dx

=_+C

13. Integrate: ∫x/(x^4+25)^1/2dx

=_ + C

14. Find ∫(−2x^2+3/x−1/x^4+4√x)dx

=_ + C

15. Find ∫(x+4)(x−6)dx

=_ + C

16. The traffic flow rate (cars per hour) across an intersection is r(t)=400+700t−270t2r(t)=400+700t-270t^2, where t is in hours, and t=0 is 6am. How many cars pass through the intersection between 6 am and 7 am?

_cars

17. A company’s marginal cost function is 17/√x where x is the number of units.

Find the total cost of the first 36 units (of increasing production from x=0 to x=36)

Total cost: $