代写MATH1062: Mathematics 1B Semester 2, 2024 Assignment 1代写留学生R程序

MATH1062: Mathematics 1B

Semester 2, 2024

Assignment 1

Submission Instructions

Solutions to Part A must be prepared in handwritten form, and uploaded as a single pdf file to https://canvas.sydney.edu.au/courses/59770/assignments/553027. Although we allow iPad/tablet writing for assignments, it’s highly recommended for students to rely on traditional pen and paper writing to mimic the exam situation.

Solutions to Part B must be prepared as a single html file and submitted to https://canvas.sydney.edu.au/courses/59770/assignments/553028. You should use the following file to complete Part B.

• An R Markdown worksheet Assignment1Worksheet.Rmd at https://canvas.sydney.edu.au/courses/59770/files/38709347.

You need to write your solutions as either embedded R code or text answers in the provided worksheet, and then generate the html file using Knit in R Studio.

Part A: Calculus Questions

Please justify your answers by showing all relevant works. Correct answers without adequate justification will not receive full marks.

1. Write the following differential equations for the unknown function, y(x) with the variable x, in the standard form. and illustrate the direction field by drawing the tangent intervals at the four points: (1, 0), (0, 1), (−1, 0) and (0, −1) in the (x, y)-plane.

For each differential equation, find the general solution, then find the particular solution satisfying y(0) = 1, finally sketch the solution curve in the above illustration of direction field. All drawing and sketch must be done manually by hand.

(a)

(b)

2. Consider a cubic water tank with the side length 4, placed horizontally. There is a hole at the bottom. Due to a pump, the water will leave the tank according to the following differential equation:

where V (t) is the volume of water remaining in the tank at the time t.

(a) Find the general solution of this differential equation.

(b) Find the particular solution if the tank is full to start with, that is, V (0) = 43 = 64.

(c) For the solution from (b), let T be the time when the tank becomes empty, that is V (T) = 0. Is T > 14 or T < 14? Explain your choice.

Part B: Statistics Questions

Part B questions are provided in the R Markdown worksheet at https://canvas.sydney.edu.au/courses/59770/files/38709347. You should answer these questions using the worksheet and use the Knit button in R Studio to generate the html file. We can only mark the html file.