2017 SP4 ENR102 Ass 02 - Essential Mathematics 2: Calculus Assignment 2 from 2017, Study Period 4 PDF

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Essential Mathematics 2: Calculus Assignment 2 from 2017, Study Period 4...

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ENR 102 Essential Mathematics 2: Calculus Assignment 2– see Unit Outline or Website for due date and time Late submission may incur a penalty. See Unit Outline for details. Total Marks: 87. • Submit online: Assignments must be submitted online. Allow sufficient time to produce a single PDF file of reasonable size and to upload it. Look at your file and make sure that it is easy to read. Photographs with poor lighting are usually unacceptable. Good quality scans are best. • Format: Use A4 sized paper. Do not use graph paper. Use the whole width of the page; do not rule a line down the middle. Do not write on both sides of the paper. Your student name should be clearly marked on each page. You can either handwrite or type your assignment. Note that typed mathematical expressions need to be notationally correct. If using Word to typeset your assignment, you should use Equation Editor. If this is not possible, write your assignment or those parts of the assignment by hand. Marks will be deducted for poorly-presented mathematical expressions. Marks will not be deducted for neatly handwritten assignments. • Show your work: Show all necessary steps so that the reader can follow your solution procedure. Write out your solutions clearly so that they are well organised and easy to follow. Use words. When you answer a mathematical problem you are telling a story, and that story should make sense (as well as being logical). • Take pride in what you hand in: Your submission should be neat and legible, and have the questions in correct order. It should not contain large sections of crossed out work. • Keep a copy: It is your responsibility to keep a copy of your assignment, and to keep your marked assignment until the final unit grades have been released. • Generally, use exact values: approximate a final answer if it makes sense to do so; for example if it is a measured quantity. • Acknowledgement of work: When submitting online, you acknowledge that the submitted assignment is your own work, unless otherwise stated. • Academic integrity: The University’s policy on academic misconduct will be strictly applied. Here are some tips to avoid academic misconduct: • Do not copy from any printed or electronic source, or from any person. • Write your own solutions. You may discuss your work with other students, but you must write up your solutions by yourself. You are not allowed to use anyone else’s written work when you are writing up your assignment. • Do not give inappropriate help. Giving inappropriate help is just as serious as receiving it, and will have the same consequences. Do not show your completed assignment to other students. Dispose of drafts so that no one can access them. • Acknowledge help and joint work. If you receive any help from another source (for example, students, tutors, friends, internet), you must make a note of it on your assignment. • Consequences. Cases of academic misconduct will be referred to the Academic Integrity Officer who will determine the suitable penalty to be applied to all parties involved.

There are four (4) marks available for presentation and communication: Excellent (4 marks) Good (3 marks) Satisfactory (2 marks) Poor (1 marks) None (0 marks)

Notation is proficient and accurate. Layout is clear and easy to follow. Diagrams are appropriate and very well presented. Presentation requirements have been met. Notation is generally appropriate, with some inaccuracies. Layout is mostly clear and easy to follow. Diagrams are mostly appropriate and well presented. Presentation requirements have been mostly met. Notation has several inaccuracies. Layout is adequate. Some attempt has been made to produce appropriate diagrams. Presentation is satisfactory. Limited accuracy of notation. Layout is poor. Limited attempt has been made to produce appropriate diagrams. Presentation is adequate. Inadequate.

Question 1 (2+3+4+2+2+2+1+4+5=25 marks) Consider the function f (x) =

1 3 x 3

− 2x2 + 3x.

(a) Show that f (x) = 0 has only two real solutions: x = 0 and x = 3. (b) Find f ′ (x), and determine the values of x for which f ′ (x) = 0. (c) Make a sign diagram to show the intervals on which the function is increasing, and on which it is decreasing. (d) Find f ′′ (x), and determine the values of x for which f ′′ (x) = 0. (e) For which values of x is the function f concave up? Concave down? (f) Find the (x, y) coordinates of any local maxima and minima of the function f . (g) Find the (x, y) coordinates of any inflection points of f . (h) Use all of the above information that you have gathered to sketch the graph of y = f (x) for −1 ≤ x ≤ 4. Z 0 −f (x) dx. Shade (lightly) the area (i) Use the Fundamental Theorem of Calculus to compute −1

corresponding to this integral on the sketch from the previous item.

Question 2 (4+4+6+6=20 marks) Find the following antiderivatives: Z (a) (3x2 + 2) sin(x3 + 2x) dx 2x2 dx + 5)3

(b)

Z

(c)

Z

(d)

Z 

(x3

 7  sin x cos x + 3 sec2 2x dx 2

3 + 2e−3x + xe−x



dx

Question 3 (10 marks) Here is what the following diagram represents: C is the location of a speed camera, placed 5 metres above the road (directly above A). V is the variable position of a vehicle moving away from A, going towards B. The pictured distance between A and V is 2.5 metres. The camera tracks the moving vehicle as it passes V. As the vehicle goes, the angle θ changes, and the faster the vehicle goes, the faster the angle changes. The speed limit of 60 km/h (kilometres per hour) corresponds to a critical angular velocity dθdt . If the vehicle goes above 60 km/h, then the angular velocity will be above critical, and the speed camera , in radians per second. will click a picture of the vehicle. Find the critical angular velocity dθ dt

Question 4 (5 marks) Verify that the function y = cos 3x + 3 sin 3x is a solution to the equation dy + y = 10 cos 3x. dx Question 5 (12 marks) A farmer wishes to fence in a rectangular field of area 3750 square metres. Let the length of each of the two sides (facing north-south) of the field be x metres, and the length of each of the other two sides (facing east-west) be y metres. The price of fencing runs to 10 dollars per metre of fencing that is laid out. Moreover, the northern edge of the fence will also need special wind protection, and that will make the northern edge of the fence twice as expensive, per metre, as the rest of the fence. Use calculus to find the dimensions of the field that will minimise the total cost. Justify that the dimensions that you have found, indeed, minimise the cost.

Question 6 (3+8=11 marks) Consider the functions y = x2 − 2 and y = 2x + 6. (a) Find the coordinates (x, y) of the two points where the graphs of these functions meet. Use algebra! (b) Find the area of the region enclosed between the two curves. Draw an appropriate sketch of the region.

Total 87 marks. [Including the four (4) marks for presentation and communication.]...