MCD2130 Assignment 2 Trimester 1, 2020 MCD2130 Functions and Their Applications Assignment 2 Trimester 1, 2020 Status: Individual Hurdle: None Weighting: 5% Word limit: No limit Due date: By 10am Monday, 11th May 2020 (Week 11) Question 1: 10 marks Question 2: 5 marks Question 3: 10 marks Question 4: 10 marks Question 5: 13 marks Mathematical communication: 3 marks Instructions: Before you start this assignment: • read your lecture notes and the relevant sections in the textbook about appropriate topics; • read through the problem sets (tutorial questions) you have completed as an aid to doing the assign- ment problems; • remember to place a cover sheet with your details on the front of your assignment, this is compulsory for all assessed work; • be aware of Monash College penalties for plagiarism and cheating (see Unit Outline); • unless decimal approximation is asked for, you must show all partial and final results in exact form; • you can use a calculator in the assignment, but not during exam; • show all working out and give clear explanations for key steps in your working; • sketch all graphs to the highest accuracy (by hand), providing all labels, legend and using different colours if appropriate; • read and understand the “Student Writing Guide” there will be marks awarded to mathemat- ical communication; • keep a copy of your assignment; • responses can be hand written, however presentation of your work matters; • a penalty of 10% applies for each day after the day due. Page: 1 of 3 MCD2130 Assignment 2 Trimester 1, 2020 1. Algebraically find all the solutions of the equation −1 + 2 sin2(x)− cos(x) = 0 for x ∈ [0, 2pi) 2. Use the first principles definition of the derivative, that is, df dx = lim ∆x→0 f(x+ ∆x)− f(x) ∆x to find the first derivative of the function f(x) = (x+ 1) 3 2 . 3. Consider the circular function f(θ) = 1 + 2 cos(2θ + pi 2 ) (a) State clearly the amplitude, phase shift, period and the equation for the midline. (b) Show working for all θ- and y-intercepts for the function y = f(θ). (c) Sketch the graph of y = f(θ), and label the axes clearly 4. Find the first derivative of each of the following functions with respect to their independent variable. Your working should clearly show only one differentiation rule per line. (a) f(s) = loge ( s2 + 1 ) (b) f(r) = r 2 e−r 2 (c) f(t) = √ t− √ 1− t2 (d) f(θ) = cos ( tan(θ) + √ θ ) 5. In this question we will find the maximum volume of a cone inscribed inside a sphere of radius 3 units. MCD2130 Assignment 3 Solutions Trimester 3 2014 (4) In this question we will find the maximum volume of a cone inscribed inside a sphere of radius 3 units. b b 3 3 y x (a) Write an equation for the volume, V , in terms of variables x and y for the inscribed cone. (b) Find an equation for x in terms of y. (c) Given your answers in part (a) and part (b) write the function V (y) representing the volume, in terms of variable y, of an inscribed cone. (d) What is the physically implied domain of V (y)? (e) Find the first derivative of V with respect to y. (f) Find critical values for the function V (y). Then write each critical point in Cartesian coordinates (x, y). (g) Find the second derivative of V with respect to y. (h) Use the second derivative test to determine the nature of any critical points found in part (f). (i) What is the value of V (y) for the end values of the domain of V ? (j) What is the maximum volume of a cone inscribed inside a sphere of radius 3 units. Solution: (a) The formula for the volume of a cone is V = pi 3 r2h, therefore V = pi 3 x2 (3 + y). ✄✂ ✁1 (b) Using the triangle we have x2 + y2 = 32. ✄✂ ✁1 Algebraically rearranging we have x = ±√9− y2, and since x represents distance we select the “+”. Therefore, x = √ 9− y2. ✄✂ ✁1 (c) The volume of the inscribed cone in terms of variable y is V (y) = pi 3 (√ 9− y2 )2 (3 + y) = pi 3 ( 9− y2) (3 + y) ✄✂ ✁1 = pi 3 (−y3 − 3y2 + 9y + 27) . (d) The physically implied domain for the volume of the cone is dom(V ) = [0, 3) . ✄✂ ✁1 Markers Notes: Accept dom(V ) = (−3, 3). Markers Notes: Including y = 3 or y = −3 is not physically possible. Page: 7 of 9 Page: 2 of 3 MCD2130 Assignment 2 Trimester 1, 2020 (a) Write an equation for the volume, V , in terms of variables x and y for the inscribed cone. (b) Find an equation for x in terms of y. (c) Given your answers in part (a) and part (b) write the function V (y) representing the volume, in terms of variable y, of an inscribed cone. (d) What is the physically implied domain of V (y)? (e) Find the first derivative of V with respect to y. (f) Find critical values for the function V (y). (g) Find the second derivative of V with respect to y. (h) Use the second derivative test to determine the nature of any critical points found in part (f). (i) Calculate the maximum volume of a cone that can be inscribed in a sphere of radius 3 units. Mathematical Communication: In this assignment marks will be awarded for mathematical communica- tion. Read the “Student Writing Guide” for assistance with this. You should be attempting to implement good mathematical communication in all of your tutorial work, assignments, tests and exams. This includes but is not restricted to; • clear explanations/reasoning for key steps in working; • write explanations/reasoning in a clear coherent sentence structure with no shorthand or mathemat- ical symbols; • take care with spelling and grammar; • notation and symbols are used correctly and used where they are meant to be. Some examples of not doing this are; not using derivative notation when you are calculating the derivative of a function, placing an equal sign between limit notation and the expression the limit is meant to be evaluating (for example lim x−→0 = x2), not dropping equal signs leaving the working look like an array of algebraic expressions with no connection, splitting up derivative notation as if it were a fraction; and so on. • each answer should have a concluding sentence which should indicate the purpose of your working and what the result means in context of the question; • where possible you should check your final result makes sense with respect to the information given in the question. End of Assignment 2 Page: 3 of 3
