MATH 1P97, WINTER 2016-17 ASSIGNMENT 3 Due Date: Monday March 6, 2017 at 11:59 p.m. Home assignments are individual submissions. While students are encouraged to work together, they cannot submit identical parts of the homework. Doing so will result in a charge of academic misconduct and a zero grade for the assignment. Further penalties will be imposed on repeat offenders. Students are expected to complete all questions on the assignment. However, only a subset of questions will be considered for marking. Marks will be deducted for incomplete assignments. Assignment submissions must be neat, legible, written on only one side of the page in pencil, blue or black ink. Students are expected to use proper paper (not torn off a booklet) and use a ruler for drawing straight lines for graphs, long fractions and answer boxes. Messy assignments will incur a penalty of 10%. Questions must be submitted in order, and a cover page must be attached to the front of the assignment, stapled on the top left corner (see sample cover page on Sakai). Be sure to write your section number on the cover page. Questions to be done by hand or with Maple are labeled HAND or MAPLE respectively. Submit clearly labelled printout for all questions which require Maple. This Assignment covers material discussed in lectures and corresponding to sections 3.3, 3.4, 3.5, 3.6, 4.1, 4.2, 4.3 and 4.4 of the Textbook as well as the Mean Value Theorem. \u0012 1. Consider the function f (x) = 3x 1 2x + 3 \u00132 . Show all details of your work on the questions below. (a) (HAND) Determine its domain of definition (that is, where the function is defined). (b) (HAND Determine infinite limits and limits at infinity and identify the vertical and horizontal asymptotes. (c) (HAND) What is the domain of continuity of f (x) (that is, where the function is continuous)? (d) (HAND) Use both the Chain Rule and the Quotient Rule to determine the first derivative of f (x). Simplify your answer to the best form. (e) (HAND) Determine the critical point(s) in the domain of f (x) and their nature (stationary point(s) (i.e., points with zero derivative) or singular point(s) (i.e., points of with no derivative)). (f) (HAND) Based on a sign study of the first derivative, determine the intervals of increase and decrease of f (x). (g) (HAND) Use (f) to determine the local extrema of f (x) (point(s) where they occur, their nature, and the value of the function f (x) at those points). (h) (HAND) Compute the second derivative of f (x). (i) (HAND) Determine the concavity (intervals of upward and downward concavity) and the points of inflection of f (x). (j) (HAND) Use the second derivative to confirm the findings in (g). (k) (MAPLE) Plot the graph of f (x) and annotate by hand with a pencil on the MAPLE printout the local extrema and the inflection point(s). 2. Consider the equation of the Conchoid of Nicomedes x2 y 2 = (x + 1)2 (4 x2 ). 1 (a) dy in terms of x, y using implicit differentiation. dx 2. (HAND) Select a particular point (x, y) of your choice on the Conchoid and write the equation of the tangent line to the Conchoid at that point. 3. (MAPLE) Use the "implicitplot" command to draw both graphs of the Conchoid and the tangent line obtained in (ii) in the window [-5,5] by [-2,2] in blue color for the Conchoid and in red color for the tangent line, with constrained scaling, and 1000 numpoints (consult Maple HELP on the color, scaling, and numpoints plot options or check the examples in the Maple folder on Sakai). 1. (HAND) Find a formula for (b) (MAPLE) Use the "implicitplot" command to draw the graph of the Heart (x2 +y 2 1)3 = x2 y 3 on an appropriate x, y window. 3. When a particle moves at high velocity, its mass m is different from its mass m0 at rest according to the following formula: m0 m= r v2 1 2 c where v = v(t) is the velocity at time t and c = 2.98 108 m/ sec is the speed of light. (a) (HAND) Verify that if v = 0 then m = m0 . What happens if v c? dv (b) (HAND) Knowing that the acceleration is the derivative of the velocity, write the Related dt dv dm in terms of . Rates formula for dt dt (c) (HAND) Compute the rate at which the mass m(t) is changing when the velocity is 2.60108 m/ sec and the acceleration is 2.30 105 m/ sec2 . 4. (MAPLE) The operating rate (expressed as a percent) of factories, mines, and utilities in a certain 1100n region of the country on the n th day of a given year is f (n) = 75 + 2 . Determine when n + 35000 the operating rate is highest during the first 300 days of that year. Justify that it is indeed a global maximum and plot the graph of the function together with the horizontal tangent line at that maximum point. 5. [BONUS] (MAPLE) Determine the value of the point c in the Mean Value Theorem for the function f (x) = (2x 1)1/3 over the interval [1, 14]. Then write the equation of the tangent line to the graph of f (x) at the point (c, f (c)). Finally plot, on the same window, the graph of f (x) (in red) and the the graph of that tangent line (in blue). Recall that the Mean Value Theorem (MVT) states that, for a differentiable function, the average rate of change over an interval equals the instantaneous rate of change at at least one point in that interval (see figure below); more precisely: Given a function y = f (x) continuous on an interval [a, b], and differentiable on (a, b), then: there exists at least one point c (a, b) with f 0 (c) = Here is an illustration of the MVT: 2 f (b) f (a) . ba Instantaneous Rate of Change at c = Average Rate of Change on [a, b] That is: Slope of tangent line at (c, f (c)) = Slope of the secant line through (a, f (a)) and (b, f (b)) Tangent line PARALLEL to Secant line 3