MATH 1013 SECTIONS A: Professor Szeptycki APPLIED CALCULUS I, FALL 009 B: Professor Toms C: Professor Szeto NAME: STUDENT #: SECTION: No ai (e.g. calculator, written notes) is allowe. Final Exam: Sat 1 Dec 009, 09:00-1:00 ANSWER AS MANY QUESTIONS AS YOU CAN. All questions carry equal marks. In all questions it is essential to explain your reasoning an to provie etails of the intermeiate steps taken in reaching your answers. Answers are to be written in this question book. Do not remove any pages. IF you nee extra space for your answers use the backs of pages, but CLEARLY inicate which answer is which! Make sure to write your name an stuent number on the paper. Stuents will not be allowe to leave uring the final 15 minutes of the examination perio in orer to avoi isruption to those continuing to work on the paper. MARKING TEMPLATE 1 3 4 5 6 7 8 9 10 TOTAL 1
Question 1. Evaluate exact values of the following quantities, if they exist. If any oes not exist, state so. a) (in raians) sin -1 (- 1 ). b) (in raians) cos -1 (cos()). c) cos(cos -1 ()). Question. Fin each of the following limits OR inicate that it oes not exist. If the limit oes not exist clearly explain why not. (i) (iii) lim x 0 lim x 3 cos x 1 (tan x) (ii) lim x 9 x 3 x 3 (iv) lim( x 9 x 3 x x + x ) x + x +1 Question 3. Consier the function f(x) = x - x a) Where is the function not continuous, if anywhere? b) Consier the following statement: f(-1) = 1, f(1) = -1. Therefore the Intermeiate Value Theorem assures us that there must be at least one value of c between -1 an 1, such that f(c)=0. Do you agree with the statement? If you o not, explain why not. c) If you believe that the Mean Value Theorem applies to f(x) on the interval [1,], fin one value of x between 1 an that satisfies the theorem. If you believe that the theorem oes not apply, explain why not. Question 4. a) Fin the inicate erivatives using any metho you like. Simplify as far as possible. (i) t [sin (tan t)] (ii) x (ln(4ex )) b) Use the limit efinition of a erivative (any other metho will receive no merit) to show that x ( x ) = x
Question 5. a) Fin y/x by implicit ifferentiation (or any other vali metho) if x 3 + 6x y 3xy = 4 b) At what values of x, if any, is y/x unefine? Question 6 Make an analysis (natural omain, range, symmetry if any, asymptotes if any, increasing/ecreasing regions, relative an absolute max an min if any, concave up/own regions, points of inflection if any, etc.) of the function f (x) = e ( 1 x ) Sketch the curve y = f(x), labelling axes, intercepts if any, asymptotes if any, points of max an min if any, an points of inflection if any. Clearly inicate intervals of x where the function is increasing or ecreasing, an the curve is concave up or concave own. Question 7 You are require to figure out the imensions of a can (i.e. a cylinrical container with a flat circular li an base) that woul minimize its cost of prouction. The volume must be 56 π cm 3. The base as well as the li is twice as costly per unit area as the cylinrical wall. [Hint: the volume of a right circular cyliner of raius r an height h is π r h. Its curve wall is πrh. You are expecte to know how to calculate the area of each en piece.] Note: you re not require to calculate the minimal cost; only the relevant imensions. Question 8. Consier the function f(x) = x - 1 over the interval [1,4]. (a) Write out an evaluate the Riemann Sum R 3 obtaine by partitioning [1,4] into 3 equal sub-intervals an choosing sample points x * i = x i (i.e. the right han en of the i-th interval). [Hint, in terms of terminology use in your text, figure out a, b, n, Δx, x 1, x, x 3. Procee from there.] (b) Generalize to N equal sub-intervals, an write an expression for R N over the same interval. You are expecte to use sigma notation to express the require sum. You are not require to evaluate the sum. 3
Question 9. cos x 1+ a) Use L'Hospital's rule to etermine lim x 0 x 4. b) Evaluate an simplify as far as possible: x i) x +1 x x ii) (sin t + cos t) t iii) sin u cos u u Question 10. Using any metho you like, evaluate the following an simplify as far as possible: 4 a) 1 ([ x ] - x) x (Reminer: [ x ] is the largest integer less than or equal to x.) b) 1 1+π (sinx + cosx) x c) x 1 x 4 sin z z END OF EXAM 4
MATH 1013 SECTION M: Professor Szeto APPLIED CALCULUS I, WINTER 011 NAME: STUDENT #: Final Exam 1 April 011, 19:00-:00 No other ais except a non-graphing calculator is allowe. ANSWER AS MANY QUESTIONS AS YOU CAN. All questions carry equal marks. In all questions it is essential to explain your reasoning an to provie etails of the intermeiate steps taken in reaching your answers. Answers are to be written in this question book. Do not remove any pages. IF you nee extra space for your answers use the backs of pages, but CLEARLY inicate which answer is which! Make sure to write your name an stuent number on the paper. Stuents will not be allowe to leave uring the final 15 minutes of the examination perio in orer to avoi isruption to those continuing to work on the paper. MARKING TEMPLATE 1 3 4 5 6 7 8 9 10 TOTAL 5
Question 1. Fin the natural omains an ranges of the functions: a) g(x) = tan(x+π/) b) f(x) = 1/(3+e x ) c) h(x) = 1/(3 - e x ) Question. Fin each of the following limits OR inicate that it oes not exist. If the limit oes not exist clearly explain why not. a) lim θ 0 c) lim x 5 5 x x 5 cosθ 1 (sinθ) b) lim t 36 ) lim u ( t 6 36 t u + 3u ) u Question 3. The function f(x) = x 3-3x - 4 refers to all parts below. a) Where is the function continuous? b) Show that the function crosses the x-axis at least once. [Hint: the Intermeiate Value Theorem may help. If you wish to use this theorem, you nee to state the conition(s) uner which IVT applies.] c) Show that the function crosses the x-axis at most once. [Hint: locate critical numbers, then apply Rolle s Theorem over an appropriate interval.] Question 4. a) Fin the inicate erivatives using any metho you like. Simplify as far as possible. (i) [cos(tan 3t)] (ii) t x (ln(5e6x )) b) Use the limit efinition of a erivative (any other metho will receive no merit) to obtain the following erivative x ( 1 x 1 ) Question 5. a) Fin y/x by implicit ifferentiation (or any other vali metho) if (x + y ) = 8 b) At what values of x, if any, is y/x unefine? 6
Question 6. Make an analysis (natural omain, range, symmetry if any, asymptotes if any, increasing/ecreasing regions, relative an absolute max an min if any, concave up/own regions, points of inflection if any, etc.) of the function y(x) = e 1x. Sketch the curve, labelling axes, intercepts if any, asymptotes if any, points of max an min if any, an points of inflection if any. Clearly inicate intervals of x where the function is increasing or ecreasing, an the curve is concave up or concave own. Question 7. a) You wish to construct a box-like builing with a volume of 64,000 m 3 an a square base. The lan costs $1400/m. The roof, floor, as well as sie-walls cost $100/m. Fin the imensions of the box that minimize the cost. b) Suppose a local regulation restricts the height to 40m or less. What imensions woul minimize the cost in this case? [Creit will be allocate to a relevant iagram, appropriately introuce variables, correct formulation of the cost function in terms of relevant variables, an correct minimization. An appropriate test (hint: use the close interval metho) must be use to show that you have ientifie a minimum rather than a maximum.] Question 8. Consier the function f(x) = x + 4 over the interval [3,6]. a) Write out an evaluate the Riemann Sum R 3 obtaine by partitioning [3,6] into 3 equal sub-intervals an choosing sample points x i * = x i (i.e. the right han en of the i-th interval). [Hint, in terms of terminology use in your text, figure out a, b, n, Δx, x 1, x, x 3. Procee from there.] b) Generalize to N equal sub-intervals, an write an expression for R N over the same interval. You are expecte to use sigma notation to express the require sum. To evaluate the sum, you may assume n i = i=1 c) Evaluate lim N R N n(n +1) n, i = i=1 n(n +1)(n +1) 6 7
Question 9. a) Use L'Hospital's rule to etermine lim x 0 b) Evaluate an simplify as far as possible: (cos x) 1 x. i) x +1 x x 3 ii) (e u + e u ) π u iii) 4 ( 4 cosθ) θ 0 cos θ Question 10. Using any metho you like, evaluate the following an simplify as far as possible: a) tan 1 (u) u x x b) 0 π/ (cos x) (cos(sin x)) x END OF EXAM 8
MATH 1013 APPLIED CALCULUS I, Winter 013 NAME: STUDENT #: Final Exam 16 April 013, 19:00-:00 No other ais except a non-graphing calculator is allowe. ANSWER ALL QUESTIONS. Questions carry equal marks. In every question it is essential to explain your reasoning an to provie etails of the intermeiate steps taken in reaching your answers. Answers are to be written in this booklet. Do not remove or insert any pages. IF you nee extra space for your answers use blank pages 1 & 13, but CLEARLY inicate, see P.1 or see P.13. If you use the flip sie of a page, ensure you inicate, see over page. Make sure you write your Name an Stuent Number on this page. You will not be allowe to leave uring the final 15 minutes of the examination perio in orer to avoi isruption to those continuing to work on the paper. MARKING TEMPLATE (answers to more than ten questions will not be grae) 1 3 4 5 6 7 8 9 10 TOTAL 9
Question 1. a) Consier the function g(x) = ln(cos(x)). Determine its omain an range. b) Above g(x) is a many-to-one mapping, an therefore oes not possess an inverse function unless its omain is restricte. Work out g -1 (x) for an appropriately restricte g(x). [You o not nee to work out the omain an range of g -1 (x).] Question. Fin each of the following limits OR inicate that it oes not exist. If the limit oes not exist clearly explain why not. a) lim θ π sinθ θ c) lim x 5 5 x x 5 b) lim u 16 4 u u 16 ) lim t ( 3t 7t t 4 ) Question 3. a) Fin the inicate erivatives using any metho you like. Simplify as far as possible. (i) t [tan(tant)] (ii) x (3 ln(e8x ) e ln x ) b) Use the limit efinition of a erivative (any other metho will receive no merit) to obtain the following erivative x ( 1 x ) Question 4. a) Fin y/x by implicit ifferentiation (or any other vali metho) if x 3 + 6x y 3xy = 4 b) At what values of x, if any, is y/x unefine? Question 5. a) If y = (sin x) x, evaluate y/x. b) Fin the equation of the tangent line to the curve x 3 / 4 + y 3 / 4 = 43 at the point (16, 81). Question 6. Make an analysis (natural omain, range, symmetry if any, asymptotes if any, increasing/ecreasing regions, relative an absolute max an min if any, concave up/own regions, points of inflection if any, etc.) of the function y = xe x. Sketch the curve, labelling axes, an any of the following if they exist: intercepts, asymptotes, max an min, an inflection. Clearly inicate intervals where the function is increasing or ecreasing, an concave up or concave own. 10
Question 7. You are an engineer in charge of builing a cylinrical builing consisting of a curve wall an the roof. (Note: no floor.) Denote its height by h, an raius r. Its volume has been specifie as V. The curve sie cost $1 per sq. m. The roof costs $N per sq. m. Write own an expression for the cost of material, C. Work out the imensions of the tank that woul minimize the cost of material, an show that the ratio of h to r is N. Note: for full creit you must specify omain, ientify critical numbers, use an appropriate test to show that a minimum is attaine, an so on. (You o NOT nee to calculate the cost.) You may assume that the surface area of the curve wall of the cyliner as the circumference of the circular en multiplie by the height. r h Question 8. Consier the function f(x) = x - over the interval [3,6]. Then, a) Write out an evaluate the Riemann Sum R 3 obtaine by partitioning [3,6] into 3 equal sub-intervals an choosing sample points x i * = x i (i.e. the right han en of the i-th interval). [Hint, in terms of terminology use in your text, figure out a, b, n, Δx, x 1, x, x 3. Procee from there.] b) Generalize to N equal sub-intervals, an write an expression for R N over the same interval. You are expecte to use sigma notation to express the require sum. To evaluate the sum, you may assume n i = i=1 c) Evaluate lim N R N n(n +1) n, i = i=1 n(n +1)(n +1) 6 Question 9. a) Use L'Hospital's rule to etermine lim x 0 x sin x 4x 3. b) Evaluate an simplify as far as possible: i) 3π x x ii) (cos θ + sin π θ) θ iii) (cosθ 1 x 3 sin θ ) θ π 4 π Question 10. Using any metho you like, evaluate the following an simplify as far as possible: π/4 4 a) tan 1 (3u 3 ) u b) x x (sec x) (cos(tan x)) x 0 END OF EXAM 11