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Math 100 Dec 02 Questions

Math 100 Dec 02 Questions - [42 1 The University of British...

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Unformatted text preview: [42] 1. The University of British Columbia Sessional Examinations - December'2002 Mathematics 100/180 Calculus I Short Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. At most one mark will be given for incorrect answers. It is not necessary to simplify your answers for this question. (a) Evaluate the following limit: . 2:2 + 32 + 2 xliIEl 9:2 —- a: - ‘2 (b) Determine an equation of the tangent line at (1,5) to the curve y = $3 _+_ 4_ (c) Calculate the derivative of 9(3) = arctan{ln(.<32 + 1)}. Note: tan”1 is alternate notation for arctan. (d). Find the second derivative of . for) : esm 2x (e) Evaluate the following limit, if it eidsts , sin 1 hm . 2-.0 tan 3a: (f) A bead is moving on a straight wire, its position a given as a function of time t below: z(t) = t3 + t + 1 Find the value a > 0 so that the average velocity of the bead over the interval [0,a] is equal to 5. (g) Suppose the function f is defined as f($) = 0ifx<0 Ax2+Bifx20 for constants A and B. For what value or values of A and B is f differentiable for all cc? (h). Find the tangent line at (1,1) to the curve zsin(zy - y2) = 2:2 -— 1. (i) Indicate on the graph below where the estimates 2:1 and $2 to the root of f ( ) will be located if calculated using Newton' 3 method starting with initial estimate( (guiss) 2:0 (j) Determine the slant asymptote of x2-4x+3 (k). If f(1) = 2- and f’(1) = 3 use a linear approximation to estimate the value f(1.15). (1) In the previous question, if it is known that |f”(:t)| < 2 for all 2:, what is the maximum error in your approximation? ' (m) Determine the first three nonzero terms in the Taylor series based at a: = 0 (Le the McLaurinseries) for cos(2;r) (n) Evaluate , cos(22:) — 1 + 2x2 hm 4 . x-eO 2‘ Hint: the answer to the previous problem will be helpful here. [10] 2. Theory Questions. Each question is worth 5 marks. Justify and simplify vour answers for full marks. Show all your work. (a) Evaluate the derivative of f (1') = «2: + 1 using the definition of the derivative, h~0 n.) z 1.... “if. + 1-:- w No cerdit will be given if you use derivative formulas. [12] [12] [12] [121 (b) Consider functions f (9:) defined on the interval [0,1] that have the following properties: (i) f is continuous in the interval [0,1]. (ii) f is differentiable in the interval (0,1). (in) m» = o and M) = 1- Find the function defined on the interval [0,1] that has these properties that makes the maximum of |f’(2:)( on the interval (0,1) as small as possible. Hint: begin by sketching some graphs that satisfy the properties above. ' Word Problems. Questions 3-6 are word problems, worth 12 marks each. Carefully justify your answers. Show all your work. It is not necessary to simplify your answers, although simplification in intermediate steps may help. 3. A bead is constrained to move on a wire bent into the shape of the graph y = sin 2:. At a certain instant in time, the bead is at a: = 1r/4 and dm/dt = ‘2. Find the rate of change of the distance between the origin and the bead at this instant in time. 4. Find the point or points on the graph y = $2 closest to (0, 3/ 2). Show that you have carefully checked all possibilities. 5. A 1 kg model boat is put into the water at rest and its engines are turned on.- Let v(t) denote the velocity of the boat in m/s, with t in seconds after the‘engines are-turned on. It is known that 'u(t) solves the differential equation d'v ‘d—t —1—’U/10 (this equation represents mass times acceleration equal to the net force of the engines and friction). - (a) Solve the difi'erential equation for v(t). (b) Evaluate limt_,¢,D v(t). 6. Sketch the graph of 9 f(2) = we“? showing all of the following if they are present: (i) y-intercept (ii) r-intercepts (roots) . . . . (iii) critical points, local maxima and minima, intervals where f is increasmg or decreasrng. (iv) inflection points and intervals where f is concave upward or downward. (v) asymptotes (horizontal, vertical, slant). Give the (x,y) coordinates for all the points of interest above. ...
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Math 100 Dec 02 Questions - [42 1 The University of British...

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