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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 x24x+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 .
xeO 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 36 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 areturned 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) yintercept (ii) rintercepts (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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