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Unformatted text preview: The University of British Columbia
December 14. 2000 Mathematics 102 Section 103, 104
Instructor: Dr. Leah Keshet Duration: 2 hours 30 minutes I Be sure that you have .10 pages in addition
to this One. 0 Put your FULL name below and on the
back of the other pages. 0 In all questions, you must show work — i.e.
display intermediate results — to get full
credit. 0 You may use a calculator and a ruler. 0 Be careful! Read the questions carefully.
Follow instructions exactly. Check all work. 0 Be neat! I will not attempt to decipher
messy calculations or interpret messy pic
tures. o All work you wish to be graded must be
placed on these sheets. Work books should
be used for scrap work only, and must not
be handed in. I will simply throw away any
other papers I receive. H
D H H 'H I... l
I. o CD ,_. H
c N First Name
Last Name
Signature Student Number H
C
C Mathematics 102 “—m Section 103. 104 __._____~__~_ page 1 out of 10 [10] 1. Find the derivatives of the following functions. Do not simplify your answers. (a) f(a:) i 111(2: + v:::2 + 1) 0)) W) = t2?” (c) f (2:) = arctan(1 — 32) (d) f($)= (COS($))/(1~Sin($)) EBE) EEQE REMOVE M, m. _ “Mn" .1» .M Im “W: “ Mathematics 102 ______ Section 103. 104 _________ page 2 out of 10 [20] 2. Each of the following questions can be done with little or no computation. Enter your answers
in the boxes provided. (a) Shown below is the graph of a function y = Rsin(w(t — b)) + c. From the graph determine the
values of the constants R, w, b, c. (b) Shown below is the graph of the velocity of a particle moving in one dimension. Indicate
DIRECTLY on the graph any time(s) at which the particle’s acceleration is zero. (c) A function f (1) satisﬁes f (1) = —1 and f’ (1) = 2. What is the equation of the tangent line of
f (x) at :c = 1? Mathematics 102 —m Section 103. 104 —________ page 3 out of 10 (d) Shown below is the graph of y = Ce"t for some constants C, k, and a tangent line. Use data
from the graph to determine 0 and k. (0, 4) “ (e) Use the data in the graph below to make the best approximation you can to f (2.01). . Mathematics 102 ' Section 103, 104 M.__ page 4 out of 10 [10] 3. Consider the curve 1:4 + y4 = 2.
Your answers to the parts below MUST be supported using calculus.
(a) What isk'the equation of the tangent line to the curve 3:4 + y“ = 2 at (1, 1)? (b) Determine if the curve 3:4 + y4 = 2 is concave up or down near the point (1, 1). Mathematics 102 Section 103. 104 page 5 out of 10 [10] 4. (a) Shown below is the graph of a function y = f(:c). Sketch the graph of f’($) on the grid to
the right. (b) Sketch on the grid below the graph of a function F(:c) such that its derivative corresponds to
the original function, f (2:), given in part (a). Mathematics 102 Section 103, 104 ____.___,__ page 6 out of 10 [8] 5. The half life of strontium 90 is 29.1 years. Suppose a pasture ﬁeld has been contaminated with
strontium and has a level of radioactivity that is 10 times the ”safe level”. How long will it take
until the level of radioactivity in the ﬁeld will return to the ”safe level” ? Mathematics 102 Wm Section 103, 104 __~_______ page 7 out of 10 [12] 6. Consider the differential equation dy/dt = 4y — y2. (a) Sketch the slope ﬁeld corresponding to this equation in the range —1 S y S 5, and sketch on
the same ﬁgure the graphs of the solutions that also satisfy y(0) = 1 and y(0) = 4. (b) Find all steady state (i.e constant or equilibrium) solutions of this differential equation. (c) Let y(t) be the solution of this differential equation with y(0) = 1. Use Euler’s method with
step size 0.1 to estimate y(0.2). ' Mathematics 102 Section 103, 104 [10] 7. Use Newton’s method to ﬁnd a value of :1: that satisﬁes
em — 122/2 = 0 . You MUST follow these instructions:
0 Use the starting value .120 = 0. a Display your answer to 4 signiﬁcant places. a Show all intermediate steps. page 8 out of 10 Mathematics 102 _________.______ Section 103. 104 ____m page 9 out of 10 [10] 8. A cannon—ball ﬁred by a cannon at ground level at angle 0 to the horizon (0 S 0 S 7r/2) will
travel a horizontal distance (called the range, R) given by the formula below: R = (1/16)vgsin0c030. Here 110 > 0, the initial velocity of the cannon—ball, is a ﬁxed constant and air resistance is neglected.
(See diagram below.) What is the maximum possible range? [10] Mathematics 102 Section 103. 104 page 10 out of 10 9. A wheel of radius 1 meter rolls on a ﬂat surface without slipping. The wheel moves from left
to right, rotating clockwise at a constant rate of 2 revolutions per second. Stuck to the rim of the
wheel isa piece of gum; as the wheel rolls along, the gum follows a path shown by the wide arc
(called a ”cycloid curve”) in the diagram. The (2:, y) coordinates of the gum are related to the
wheel’s angle of rotation 0 by the formulae 22:9—sin0, yzl—cosﬁ, where 0 S 0 S 27r. How fast is the gum moving horizontally at the instant that it reaches its
highest point? How fast is it moving vertically at that same instant? ...
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 Winter '09
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