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Unformatted text preview: Math 137 ASSIGNMENT 5 Fall 2008 Submit all boxed problems and all extra problems by 8:20 am. October 17.
All solutions must be clearly stated and fully justiﬁed. Use the format given on UWACE
under Content, in the ﬁle Assignment Format for Math 135 and Math 137. NOTE: Much of this assignment should be familiar to you. However, the depth to which
this material was covered varies in your high school mathematics courses. Do enough of
the unboxed problems to ensure that you are adept with these concepts and techniques. In
writing your solutions, reference the derivative rules used by DSR, DPR, DQR, DCR for
sum, product, quotient, and chain rules. TEXT PROBLEMS: Section 2.8 — 3, I, I, 10, 11, 30, Section 3.1 — 7, 17, I, 24, 30, I, 42, 46, 51, I, I, 64 Section 3.2  I, 6, I, 17, I, 25, 33, 43, 47, 50, 51, Section 3.4 — I, 19, I, 21, I, I, 48, I, 61, 69, I, I,
Section 3.5 — 3, LE], 10, 15, I, 21, I, I, Pages 266269 — 3, 4, 10, I, EXTRA PROBLEMS: E5.1‘ Data given by A. Nilsson (in Greenhouse Earth, New York: Wiley, 1992) on CFCS in
the atmosphere give an approximate model for the concentration of CFC—11 in ppb
(parts per billion) of the form C(t) = 0.05(1.040)‘ ,
where t is the number of years since 1950. a) Find the rates at which C(t) was increasing in 1950, 1980 and 2000. How many
times faster is the concentration of CFC11 increasing in 2000 than in 1950? b) Show that C(t) satisﬁes the equation C’(t) = 160, 0(0) = 00, where k and Co are
constants you should find. E52 The function t 3(1) = 160 (Z — 1 + 6‘“) gives the distance 3 metres fallen by a sky—diver in t seconds. (i) Find her velocity v(t), and state the units of v. (ii) Her ‘terminal velocity’ w is approached as her acceleration approaches zero, i.e.
when the force of gravity is exactly balanced by air resistance. Show that this
happens as t ——> +00, and ﬁnd 07‘. (iii) Find the distance she has fallen when her velocity 1) reaches 95% of ’01. (iv) Sketch v(t), and explain how the shape of the graph depicts the physical situation
represented here. E5.3 Consider the curve with implicit equation :82 + y2 = (:1:2 + y2  :1:)2. a) Find the a: and y intercepts.
b) Show that the curve is symmetric about the :raxis. c) Find the equation of the tangent line to this curve at the upper y—intercept. Using
symmetry, state the equation for the tangent line at the lower y—intercept.
(:1:2 + y2)(2:r  1) — 2a:2 HINT: Implicit differentiation should yield y’ = 1/(1 + 2:1: _ 2(32 + 2,2» 2 2
d) Show that horizontal tangents occur where m2 +y2 = 2:: 1, and then substitute
this into the curve equation to ﬁnd the two points where this occurs.
1 3 1 — 3
e) Use y’ from part c) to show that the points (2,0), (—1, §), and (—Z’ T“) give ly’l —» oo, i.e., vertical tangents. f) Sketch the curve carefully, using its symmetry about y = 0, its intercepts, in part
c), d), e). ...
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 Fall '08
 SPEZIALE
 Calculus

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