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Unformatted text preview: 1. Evaluate . 932—4
(a) zll—in232—2a: (b) 1m («332? :r _ x) $—>O'0 2. Calculate and simplify where possible the derivatives of the following functions. (a) 3:: (b) (1+ 3:2) arctanm (c) In (x + W) 3. Find g’(2) ifg(a:) = z3h(x2), where h(4) = 2 and h’(4) = —2. 4. Find f”(:1:) iff(x) = zcos z.
5. At what points (z, y) does the curve y = ze‘(“2‘1)/2 have a horizontal tangent line? 6. What is the slope of the curve $31; + 2:2:y3 = 3 at (1, 1)? 22:
7. Let f(x) = $2 +3. (a) Write an equation of the tangent line to the curve y = f (m) at :2: = 1.
(b) Use linear approximation to give an approximate value for f (1.2). 8. A particle moves along the x-axis so that its position at time t is given by a:
t3 — 4t2 + 1.
(a) At t = 2, what is the particle’s speed?
(b) At t = 2, in what direction is the particle moving?
(c) At t = 2, is the particle’s speed increasing or decreasing? 9. Let f(z) = mm for :1: > 0.
(a) Find f’(a:).
(b) At what value of a; does the curve y = f (2:) have a horizontal tangent line?
(c) Does the function f have a local minimum, a local maximum, or neither of these
at the point :2: found in part (b)? 10. You are using Newton’s Method to locate a. root of the equation f (as) = 0, and you
make an initial guess $0 = 2 for the root. The tangent line to y = f (:3) at z = 2 has the equation 3y = 10:1: — 19. What is the next approximation 1:1 to the root that will
be yielded by Newton’s Method? 11. (a) Write the Taylor series for sin(2z) in powers of 2:, showing explicitly all terms of
order up to and including 9:5.
‘ 1 2 (b) Write the Taylor series for 2126""152 in powers of at, showing explicitly all terms of
order up to and including 1:5. ' _ —z
(c) Evaluate lim sm(21:) 2:1:e
:c—>0 2:3 2 .12. Let g(x) = arcsin(cos m).
(a) Calculate and simplify the derivative g’(a:).
(b) At what points does g’(:r) fail to exist? For questions 13-16 show all your work in the space provided. 13. You are in a dune buggy at point P in the desert, 12 km due south of the nearest
point A on a straight east-west road. You want to get to a town B on the road 18
km east of A. If your dune buggy can travel at an average speed of 15 km/h through the desert, and _30 km/ h along the road, towards what point on the road should you
head to minimize your travel time from P to B? 14. A water tank has the shape of a circular cone with top radius 6 metres and vertical
depth 9 metres. Water is pouring into the tank at a constant rate of 141r cubic metres
per hour, and is simultaneously leaking out of a large hole at the bottom at a rate of 21rh cubic metres per hour when the vertical depth of water in the tank is h metres.
(a) How fast is the depth h changing when it is 3 metres?
(b) Will the water ever overﬂow from the tank? Explain. 15. According to Newton’s Law of Cooling, the temperature T(t) at time t of a hot object
introduced into cooler surroundings changes (drops) at a rate proportional to the
amount by which its temperature exceeds the temperature K of its surroundings. (a) Express this fact as a differential equation that must be satisﬁed by T(t). (b) Show that if U (t) = T(t) — K, then U (t) satisﬁes the differential equation of
exponential decay. (c) A cup of coffee is poured in a room being maintained at 20°C. It cools from 80°C
to 50°C in 5 minutes. How much longer will it take to cool to 40°C? 16. A function f(:1:) deﬁned on the whole real line satisﬁes the following conditions:
f(0)=0, f(2)=2 1imf($)=0 I400 f'(a:) = K (2:1: — 2:2)e‘” for some positive constant K. (a) Determine the intervals on which f is increasing and decreasing and the location
of any local maximum and minimum values of f. (b) Determine the intervals on which f is concave upward or downward and the a:-
coordinatm of any inﬂection points of f. (c) Determine lim f (ac). z—y—oo . _
(d) Sketch the graph of y = f (2:), showing any asymptotes and the information
determined in parts (a)-(c). ...
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- Fall '08