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Unformatted text preview: MATH 100/180
December 2001 Question 1 [54 marks, 3 marks each] a) b)
C) d) g) h) j) k) 1) x2 ~x—2 Evaluate lim 2 X—>2x +x——6 Determine an equation of the tangent line at (1,2) to the curve y = x2 — 4x + 5 . Calculate the derivative of (l —— x2)sin‘l x 2 —
Calculate and simplify the derivative of 32—3.
x — 2x + 2
If f(0) = 5 and f'(0) =% ﬁnd the derivative of g(x) = 1+|f(x)|2 at x = 0. Acosx if x20 Suppose the function f is deﬁned as f (x) = _ for constants A and B. For what
1— Bx If x < 0 values of A and B is f differentiable for all x. Find the third derivative of f (x) = sin ax where a is a constant. Find the tangent line at (0,0) to the curve tan‘1 (x + y) + x2 — 2 = 0 . A bead is moving on a straight wire and its position is given by x(t) = 4t3 —t . Find the positive time t at which its instantaneous speed is equal to its average speed over the time
interval [0,1]. A bead is moving on a straight wire and its position is given by x(t) = t4 — t2 . Find all times I when the acceleration is zero. If f (3) = 5 and f ' (3) = 2 use a linear approximation to estimate the value f (2.8) . A spherical balloon is being ﬁlled with air at a constant rate of 6 cubic centimeters per second.
When its radius is 10 cm, how fast is the radius increasing. Ignore any compressibility effects
of the air. A bacteria colony is established and then grows exponentially. If there are 10,000 bacteria
after 1 day and 20,000 after 2 days, how many will there be after 5 days? Newton’s Law of Cooling states that temperature T(t) of an object changes with time t at a rate
proportional to the difference between T(t) and the ambient temperature To (the temperature of
the surroundings). Express this law as a differential equation for Tusing a positive constant k.
Do not attempt to solve this equation. Indicate on the g raph below where the estimates x1 and x; to the roots of f(x) will be located if
calculated using Newton’s Method starting with initial estimate (guess) x0 f(x) X0 X
p) Determine the ﬁrst three nonzero terms in the Taylor series based at x=0 for e)‘2 .
q) Determine the Taylor series based at x=0 for (x + 1) sin x showing all terms upto and including
the x3 term.
. c0 x —
r) Evaluate 11m ———2——
x—>0 x Question 2 [10 marks] You are ﬂying a kite. At a certain time, the kit is 30m high and 40m horizontally away from you and
is moving horizontally away from you at a rate of 10 meters per minute. Assume the string lies on a
straight line between you and the kite at all times. a) How fast are you letting out the string at that time. b) How fast is the angle between the string and the ground changing at that time. Question 3 [12 marks]
A window is in the shape of a rectangle with the top edge replaced by a semicircle and has perimeter
10m. Find the dimensions of the rectangle that gives the window of greatest area. Question 4 [12 marks]
A certain radioactive substance is known to have a half live of 120 years. Initially, 100 kg were placed in an underground storage facility in 1980. In 2000, another 100 kg were added in a single shipment
and in 2020 another 100 kg will be added in a single shipment. How much of the radioactive
substance will there be in the facility in 2030'? Question 5 [12 marks]
A function f(x) is known to be deﬁned and have continous ﬁrst and second order derivatives for all x. It is also known to have the following properties: i) f(x) : —f(—x) for all x (ie the function is odd)
ii) the graph of f has a slant asymptote y=x iii) the only value x>0 for which f '(x) = 0is x=1 iv) the only value x>0 for which f ” (x) = 0 is x=2 v) f(1)=_1,f(2)=1,[email protected]=o Sketch a graph of f(x) that satisﬁes all of the conditions above showing all root(s), local minina(s),
local maxima(s), and inﬂection point(s). ' ...
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- Fall '08