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Unformatted text preview: MATH 140 Dr. Wolfe FINAL EXAM July 25, 2003
1. (25 points)
(a) Find
9:2 + 3
l' — .
.22.” x _ 1 }
(b) Let
1 + x
= l .
9(17) n 1 _ 38
Find the domain of g and also ﬁnd the derivative of g.
(c) Find
lim sinw
:c—+0+ «E
(d) Suppose f’ (2) = 5. Evaluate
lim 2: — 2 9H2 f(<13)— f(2)' 2. (15 points) A helicopter is rising vertically at a constant speed. An observer on the
ground 800 ft. from the point at which the helicopter took off shines a beam of light
at the helicopter. When the helicopter is 600 ft. above the ground, the angle the beam makes with the ground is increasing at the rate of .02 radians per second. How fast is the
helicopter rising ? 3. (15 points) Let f (:13) = 3:643”. Find all critical numbers of f and determine if f has a relative maximum, a relative minimum or neither at each of these critical numbers. Be
sure to explain why. 4. (15 points) Sketch the graph of a function 9 deﬁned on (——00, 1) U (1, 00), with all the following properties. Label the graph with all pertinent information. (a) limx_,_oo g(a:) = ——1 and limxnoo g(x) = 2. (b) limm_,1_ g(a:) 2 00 and 1imzn1+g(x) == 00. (c) The graph of g is concave upward on (~00, 1) and on (1,5), and is concave‘downward
on (5,00). (d) 9(3) = 0 is a relative minimum of g. . 5. (20 points) A rectangular ﬁeld having an area of 1200 square meters is to be enclosed by a fence and an additional fence is to be used to divide the ﬁeld down the middle. If the
cost of the fence down the middle is $2 per meter and the fence along the outside costs $3 per meter, ﬁnd the dimensions of the ﬁeld such that the cost of the fencing will be as
small as possible. A 6. (25 points) Evaluate the following integrals: 7. (20 points)
(a) Sketch the ellipse .732 + 4y2 = 4 and locate the foci and vertices.
(b) Find the equation of the line L which is tangent to the ellipse at (1, —\/§ / 2) (c) Express the area A of the region inside the ellipse of part (a) as an integral. Do not
attempt to evaluate the integral. 8. (15 points) Let f(m) 2 cc — 51n at.
(a) Show that there is exactly one solution a to the equation f (ac) = 0 lying in (1,2). Note
ln2 z .69. (b) We wish to ﬁnd oz by using the NewtonRaphson method. If our initial guess is $0 = 1,
what is $1 ? ...
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 Algebra

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