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werwr - MATH 140 Dr Wolfe FINAL EXAM 1(25 points(a Find 9:2...

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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 Newton-Raphson method. If our initial guess is $0 = 1, what is $1 ? ...
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