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Unformatted text preview: MATHEMATICS 295 FINAL EXAMINATION — SPRING 2002
Print Your Name Signature Print Your Instructor’s Name Section # , Recitation Section # Student Identiﬁcation Number INSTRUCTIONS. This examination has 11 problems and 10 printed pages. (There are '2 additional pages for scrap work.) Make sure your examination copy is complete before
you begin work. There are 200 points available on this examination. The point values are indicated for each of the 11 problems. All work for which you seek credit must be written on the printed pages in the appropriate places. The last two scrap pages of this booklet will not be graded! All answers must be justiﬁed. Calculators may be used to check your answers but not to justify
them. Do not write below this line 1 7. 2 8. 3 9 4 10. 5 11.
6. Total 2 1. (8 points each) Evaluate the following limits. If a limit does not exist, say so and explain
why not:
, 21:2 —~ 41‘ + 2
a) hm —~—~.——
z—u $3 — 2:132 + x 224+:c3 2240012135 +128 . 227
c) 11111 ,
z—m tan 51‘ , [rt—1f
(nil—{n1 z—l 3 2. (10 points each) Find the derivatives of each of the followin I sing:
a $=$e——+ .
)f() 1.9 $4+5 g functions. Do not simplify. (3) f(x) : tan(sin 2:) . . . . . . , . ($2 + I)“
d) Usmg logarithmic differentiation, find f (.73) if f(1:) : 7 .
2:5 (7 + sec x)4 5 5. (15 points) A shallow concrete reservior is in the shape of an inverted cone of radius 45
feet and height 6 feet. Water is leaking from the bottom (the vertex) at the rate of 50 cubic feet
per minute. How fast is the water level falling when the water is 5 feet deep? The volume V of a cone is given by V 2 énrzh where r is the radius of the cone and h is its height. 6 6. (15 points) A right triangle has one vertex at the origin, its right angle on the positive :L'axis, and its third vertex P on the graph of y = ( +
a:
P which causes the triangle to have maximum area. For full credit you must justify your answer using either the ﬁrst or second derivative test. 1 . .
W. Find the x coordinate of the pomt 2 ' « — 12
7. (5 points each) Consider the following function: f(z) 2 m. 2
x
a) Determine the horizontal asymptote(s) of f or tell why it does not have any. b) Determine the vertical asymptote(s) of f and the left hand and right hand limits of f(:z:)
at each vertical asymptote. c) Find f’(.1:) (the ﬁrst derivative of f). 8 d) Determine the intervals where f is decreasing or increasing; ﬁnd any local maxima and inininia off. 12(2: — 6) e) The second derivative of f is given by f”(:r) = T. (Do NOT verify this. You will run out of time if you do.) Determine the intervals where f is concave up or down, and, find any inflection points of f. f) Using the ABOVE, sketch the graph of f showing and LABELING all of its interesting features. NO credit will be given for only copying a picture from your graphing calculator. 9 8. (9 points) Use the deﬁnition of the derivative to ﬁnd the derivative of f(:c) = 3x2 + 4 as
a function of x. 9. (8 points) Find the derivative of $3 f(x) = /ewstdt. 0 10 10. (11 points each) Evaluate the following indeﬁnite integrals: a)/ $+2 dx
(:52 +41:+1)2 b) /__sm_x_d$
2+cosa: 1
11. (11 points) Evaluate the deﬁnite integral: /2:sin 7m:2 d2: 0 ...
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