Math 53
2
MATH 53 Sampl e Long Exam 1 EAC Arances December 3, 2010 DIRECTIONS: Do as indicated. Show the necessary solution steps. Use only the right pages of your bluebooks for your solution steps and answers. Use the left pages for scratch work . Box all final answers when necessary. I. TRUE or FALSE. Write true if the statement is always true. Otherwise, write false. 1. If lim x→a + f (x) and lim x→a − f (x) both exist, then the limit of f (x) as x → a exists. 2. Let f be a function defined on the closed interval [a, b] such that f (a) < 0 and f (b) > 0, then f has a zero in the interval ( a, b). 3. If lim x→a f (x) = 0 and g (x) is a function defined at x = a, then lim x→a f g (x) = 0. 4. If lim x→+∞ f (x) = 0 and lim x→+∞ g(x) = +∞, then lim x→+∞ (f g)(x) = 0. 5. There is a functio n, not identic ally zero, tha t is both odd and even. II. LIMIT EVALUATION. Answer completely . Make sure to use proper limit notation. 1. lim x→ π 12 + 1 − tan3x 1 − cos x 2. lim x→7 − 5 − √ 4 + 3x |7 − x| 3. lim x→−∞ 3x 2 + 8x + 6 − 3x 2 + 3x + 1 4. lim x→2 − x − 2 tan π 2 x 5. lim x→−1 − x 3 − x 2 − 10x − 8 x 3 + 5x 2 + 7x + 3 6. lim x→−∞ sin 2 3x x 2 III. DO AS INDICATED. Given the function f , defined as f (x) = 2 − x x + 4 x < −2 [ [x + 4]] −2 ≤ x < 0 x 2 + 3x x + x 2 0 ≤ x 1. Ident ify all possible point s of disco ntinui ty of f . 2. T est the continuity of f at each of those point s. If a discontinuity occurs, classify it. IV. Given the syste m of equat ions y = 6sin x y = 4 − (2x) 3 , use the intermediate value theorem to show that the system has a solution in x in the interval 0, π 6 . END OF EXAM! Total: ∞ points
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Transcript of Math 53
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MATH 53 Sample Long Exam 1EAC Arances December 3, 2010
DIRECTIONS: Do as indicated. Show the necessary solution steps. Use only the right pages of yourbluebooks for your solution steps and answers. Use the left pages for scratch work. Box all final answerswhen necessary.
I. TRUE or FALSE. Write true if the statement is always true. Otherwise, write false.
1. If limxa+
f(x) and limxa
f(x) both exist, then the limit of f(x) as x a exists.2. Let f be a function defined on the closed interval [a, b] such that f(a) < 0 and f(b) > 0,
then f has a zero in the interval (a, b).
3. If limxaf(x) = 0 and g(x) is a function defined at x = a, then limxa
(f
g
)(x) = 0.
4. If limx+f(x) = 0 and limx+g(x) = +, then limx+(fg)(x) = 0.
5. There is a function, not identically zero, that is both odd and even.
II. LIMIT EVALUATION. Answer completely. Make sure to use proper limit notation.
1. limx pi
12+
1 tan 3x1 cosx
2. limx7
54 + 3x|7 x|
3. limx
(3x2 + 8x+ 6
3x2 + 3x+ 1
)
4. limx2
x 2tan pi2x
5. limx1
x3 x2 10x 8x3 + 5x2 + 7x+ 3
6. limx
sin2 3x
x2
III. DO AS INDICATED.Given the function f , defined as
f(x) =
2 xx+ 4
x < 2[[x+ 4]] 2 x < 0x2 + 3x
x+ x20 x
1. Identify all possible points of discontinuity of f .
2. Test the continuity of f at each of those points. If a discontinuity occurs, classify it.
IV. Given the system of equations {y = 6 sinx
y = 4 (2x)3 ,
use the intermediate value theorem to show that the system has a solution in x in the interval(0, pi6
).
END OF EXAM!Total: points
