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MATH 161 Quiz 10: Calculus I at Millersville Univ. - Answers & Population Growth, Quizzes of Calculus

Millersville University of Pennsylvania (MU)Calculus

The tenth quiz for calculus i (math 161) at millersville university, including the questions, answer key, and a problem about bacterial population growth.

Typology: Quizzes

Pre 2010

Uploaded on 08/19/2009

koofers-user-x5k
koofers-user-x5k ๐Ÿ‡บ๐Ÿ‡ธ

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Millersville University Name Answer Key
Department of Mathematics
MATH 161, Calculus I, Quiz 10
April 24, 2009
Please answer the following questions. Your answers will be evaluated on their correctness,
completeness, and use of mathematical concepts we have covered. Please show all work and
write out your work neatly. Answers without supporting work will receive no credit.
1. Evaluate the following definite integral.
Z1
0
exโˆ’1
e2xdx
Z1
0
exโˆ’1
e2xdx =Z1
0๎˜’ex
e2xโˆ’1
e2x๎˜“dx
=Z1
0๎˜eโˆ’xโˆ’eโˆ’2x๎˜‘dx
=Z1
0
eโˆ’xdx โˆ’Z1
0
eโˆ’2xdx
=๎˜โˆ’eโˆ’x๎˜‘๎˜Œ
๎˜Œ
๎˜Œ
1
0
โˆ’๎˜’โˆ’1
2eโˆ’2x๎˜“๎˜Œ
๎˜Œ
๎˜Œ
๎˜Œ
1
0
=๎˜โˆ’eโˆ’1+e0๎˜‘โˆ’๎˜’โˆ’1
2eโˆ’2+1
2e0๎˜“
= 1 โˆ’1
e+1
2e2โˆ’1
2
=1
2โˆ’1
e+1
2e2
pf2

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Millersville University Name Answer Key Department of Mathematics MATH 161, Calculus I, Quiz 10 April 24, 2009

Please answer the following questions. Your answers will be evaluated on their correctness, completeness, and use of mathematical concepts we have covered. Please show all work and write out your work neatly. Answers without supporting work will receive no credit.

  1. Evaluate the following definite integral. โˆซ (^1)

0

ex^ โˆ’ 1 e^2 x^

dx

โˆซ (^1) 0

ex^ โˆ’ 1 e^2 x^

dx =

โˆซ (^1) 0

( (^) ex

e^2 x^

e^2 x

) dx

=

โˆซ (^1) 0

( eโˆ’x^ โˆ’ eโˆ’^2 x

) dx

=

โˆซ (^1)

0

eโˆ’x^ dx โˆ’

โˆซ (^1)

0

eโˆ’^2 x^ dx

( โˆ’eโˆ’x

)โˆฃโˆฃ โˆฃ

1 0 โˆ’

( โˆ’

eโˆ’^2 x

)โˆฃโˆฃ โˆฃโˆฃ

1 0 =

( โˆ’eโˆ’^1 + e^0

) โˆ’

( โˆ’

eโˆ’^2 +

e^0

)

e

2 e^2

e

2 e^2

  1. Suppose a bacteria culture initially has 400 cells. After 1 hour, the population has increased to 800. Find an equation for the population at any time. What will the population be after 10 hours? The equation for exponential growth is y(t) = Aekt. Since the initial population of the culture is 400 cells then A = 400. The doubling time is 1 hour, so

ln 2 k

=โ‡’ k = ln 2.

Thus y(t) = 400et^ ln 2^ = 400eln(2t^ )^ = 400(2t). Consequently y(10) = 400(2^10 ) = 400(1024) = 409, 600.