The University of Sydney
School of Mathematics and Statistics
Tutorial for Week 11
MATH1011: Applications of Calculus
Semester 1, 2020
Web Page:
www.maths.usyd.edu.au/MATH1011
Lecturer: Clio Cresswell, Brad Roberts
Tutorial exercises:
1.
After
t
hours, an organism is producing cells at the rate of 20e
t
cells per hour.
(a)
If
N
(
t
) gives the number of cells produced after time
t
, estimate the number of
cells produced in the first four hours by dividing the interval 0
≤
t
≤
4 into eight
equal subintervals and evaluating the sum
8
X
i
=1
N
0
(
t
*
i
)
×
4

0
8
, where each
t
*
i
is at
the midpoint of the
i
th subinterval.
(b)
Find the number of cells produced in the first four hours by evaluating an appro
priate definite integral.
2.
An animal population is increasing at a rate of 100+40
t
+3
t
2
individuals per year (where
t
is measured in years). By how much does the animal population increase between the
seventh and the tenth years?
3.
Suppose that a person is injected with 40 mg of a certain drug, and that the drug is
then continuously eliminated from the body, at the rate of 12
.
5e

0
.
06
t
mg per hour after
t
hours, until it is eliminated completely.
(a)
Let
A
(
t
) be the number of mg of drug remaining in the body
t
hours after the
drug was injected into the patient.
What does the given information say about
the derivative,
A
0
(
t
), of
A
(
t
)? (Note that
A
(
t
) is
decreasing
.)
(b)
Find a formula for
A
(
t
) in terms of
t
.
(c)
How much drug has been eliminated from the body after 1 hour?
(d)
After how many minutes will the person be drugfree?
4.
The rate of growth of a bacterial population,
P
(
t
), is 10(e
t
+ 1), where
t
is in hours.
(a)
By how much does the population increase in the first ten hours?
(b)
Find a formula for
P
(
t
) if initially the population is 1000.
5.
Consider a rightcircular cone of radius
R
and height
h
.
Use integration to find the
volume of this cone.
6.
Suppose
A
(
x
) =
5
√
x
2
describes the area of a crosssection of a solid at
x
(perpendicular
to the xaxis). Find the volume of this solid from
x
= 0 to
x
= 1.