Final Exam, Calculus with Life Sciences Applications
Spring 2005, Monday May 9, 2005
Please show all your work, not a calculator answer. And relax, relax, relax, ...
1. Twelve persons are to be equally divided among four laboratories. How many ways
can this be done?
2. Suppose 40% of the seeds of a plant germinate. A researcher plants 7 pots with 8
seeds in each. What is the probability that at least one pot will have no seeds germinate?
3. Suppose the rainfall in a region is normally distributed with mean 100 cm. and
standard deviation 15 cm. What is the probability that in a given year the rainfall does
not exceed 70 cm.?
You may use the following table of areas under the standard normal curve to the left of
the given z-value:
z
Area
0
0.50
0.5
0.69
1
0.84
1.5
0.93
2
0.98
2.5
0.99
4. Let X be a random variable with probability mass function described by the following
table:
x
P(X=x)
-1
0.1
0
0.3
2
0.2
3
0.1
4
0.2
6
0.1
Find E(X) and Var(X).
5. Suppose we cross dominant phenotype red flowering pea plants with recessive
phenotype white flowering pea plants. Suppose that the offspring are 20% white
flowering. What percentage of the original red flowering plants had a pure RR genotype?
6. Let X be a continuous random variable with density function f(t) = 0.5 exp(-0.5 t) for
t >0 and f(t) = 0 otherwise.
(a.) Verify that this indeed has the properties of a probability density function.
(b.) Find E(X).
7. To solve a system of linear equations efficiently on a computer, we set up an
augmented matrix and use row reduction. Suppose we have the system of equations
3 x2 yz9, 2 x4 y3 z8, 5 y2 z1
(a.) Set up the augmented matrix used to solve this system of linear equations.
(b.) Carry out the first two steps of the row reduction for this matrix.
8. Here is a slope field:
(a.) Does the origin represent a stable or unstable equilibrium?
(b.) It corresponds to one of the following four sets of differential equations.
Which one, and why?
(i.) d x1 / dt = +0.5 x1 + 0.8 x2
d x2 / dt = +0.5 x1 – 0.8 x2
(ii.) d x1 / dt = +0.5 x1 – 0.8 x2
d x2 / dt = +0.5 x1 + 0.8 x2
(iii.) d x1 / dt = -0.5 x1 – 0.8 x2
d x2 / dt = -0.5 x1 + 0.8 x2
(iv.) d x1 / dt = -0.5 x1 + 0.8 x2
d x2 / dt = +0.5 x1 + 0.8 x2
9. In the Gompertz growth model, the per capita growth rate depends on the population
density. Gompertz argued that the population N(t) satisfies a differential equation
dN / dt = k N ( ln(K) – ln(N) )
with initial condition N(0) = N0 where k and K are positive constants. He then showed
that the solution to this equation is given by
N(t) = K exp( - ln( K/N0 ) exp( -k t) )
Verify that this N(t) is a solution to Gompertz’ differential equation and initial condition.
4
10. Find 
 z 2 z5 dz
3
5

1

dx
11. Find 
2

x 2 x

3
12. Use the midpoint rule with four boxes to estimate this integral:
0.3 
2
 x 




 2 

e
dx


-0.1