Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 37
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41Worksheet #28: Exponential Growth and Decay, Area Between Curves
1. Solve the following equations for α:
20
(a) 500 = 1000e
ln(2)
10
(b) 40 = αe
, where k =
.
7
0 06
(c) 100, 000 = 40, 000e
.
ln(0 5)
36
(d) α = 2, 000e
, where k =
.
18
2. The mass of substance X decays exponentially. Let m(t) denote the mass of substance X at time t
where t is measured in hours. If we know m(1) = 100 grams and m(10) = 50 grams, find an expression
for the mass at time t.
3. A certain cell culture grows at a rate proportional to the number of cells present. If the culture contains
500 cells initially and 800 after 24 hours, how many cells will there be after a further 12 hours?
4. Suppose that the rate of change of the mosquito population in a pond is directly proportional to the
number of mosquitoes in the pond.
dP
= KP
dt
where P = P (t) is the number of mosquitoes at time t, t is measured in days and the constant of
proportionality K = .007
(a) Give the units of K.
(b) If the population of mosquitoes at time t = 0 is P (0) = 200. How many mosquitoes will there be
after 90 days?
5. Suppose that P (t) gives the number of individuals in a population at time t where t is measured in
years. Each year 23 out of 1000 individuals give birth and 11 out of 1000 individuals die.
Find a differential equation of the form P = kP that the function P satisfies.
6. Suppose that f is a solution of the differential equation f = kf on an open interval (a, b) where k is
a constant. Compute the derivative of g(x) = e
f (x) and show that g is constant.
Explain why f (x) = Ae ?
2
3
7. Find the area of the region between the graphs of y = x
and y = x
.
1
3
8. Find the area of the regions enclosed by the graphs of y =
x and y =
x +
in two ways. 1) By
4
4
writing an integral in x. 2) Solve each equation to express x in terms of y and write an integral with
respect to y.
3
2
9. Find the area of the region enclosed by the graphs of y = x + 1 and y = x
+ x
x + 1.
10. (a) Show that lim
ln (1 + hr)
= rt. (Hint: Use L’Hospital).
0
(b) From (a) show that lim
(1 + hr)
= e . (Hint: Use your previous result and properties of
0
logarithms.)
(c) The compound interest formula is
r
P (t) = A
1 +
0
n
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