Chapter 8 Exponential functions - Ugrad.math.ubc.ca

Chapter 8
Exponential functions
8.1
Graph the following functions:
(a) f(x) = x 2 e −x
(b) f(x) = ln(x 2 + 3)
(c) f(x) = ln(e 2x )
8.2
Express the following in terms of base e:
(a) y = 3 x
(b) y = 1
7 x
(c) y = 15 x2 +2
Express the following in terms of base 2:
(d) y = 9 x
(e) y = 8 x
(f) y = −e x2 +3
Express the following in terms of base 10:
(g) y = 21 x
(h) y = 1000 −10x
(i) y = 50 x2 −1
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Math 102 Problems Chapter 8
8.3
Compare the values of each pair of numbers (i.e. indicate which is larger):
(a) 5 0.75 , 5 0.65
(b) 0.4 −0.2 , 0.4 0.2
(c) 1.001 2 , 1.001 3
(d) 0.999 1.5 , 0.999 2.3
8.4
Rewrite each of the following equations in logarithmic form:
(a) 3 4 = 81
(b) 3 −2 = 1 9
(c) 27 −1 3 =
1
3
8.5
Solve the following equations for x:
(a) ln x = 2 ln a + 3 lnb
(b) log a x = log a b − 2 3 log a c
8.6 Reflections and transformations
What is the relationship between the graph of y = 3 x and the graph of each of the following
functions?
(a) y = −3 x (b) y = 3 −x (c) y = 3 1−x
(d) y = 3 |x| (e) y = 2 · 3 x (e) y = log 3 x
8.7
Solve the following equations for x:
(a) e 3−2x = 5
(b) ln(3x − 1) = 4
(c) ln(ln(x)) = 2
(d) e ax = Ce bx , where a ≠ b and C > 0.
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Math 102 Problems Chapter 8
8.8
Find the first derivative for each of the following functions:
(a) y = ln(2x + 3) 3
(b) y = ln 3 (2x + 3)
(c) y = ln(cos 1 x) 2
(d) y = log a (x 3 − 2x) (Hint :
d
dx (log a x) = 1
x ln a )
(e) y = e 3x2
(f) y = a −1 2 x
(g) y = x 3 · 2 x
(h) y = e ex
(i) y = et − e −t
e t + e −t
8.9
Find the maximum and minimum points as well as all inflection points of the following functions:
(a) f(x) = x(x 2 − 4)
(b) f(x) = x 3 − ln(x), x > 0
(c) f(x) = xe −x
(d) f(x) = 1
1−x + 1
1+x , −1 < x < 1
(e) f(x) = x − 3 3√ x
(f) f(x) = e −2x − e −x
8.10
Shown in Figure 8.10 is the graph of y = Ce kt for some constants C, k, and a tangent line. Use
data from the graph to determine C and k.
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Math 102 Problems Chapter 8
(0, 4)
y = Ce kt
(2, 0)
Figure 8.1: Figure for Problem 8.10
8.11
Consider the two functions
(a) y 1 (t) = 10e −0.1t ,
(b) y 2 (t) = 10e 0.1t .
Which one is decreasing and which one is increasing? In each case, find the value of the function at
t = 0. Find the time at which the increasing function has doubled from this initial value. Find the
time at which the decreasing function has fallen to half of its initial value. [Remark: these values
of t are called, the doubling time, and the half-life, respectively]
8.12 Shannon Entropy
In a recent application of information theory to the field of genomics, a function called the Shannon
entropy, H, was considered. A given gene is represented as a binary device: it can be either “on”
or “off” (i.e. being expressed or not). If x is the probability that the gene is “on” and y is the
probability that it is “off”, the Shannon entropy function for the gene is defined as
H = −x log(x) − y log(y)
[Remark: the fact that x and y are probabilities, just means that they satisfy 0 < x ≤ 1, and
0 < y ≤ 1.] The gene can only be in one of these two states, so x + y = 1. Use these facts to show
that the Shannon entropy for the gene is greatest when the two states are equally probable, i.e. for
x = y = 0.5.
8.13 A threshold function
The response of a regulatory gene to inputs that affect it is not simply linear. Often, the following
so called “squashing function” or “threshold function” is used to link the input x to the output y
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Math 102 Problems Chapter 8
of the gene.
where a, b are constants.
y = f(x) =
1
1 + e (ax+b)
(a) Show that 0 < y < 1.
(b) For b = 0 and a = 1 sketch the shape of this function.
(c) How does the shape of the graph change as a increases?
8.14
Sketch the graph of the function y = e −t sin πt.
8.15 The Mexican Hat
Find the critical points of the function
y = f(x) = 2e −x2 − e −x2 /3
and determine the value of f at those critical points. Use these results and the fact that for very
large x, f → 0 to draw a rough sketch of the graph of this function. Comment on why this function
might be called “a Mexican Hat”. (Note: The second derivative is not very informative here, and
we will not ask you to use it for determining concavity in this example. However, you may wish to
calculate it just for practice with the chain rule.)
8.16 The Ricker Equation
In studying salmon populations, a model often used is the Ricker equation which relates the size of
a fish population this year, x to the expected size next year y. (Note that these populations do not
change continuously, since all the parents die before the eggs are hatched.) The Ricker equation is
y = αxe −βx
where α, β > 0. Find the value of the current population which maximizes the salmon population
next year according to this model.
8.17 Spacing in a fish school
Life in a social group has advantages and disadvantages: protection from predators is one advantage.
Disadvantages include competition with others for food or resources. Spacing of individuals in a
school of fish or a flock of birds is determined by the mutual attraction and repulsion of neighbors
from one another: each individual does not want to stray too far from others, nor get too close.
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Math 102 Problems Chapter 8
Suppose that when two fish are at distance x > 0 from one another, they are attracted with
“force” F a and repelled with “force” F r given by:
F a = Ae −x/a
F r = Re −x/r
where A, R, a, r are positive constants. A, R are related to the magnitudes of the forces, and a, r to
the spatial range of these effects.
(a) Show that at the distance x = a the first function has fallen to (1/e) times its value at the
origin. (Recall e ≈ 2.7.) For what value of x does the second function fall to (1/e) times
its value at the origin? Note that this is the reason why a, r are called spatial ranges of the
forces.
(b) It is generally assumed that R > A and r < a. Interpret what this mean about the comparative
effects of the forces and sketch a graph showing the two functions on the same set of axes.
(c) Find the distance at which the forces exactly balance. This is called the comfortable distance
for the two individuals.
(d) If either A or R changes so that the ratio R/A decreases, does the comfortable distance
increase or decrease? (Give reason.)
(e) Similarly comment on what happens to the comfortable distance if a increases or r decreases.
8.18 Seed distribution
The density of seeds at a distance x from a parent tree is observed to be
D(x) = D 0 e −x2 /a 2 ,
where a > 0, D 0 > 0 are positive constants. Insects that eat these seeds tend to congregate near
the tree so that the fraction of seeds that get eaten is
F(x) = e −x2 /b 2
where b > 0. (Remark: These functions are called Gaussian or Normal distributions. The parameters
a, b are related to the “width” of these bell-shaped curves.) The number of seeds that survive
(i.e. are produced and not eaten by insects) is
S(x) = D(x)(1 − F(x))
Determine the distance x from the tree at which the greatest number of seeds survive.
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Math 102 Problems Chapter 8
8.19 Euler’s “e”
In 1748, Euler wrote a classic book on calculus (“Introductio in Analysin Infinitorum”) in which
he showed that the function e x could be written in an expanded form similar to an (infinitely long)
polynomial:
e x = 1 + x + x2
1 · 2 + x3
1 · 2 · 3 + ...
Use as many terms as necessary to find an approximate value for the number e and for 1/e to 5
decimal places. Remark: we will see later that such expansions, called power series, are central to
approximations of many functions.
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