10.6: Exercises

10.1. Polymerase Chain reaction (PCR). The polymerase chain reaction (PCR) was invented by Mullis in 1983 to amplify DNA. It is based on the fact that each strand of (double-stranded) DNA can act as a template for the synthesis of the second ("complementary") strand. The method consists of repeated cycles of heating (which separates the DNA strands) and cooling (allowing for new DNA to be assembled on each strand). The reaction mixture includes the original DNA to be amplified, plus enzymes and nucleotides, the components needed to form the new DNA). Each cycle doubles the amount of DNA.

A particular PCR experiment consisted of 35 cycles.

(a) By what factor was the original DNA amplified? Give your answer both in terms of powers of 2 and in approximate decimal (powers of ten) notations.

(b) Use the approximation in the caption of Table \(10.1\) (rather than a scientific calculator) to find the decimal approximation.

10.2. Invention of the game of chess. According to some legends, the inventor of the game of chess (who lived in India thousands of years ago) so pleased his ruler, that he was asked to chose his reward.

"I would be content with grains of wheat. Let one grain be placed on the first square of my chess board, and double that number on the second, double that on the third, and so on," said the inventor. The ruler gladly agreed.

A chessboard has \(8 \times 8\) squares. How many grains of wheat would be required for the last square on that board? Give your answer in decimal notation.

Note: in the original wheat and chessboard problem, we are asked to find the total number of wheat grains on all squares. This requires summing a geometric series, and is a problem ideal for early 2 nd term calculus.

10.3. Computing powers of 2. In order to produce the graph of the continuous function \(2^{x}\) in Figure 10.2, it was desirable to generate many points on that graph using simple calculations. Suppose you have an ordinary calculator with the operations \(+,-, \times, /\). You also know that \(\sqrt{2} \approx 1.414\).

How would you compute \(2^{x}\) for the values \(x=7 / 2, x=-1 / 2\), and \(x=-5\) ?

10.4. Exponential base requirement. Explain the requirement that \(a\) must be positive in the exponential function \(y=a^{x}\). What could go wrong if \(a\) was a negative base?

10.5. Derivative of \(3^{x}\). Find the derivative of \(y=3^{x}\). What is the value of the multiplicative constant \(C_{3}\) that shows up in your calculation?

10.6. Graphing functions. Graph the following functions:

(a) \(f(x)=x^{2} e^{-x}\)

(b) \(f(x)=\ln \left(e^{2 x}\right)\).

10.7. Changing bases. Express the following in terms of base \(e\) :

(a) \(y=3^{x}\)

(b) \(y=\frac{1}{7^{x}}\)

(c) \(y=15^{x^{2}+2}\)

Express the following in terms of base 2:

(d) \(y=9^{x}\)

(e) \(y=8^{x}\)

(f) \(y=-e^{x^{2}+3}\).

Express the following in terms of base 10 :

(g) \(y=21^{x}\)

(h) \(y=1000^{-10 x}\)

(i) \(y=50^{x^{2}-1}\)

10.8. Comparing numbers expressed using exponents. 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) \(0.999^{1.5}, 0.999^{2.3}\)

10.9. Logarithms. Rewrite each of the following equations in logarithmic form:

(a) \(3^{4}=81\)

(b) \(3^{-2}=\frac{1}{9}\)

(c) \(27^{-\frac{1}{3}}=\frac{1}{3}\).

10.10. Equations with logarithms. Solve the following equations for \(x\) :

(a) \(\ln x=2 \ln a+3 \ln b\)

(b) \(\log _{a} x=\log _{a} b-\frac{2}{3} \log _{a} c\)

10.11. 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) \(y=2 \cdot 3^{x}\)

(f) \(y=\log _{3} x\)

10.12. Equations with exponents and logarithms. Solve the following equations for \(x\) :

(a) \(e^{3-2 x}=5\)

(b) \(\ln (3 x-1)=4\)

(c) \(\ln (\ln (x))=2\)

(d) \(e^{a x}=C e^{b x}\), where \(a \neq b\) and \(C>0\).

10.13. Derivative of exponential and logarithmic functions. Find the first derivative for each of the following functions:

Formula: \(\frac{d}{d x}\left(\log _a x\right)=\frac{1}{x \ln a}\)

(a) \(y=\ln (2 x+3)^{3}\)

(b) \(y=\ln ^{3}(2 x+3)\)

(c) \(y=\ln \left(\cos \frac{1}{2} x\right)\)

(d) \(y=\log _{a}\left(x^{3}-2 x\right)\)

(e) \(y=e^{3 x^{2}}\)

(f) \(y=a^{-\frac{1}{2} x}\)

(g) \(y=x^{3} \cdot 2^{x}\)

(h) \(y=e^{e^{x}}\)

(i) \(y=\frac{e^{t}-e^{-t}}{e^{t}+e^{-t}}\)

10.14. Maximum, minimum and inflection points. Find the maximum and minimum points as well as all inflection points of the following functions:

(a) \(f(x)=x\left(x^{2}-4\right)\)

(b) \(f(x)=x^{3}-\ln (x), x>0\)

(c) \(f(x)=x e^{-x}\)

(d) \(f(x)=\frac{1}{1-x}+\frac{1}{1+x},-1<x<1\)

(e) \(f(x)=x-3 \sqrt[3]{x}\)

(f) \(f(x)=e^{-2 x}-e^{-x}\)

10.15. Using graph information. Shown in Figure 10.9 is the graph of \(y=C e^{k t}\) for some constants \(C, k\), and a tangent line. Use data from the graph to determine \(C\) and \(k\).

Figure 10.9: Figure for Exercise 10.15; \(y=C e^{k t}\) and a tangent line.

10.16. Comparing exponential functions. Consider the two functions

1. \(y_{1}(t)=10 e^{-0.1 t}\),
2. \(y_{2}(t)=10 e^{0.1 t}\).

Answer the following:

(a) Which one is decreasing and which one is increasing?

(b) In each case, find the value of the function at \(t=0\).

(c) Find the time at which the increasing function has doubled from this initial value.

(d) Find the time at which the decreasing function has fallen to half of its initial value.

Note: these values of \(t\) are called the doubling time, and half-life, respectively

10.17. Invasive species. An ecosystem with mature trees has a relatively constant population of beetles (species 1 ) - around \(10^{9}\). At \(t=0\), a single reproducing invasive beetle (species 2 ) is introduced accidentally.

Formula:

\[ \frac{d}{d x}\left(\log _{a} x\right)=\frac{1}{x \ln a}  \nonumber \]

If this population grows at the exponential rate

\[N_{2}(t)=e^{r t}, \quad \text { where } r=0.5 \text { per month, } \nonumber \]

how long does it take for species 2 to overtake the population of the resident species 1? Assume exponential growth for the entire duration.

10.18. Human population growth. It is sometimes said that the population of humans on Earth is growing exponentially. This means

\[P(t)=C e^{r t}, \quad \text { where } r>0 . \nonumber \]

We investigate this claim. To this end, we consider the human population beginning in year \(1800(t=0)\). Hence, we ask whether the data in Table \(10.5\) fits the relationship

\[P(t)=C e^{r(t-1800)}, \quad \text { where } t \text { is time in years and } r>0 ? \nonumber \]

Table 10.5: The human population (billions) over the years AD 1 to AD 2020.

year	human population (billions)
1	\(0.2\)
1000	\(0.275\)
1500	\(0.45\)
1650	\(0.5\)
1750	\(0.7\)
1804	1
1850	\(1.2\)
1900	\(1.6\)
1927	2
1950	\(2.55\)
1960	3
1980	\(4.5\)
1987	5
1999	6
2011	7
2020	\(7.7\)

(a) Show that the above relationship implies that \(\ln (P)\) is a linear function of time, and that \(r\) is the slope of the linear relationship (hint: take the natural logarithm of both sides of the relationship and simplify).

(b) Use the data from Table \(10.5\) for the years 1800 to 2020 to investigate whether \(P(t)\) fits an exponential relationship (hint: plot \(\ln (P)\), where \(P\) is human population (in billions) against time \(t\) in years we refer to this process as "transforming the data".

(c) A spreadsheet can be used to fit a straight line through the transformed data you produced in (b).

(i) Find the best fit for the growth rate parameter \(r\) using that option.

(ii) What are the units of \(r\) ?

(iii) What is the best fit value of \(C\) ?

(d) Based on your plot of \(\ln (P)\) versus \(t\) and the best fit values of \(r\) and \(C\), over what time interval was the population growing more slowly than the overall trend, and when was it growing more rapidly than this same overall trend?

(e) Under what circumstances could an exponentially growing population be sustainable?

10.19. A sum of exponentials. Researchers that investigated the molecular motor dynein found that the number of motors \(N(t)\) remaining attached to their microtubule tracks at time \(t\) (in sec) after a pulse of activation was well described by a double exponential of the form

\[N(t)=C_{1} e^{-r_{1} t}+C_{2} e^{-r_{2} t}, \quad t \geq 0 . \nonumber \]They found that \(r_{1}=0.1, r_{2}=0.01\) per second, and \(C_{1}=75, C_{2}=25\) percent.

(a) Plot this relationship for \(0<t<8\) min. Which of the two exponential terms governs the behavior over the first minute? Which dominates in the later phase?

(b) Now consider a plot of \(\ln (N(t))\) versus \(t\). Explain what you see and what the slopes and other aspects of the graph represent.

10.20. Exponential Peeling. The data in Table \(10.6\) is claimed to have been generated by a double exponential function of the form

\[N(t)=C_{1} e^{-r_{1} t}+C_{2} e^{-r_{2} t}, \quad t \geq 0 . \nonumber \]

Use the data to determine the values of the constants \(r_{1}, r_{2}, C_{1}\), and \(C_{2}\).

Table 10.6: Table for Exercise 20; data to be fit to a function of the form \(N(t)=\) \(C_{1} e^{-r_{1} t}+C_{2} e^{-r_{2} t}, \quad t \geq 0\).

time	\(N(t)\)
\(0.0000\)	\(100.0000\)
\(0.1000\)	\(57.6926\)
\(0.2000\)	\(42.5766\)
\(0.3000\)	\(35.8549\)
\(0.4000\)	\(31.8481\)
\(0.5000\)	\(28.8296\)
\(2.5000\)	\(4.7430\)
\(4.5000\)	\(0.7840\)
\(6.0000\)	\(0.2032\)
\(8.0000\)	\(0.0336\)

10.21. Shannon Entropy. In a recent application of information theory to the field of genomics, a function called the Shannon entropy, \(H\), was considered. In it, 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

\[H=-x \log (x)-y \log (y) \nonumber \]

Note that

• \(x\) and \(y\) being probabilities just means that they satisfy \(0<x \leq 1\), and \(0<y \leq 1\) and
• 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 \(x=y=0.5\).

10.22. 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 \(y\) of the gene:

\[y=f(x)=\frac{1}{1+e^{(a x+b)}}, \nonumber \]

where \(a, b\) are constants.

(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?

10.23. Graph sketching. Sketch the graph of the function \(y=e^{-t} \sin \pi t\).

10.24. The Mexican Hat. Consider the function

\[y=f(x)=2 e^{-x^{2}}-e^{-x^{2} / 3} \nonumber \]

(a) Find the critical points of \(f\).

(b) Determine the value of \(f\) at those critical points.

(c) Use these results and the fact that for very large \(x, f \rightarrow 0\) to draw a rough sketch of the graph of this function.

(d) Comment on why this function might be called "a Mexican Hat".

Note: The second derivative is not very informative here, and we do not ask you to use it for determining concavity in this example. However, you may wish to calculate it for practice with the chain rule.

10.25. 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 \(y\). The Ricker equation is

\[y=\alpha x e^{-\beta x} \nonumber \]

where \(\alpha, \beta>0\).

(a) Find the value of the current population which maximizes the salmon population next year according to this model.

(b) Find the value of the current population which would be exactly maintained in the next generation.

(c) Explain why a very large population is not sustainable.

Note: these populations do not actually change continuously, since all the parents die before the eggs are hatched.

10.26. Spacing in a fish school. Life in a social group has advantages and disadvantages: protection from predators is one advantage. Disadvantages include competition 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 neighbours from one another: each individual does not want to stray too far from others, nor get too close. 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:

\[\begin{gathered} F_{a}=A e^{-x / a} \\ F_{r}=R e^{-x / r} \end{gathered} \nonumber \]

where \(A, R, a, r\) are positive constants.

Note: \(A, R\) are related to the magnitudes of the forces, while \(a, r\) to the spatial range of these effects.

(a) Show that at distance \(x=a\), the first function has fallen to \((1 / e)\) times its value at the origin. (Recall \(e \approx 2.7\).)

(b) 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.

(c) It is generally assumed that \(R>A\) and \(r<a\). Interpret what this mean about the comparative effects of the forces.

(d) Sketch a graph showing the two functions on the same set of axes.

(e) Find the distance at which the forces exactly balance. This is called the comfortable distance for the two individuals.

(f) If either \(A\) or \(R\) changes so that the ratio \(R / A\) decreases, does the comfortable distance increase or decrease? Justify your response.

(g) Similarly comment on what happens to the comfortable distance if \(a\) increases or \(r\) decreases.

10.27. Seed distribution. The density of seeds at a distance \(x\) from a parent tree is observed to be

\[D(x)=D_{0} e^{-x^{2} / a^{2}} \nonumber \]

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

\[F(x)=e^{-x^{2} / b^{2}} \nonumber \]

where \(b>0\)

Note: 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

\[S(x)=D(x)(1-F(x)) \nonumber \]

Determine the distance \(x\) from the tree at which the greatest number of seeds survive.

10.28. Euler’s ’ \(e\) ’. In 1748 , Euler wrote a classic book on calculus, "Introductio in Analysin Infinitorum" [Euler, 1748] in which he showed that the function \(e^{x}\) could be written in an expanded form similar to an (infinitely long) polynomial:

\[e^{x}=1+x+\frac{x^{2}}{1 \cdot 2}+\frac{x^{3}}{1 \cdot 2 \cdot 3}+\ldots \nonumber \]

Use as many terms as necessary to find an approximate value for the number \(e\) and for \(1 / e\) to 5 decimal places.

Note: in other mathematics courses we see that such expansions, called power series, are central to approximations of many functions.