# Untitled Page 13

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## Homework Problems

**1**.
Carry out a calculation like the one in example 9 on page 26 to
show that the derivative of *t*^{4} equals 4*t*^{3}.
(solution in the pdf version of the book)

**2**.
Example 12 on page 29 gave a tricky argument to show that the derivative of
cos *t* is -sin *t*. Prove the same result using the method of example 11 instead.
(solution in the pdf version of the book)

**3**.
Suppose *H* is a big number. Experiment on a calculator to figure out whether √H+1-√H-1
comes out big, normal, or tiny. Try making *H* bigger and bigger, and see if you observe a trend.
Based on these numerical examples, form a conjecture about what happens to this expression when *H* is infinite.
(solution in the pdf version of the book)

**4**.
Suppose *dx* is a small but finite number. Experiment on a calculator to figure out how √dx
compares in size to *dx*. Try making *dx* smaller and smaller, and see if you observe a trend.
Based on these numerical examples, form a conjecture about what happens to this expression when *dx* is
infinitesimal.
(solution in the pdf version of the book)

**5**.
To which of the following statements can the transfer principle be applied? If you think it can't be applied
to a certain statement, try to prove that the statement is false for the hyperreals, e.g., by giving a
counterexample.

(a) For any real numbers *x* and *y*, *x*+*y*=*y*+*x*.

(b) The sine of any real number is between -1 and 1.

(c) For any real number *x*, there exists another real number *y* that is greater than *x*.

(d) For any real numbers *x*≠ *y*, there exists another real number *z* such that *x*<z<y.

(e) For any real numbers *x*≠ *y*, there exists a rational number *z* such that *x*<z<y. (A rational number is
one that can be expressed as an integer divided by another integer.)

(f) For any real numbers *x*, *y*, and *z*, (*x*+*y*)+*z*=*x*+(*y*+*z*).

(g) For any real numbers *x* and *y*, either *x*<y or *x*=*y* or *x*>y.

(h) For any real number *x*, *x*+1≠ *x*.
(solution in the pdf version of the book)

**6**.
If we want to pump air or water through a pipe, common sense tells us that it will be easier
to move a larger quantity more quickly through a fatter pipe. Quantitatively, we can define
the resistance, *R*, which is the ratio of the pressure difference produced by the pump to the
rate of flow. A fatter pipe will have a lower resistance. Two pipes can be used in parallel,
for instance when you turn on the water both in the kitchen and in the bathroom, and in this
situation, the two pipes let more water flow than either would have let flow by itself, which
tells us that they act like a single pipe with some lower resistance. The equation for their
combined resistance is *R*=1/(1/*R*_{1}+1/*R*_{2}). Analyze the case where one resistance is finite,
and the other infinite, and give a physical interpretation. Likewise, discuss the case where
one is finite, but the other is infinitesimal.
(solution in the pdf version of the book)

**7**.
Naively, we would imagine that if a spaceship traveling at *u*=3/4 of the speed of light was to shoot a missile
in the forward direction at *v*=3/4 of the speed of light (relative to the ship), then the missile would be traveling at *u*+*v*=3/2 of the
speed of light. However, Einstein's theory of relativity tells us that this is too good to be true, because
nothing can go faster than light. In fact, the relativistic equation for combining velocities in this way
is not *u*+*v*, but rather (*u*+*v*)/(1+*uv*). In ordinary, everyday life, we never travel at speeds anywhere near
the speed of light. Show that the nonrelativistic result is recovered in the case where both *u* and *v* are
infinitesimal.
(solution in the pdf version of the book)

**8**.
Differentiate (2*x*+3)^{100} with respect to *x*.
(solution in the pdf version of the book)

**9**.
Differentiate (*x*+1)^{100}(*x*+2)^{200} with respect to *x*.
(solution in the pdf version of the book)

**10**.
Differentiate the following with respect to *x*: *e*^{7x}, *e*^{ex}. (In the latter expression,
as in all exponentials nested inside exponentials, the evaluation proceeds from the top down, i.e.,
*e*^{(ex)}, not (*e*^{e})^{x}.)
(solution in the pdf version of the book)

**11**.
Differentiate *a*sin(*bx*+*c*) with respect to *x*.
(solution in the pdf version of the book)

**12**.
Let *x*=*t*^{p/q}, where *p* and *q* are positive integers. By a technique similar to the one
in example 21 on p. 38,
prove that the differentiation rule for *t*^{k} holds when *k*=*p*/*q*.qwe
(solution in the pdf version of the book)

**13**.
Find a function whose derivative with respect to *x* equals *a*sin(*bx*+*c*). That is, find an
integral of the given function.
(solution in the pdf version of the book)

**14**.
Use the chain rule to differentiate ((*x*^{2})^{2})^{2}, and show that you get the
same result you would have obtained by differentiating *x*^{8}.
[M. Livshits]
(solution in the pdf version of the book)

**15**.
The range of a gun, when elevated to an angle θ, is given by

Find the angle that will produce the maximum range. (solution in the pdf version of the book)

**16**.
Differentiate sin cos tan *x* with respect to *x*.

**17**.
The hyperbolic cosine function is defined by

Find any minima and maxima of this function. (solution in the pdf version of the book)

**18**.
Show that the function sin(sin(sin *x*)) has maxima and minima at all the same places
where sin *x* does, and at no other places.
(solution in the pdf version of the book)

**19**.
Let *f*(*x*)=|*x*|+*x* and *g*(*x*)=*x*|*x*|+*x*. Find the derivatives of these functions at *x*=0 in terms of
(a) slopes of tangent lines and (b) infinitesimals.
(solution in the pdf version of the book)

**20**.
In free fall, the acceleration will not be exactly constant, due to air resistance. For example,
a skydiver does not speed up indefinitely until opening her chute, but rather approaches a certain
maximum velocity at which the upward force of air resistance cancels out the force of gravity.
The expression for the distance dropped by of a free-falling object, with air resistance, is^{8}

where *g* is the acceleration the object would have without air resistance, the function cosh
has been defined in problem 17, and
*A* is a constant that depends on the size, shape, and mass of the object, and the density of
the air. (For a sphere of mass *m* and diameter *d* dropping in air, *A*=4.11*m*/*d*^{2}.
Cf. problem 10, p. 115.)

(a) Differentiate this expression to find the velocity. Hint: In order to simplify the writing,
start by defining some other symbol to stand for the constant √g/A.

(b) Show that your answer can be reexpressed in terms of the function tanh defined by
tanh *x*=(*e*^{x}-*e*^{-x})/(*e*^{x}+*e*^{-x}).

(c) Show that your result for the velocity approaches
a constant for large values of *t*.

(d) Check that your answers to parts b and c have units of velocity.
(solution in the pdf version of the book)

**21**.
Differentiate tanθ with respect to θ.
(solution in the pdf version of the book)

**22**.
Differentiate √[3]{x} with respect to *x*.
(solution in the pdf version of the book)

**23**.
Differentiate the following with respect to *x*:

(a) y=√x^{2}+1

(b) y=√x^{2}+a^{2}

(c) y=1/√a+x

(d) y=a/√a-x^{2}

[Thompson, 1919]
(solution in the pdf version of the book)

**24**.
Differentiate ln(2*t*+1) with respect to *t*.
(solution in the pdf version of the book)

**25**.
If you know the derivative of sin *x*, it's not necessary to use the product rule in order
to differentiate 3sin *x*, but show that using the product rule gives the right result anyway.
(solution in the pdf version of the book)

**26**.
The Γ function (capital Greek letter gamma) is a continuous mathematical function that
has the property Γ(*n*)=1⋅2⋅…⋅(*n*-1) for *n* an integer. Γ(*x*) is also well
defined for values of *x* that are not integers, e.g., Γ(1/2) happens to be √π.
Use computer software that is capable of evaluating the Γ function to determine numerically
the derivative of Γ(*x*) with respect to *x*, at *x*=2. (In Yacas, the function is called Gamma.)
(solution in the pdf version of the book)

**27**.
For a cylinder of fixed surface area, what proportion of length to radius will give the maximum volume?
(solution in the pdf version of the book)

**28**.
This problem is a variation on problem 11 on page 21. Einstein found that the
equation *K*=(1/2)*mv*^{2} for kinetic energy was only a good approximation for speeds much less than
the speed of light, *c*. At speeds comparable to the speed of light, the correct equation is

(a) As in the earlier, simpler problem, find the power *dK*/*dt* for an object accelerating
at a steady rate, with *v*=*at*.

(b) Check that your answer has the right units.

(c) Verify that the power required becomes infinite in the limit as *v* approaches *c*, the speed of
light. This means that no material object can go as fast as the speed of light.
(solution in the pdf version of the book)

**29**.
Prove, as claimed on page 42,
that the derivative of ln |*x*| equals 1/*x*, for both positive and negative *x*.
(solution in the pdf version of the book)

**30**.
On even function is one with the property *f*(-*x*)=*f*(*x*). For example, cos *x* is an even function, and *x*^{n} is an even function if
*n* is even. An odd function has *f*(-*x*)=-*f*(*x*). Prove that the derivative of an even function is odd.
(solution in the pdf version of the book)

**31**.
Suppose we have a list of numbers *x*_{1},… *x*_{n}, and we wish to find some number *q* that is as close as possible
to as many of the *x*_{i} as possible. To make this a mathematically precise goal, we need to define some numerical
measure of this closeness. Suppose we let *h*=(*x*_{1}-*q*)^{2}+…+(*x*_{n}-*q*)^{2}, which can also be notated using Σ, uppercase Greek
sigma, as h=\sum_{i=1}^{n} (x_{i}-q)^{2}. Then minimizing *h* can be used as a definition of optimal closeness.
(Why would we not want to use h=\sum_{i=1}^{n} (x_{i}-q)?) Prove that the value of *q* that minimizes *h* is
the average of the *x*_{i}.

**32**.
Use a trick similar to the one used in example 16 to prove that the power
rule d(*x*^{k})/*dx*=*kx*^{k-1} applies to cases where *k* is an integer less than 0.
(solution in the pdf version of the book)

**33**.
The plane of Euclidean geometry is today often described as the set of all coordinate pairs (*x*,*y*), where
*x* and *y* are real. We could instead imagine the plane F that is defined in the same way, but with *x* and *y* taken from the
set of hyperreal numbers. As a third alternative, there is the plane G in which the finite hyperreals are used.
In E, Euclid's parallel postulate holds: given a line and a point not on the line, there exists exactly one line passing
through the point that does not intersect the line. Does the parallel postulate hold in F? In G?
Is it valid to associate only E with the plane described by Euclid's axioms?
(solution in the pdf version of the book)

**34**.
Discuss the following statement: *The repeating decimal 0.999… is infinitesimally less than one*.
(solution in the pdf version of the book)

**35**.
Example 20 on page 38
expressed the chain rule without the Leibniz notation, writing a function *f* defined by
*f*(*x*)=*g*(*h*(*x*)). Suppose that you're trying to remember the rule, and two of the possibilities
that come to mind are *f*'(*x*)=*g*'(*h*(*x*)) and *f*'(*x*)=*g*'(*h*(*x*))*h*(*x*). Show that neither of these
can possibly be right, by considering the case where *x* has units. You may find it helpful
to convert both expressions back into the Leibniz notation.
(solution in the pdf version of the book)

**36**.
When you tune in a radio station using an old-fashioned rotating dial you don't have to be
exactly tuned in to the right frequency in order to get the station. If you did, the
tuning would be infinitely sensitive, and you'd never be able to receive any signal at
all! Instead, the tuning has a certain amount of “slop” intentionally designed into it.
The strength of the received signal *s* can be expressed in terms of the dial's setting
*f* by a function of the form

where *a*, *b*, and *f*_{o} are constants. This functional form is in fact very general, and
is encountered in many other physical contexts. The graph below shows the resulting bell-shaped
curve. Find the frequency *f* at which the maximum response occurs, and show that if *b* is small,
the maximum occurs close to, but not exactly at, *f*_{o}.
(solution in the pdf version of the book)

**37**.
In a movie theater, the image on the screen is formed by a lens in the projector, and originates from one of the frames on the strip of celluloid film (or,
in the newer digital projection systems, from a liquid crystal chip).
Let the distance from the film to the lens be *x*, and let the distance from the lens to the screen be *y*. The projectionist
needs to adjust *x* so that it is properly matched with *y*, or else the image will be out of focus. There is therefore
a fixed relationship between *x* and *y*, and this relationship is of the form

where *f* is a property of the lens, called its focal length. A stronger lens has a shorter focal length.
Since the theater is large, and the projector is relatively small, *x* is much less than *y*.
We can see from the equation that if *y* is sufficiently large, the left-hand side of the equation
is dominated by the 1/*x* term, and we have *x* ≈ *f*. Since the 1/*y* term doesn't completely
vanish, we must have *x* slightly greater than *f*, so that the 1/*x* term is slightly less than
1/*f*. Let *x*=*f*+*dx*, and approximate *dx* as being infinitesimally small.
Find a simple expression for *y* in terms of *f* and *dx*.
(solution in the pdf version of the book)

**38**.
Why might the expression 1^{∞} be considered an indeterminate form?
(solution in the pdf version of the book)

(c) 1998-2013 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version.

### Footnotes

*finite*sum obtained by leaving out the “...” has only higher powers of

*t*. This is taken up in more detail in ch. 7. Note that the series only gives the right answer for

*t*<1. E.g., for

*t*=1, it equals 1+1+1+…, which, if it means anything, clearly means something infinite.

*a*, there is no real number

*b*such that

*a*=0

*b*.” This means that we can't divide

*a*by 0 and get

*b*. Applying the transfer principle to this statement, we see that the same is true for the hyperreals: division by zero is undefined. However, we can divide a finite number by an infinitesimal, and get an infinite result, which is almost the same thing.

*x*shouldn't be broken up into two lines as shown in the listing.

*The Physics Teacher*, 43 (2005) 432.