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2.8. Homework Problems

1. Carry out a calculation like the one in Example 9 on page 26 to show that the derivative of t4 equals 4t3.

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.

3. Suppose H is a big number. Experiment on a calculator to figure out whether \(\sqrt{H+1}-\sqrt{H-1}\) 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.

4. Suppose dx is a small but finite number. Experiment on a calculator to figure out how \(\sqrt{dx}\) 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.

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 xy, there exists another real number z such that x<z<y.
(e) For any real numbers xy, 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.

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/R1+1/R2). 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.

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.

8. Differentiate (2x+3)100 with respect to x.

9. Differentiate (x+1)100(x+2)200 with respect to x.

10. Differentiate the following with respect to x: e7x, eex. (In the latter expression, as in all exponentials nested inside exponentials, the evaluation proceeds from the top down, i.e., e(ex), not (ee)x.)

11. Differentiate asin(bx+c) with respect to x.

12. Let x=tp/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 tk holds when k=p/q.

13. Find a function whose derivative with respect to x equals asin(bx+c). That is, find an integral of the given function.

14. Use the chain rule to differentiate ((x2)2)2, and show that you get the same result you would have obtained by differentiating x8. [M. Livshits]

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

\begin{align} R=\frac{2v^2}{g}\sin\theta\:\cos\theta . \end{align}

Find the angle that will produce the maximum range.

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

17. The hyperbolic cosine function is defined by

\begin{align} \cosh x = \frac{e^x+e^{-x}}{2} . \end{align}

Find any minima and maxima of this function.

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.

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.

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]

\[ d = A \ln\left[\cosh\left(t\sqrt{\frac{g}{A}}\right)\right] , \]

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=4.11m/d2. 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 \(\sqrt{g/A}\) .
(b) Show that your answer can be reexpressed in terms of the function tanh defined by tanh x = (ex-e-x)/(ex+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.

[8] Jan Benacka and Igor Stubna, The Physics Teacher, 43 (2005) 432.

21. Differentiate tanθ with respect to θ.

22. Differentiate \(\sqrt[3]{x}\) with respect to x.

23. Differentiate the following with respect to x:
(a) \(y=\sqrt{x^2+1}\)
(b) \(y=\sqrt{x^2+a^2}\)
(c) \(y=1/\sqrt{a+x}\)
(d) \(y=a/\sqrt{a-x^2}\)
[Thompson, 1919]

24. Differentiate ln(2t+1) with respect to t.

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.

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 \(\sqrt{\pi}\). 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.)

27. For a cylinder of fixed surface area, what proportion of length to radius will give the maximum volume?

28. This problem is a variation on problem 11 on page 21. Einstein found that the equation K=(1/2)mv2 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

\[ K = \frac{\frac{1}{2}mv^2}{\sqrt{1-v^2/c^2}} . \]

(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.

29. Prove, as claimed on page 42, that the derivative of ln |x| equals 1/x, for both positive and negative x.

30. On even function is one with the property f(-x) = f(x). For example, cos x is an even function, and xn 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.

31. Suppose we have a list of numbers x1,… xn, and we wish to find some number q that is as close as possible to as many of the xi as possible. To make this a mathematically precise goal, we need to define some numerical measure of this closeness. Suppose we let h = (x1-q)2+…+(xn-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 xi.

32. Use a trick similar to the one used in Example 16 to prove that the power rule d(xk)/dx = kxk-1 applies to cases where k is an integer less than 0.

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?

34. Discuss the following statement: The repeating decimal 0.999… is infinitesimally less than one.

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.

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

\[ s = \frac{1}{\sqrt{a(f^2-f_\text{o}^2)^2+bf^2}} , \]

where a, b, and fo 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, fo.

resonance

Figure M. The function of problem 36, with a=3, b=1, and fo=1.

hw-near-focal-point

Figure N. Problem 37. A set of light rays is emitted from the tip of the glamorous movie star's nose on the film, and reunited to form a spot on the screen which is the image of the same point on his nose. The distances have been distorted for clarity. The distance y represents the entire length of the theater from front to back.

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

\[ \frac{1}{x}+\frac{1}{y} = \frac{1}{f} , \]

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 xf. 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.

38. Why might the expression 1 be considered an indeterminate form?

 

[Solutions of these problems are in the pdf version of the book]