REAL NUMBERS AND LAWS OF ALGEBRA
1.62. Prove that (a) v% (&) V3, (c) V2 + A/3 are irrational numbers.
1.63. Arrange in order of increasing value: 11/9, \fí, VÏÏ ^/5/^/Z fy(N.
1.64. Explain why (a) 1/0, (6) 0/0 cannot represent unique numbers.
1.65. Explain why we must define (a) (-5)(3) = -15, (6) (-2)(-3) = 6 if the rules of algebra on page 1 are to hold.
FUNCTIONS
1.66. If /(») = (8«« + 2<B-6)/(«-4), find (o)/(2), (6) /(-I), («) /(3/2), (d) f(-x), (e)/(\/2).
1.67. An odd function is one for which /(—a;) = — /(«) while an ei>ew function is one for which /(—a;) = /(a;). Classify each of the following according as they are even or odd: (a) cos 2a;, (6) sin 3ic, (c) tan x, (d) ex, (e) ex — e~x, (/) ex + e~x.
1.68. (a) Prove that eolnl) = &«. (6) Find e-2ta«.
1.69. Prove that (a) sin 3* = 3 sin * - 4 sin3 x, (b) cos Sx - 4 cos3 x - 3 cos x.
1.70. Prove that (a) coth2 x - csch2 x = 1, (6) cosh2 x + sinh2 * = cosh 2x, (c) 1.71. If cos x = 8/17, find (a) sin 2», (6) cos 2a, (c) sin (a/2).
1.72. Find tanh (In 3).
LIMITS AND CONTINUITY
1.73. Find (a) (b) (c) and prove your conclusions.
1.74. If prove that
1.75. If and prove that (a) (e)
(b) (d)
1.76. If f(x) = \x\/x, prove that lim f(x) does not exist.
z-»0
1.77. Using the definition prove that f(x) = 3 — x2 is continuous at (a) x = 1, (b) x = a.
1.78. If f(x) is continuous at a, prove that [/(œ)]2 is also continuous at a.
1.79. If f1(x) and fz(x) are continuous at a prove that (a) f1(x) + f2(x), (b) fi(x) —f2(x), (c) f1(x) f2(x), (d) fi(x)/f2(x), fz(a)¥=0 are also continuous at a. Revise the statement if a is replaced by an interval such as a = x = b or a < x < b.
1.80. If prove that f(x) is continuous (a) at x = 0, (6) in any interval.
1.81. Prove the statement at the end of Problem 1.12(6).
DERIVATIVES
1.82. Using the definition, find the derivatives of (a) f(x) = x2 - 2x + 5 and (6) f ( x ) = (x — l)/(x + 1).
1.83. Prove the differentiation formulas (a) 2, (b) 4 on page 4.
1.84. Assuming that prove the formulas (a) 12, (6) 13, (c) 14 on page 5.
1.85. Prove formulas (a) 15, (b) 16, (c) 18, (d) 19, (e) 20 on page 5.
1.86. Find (/)
1.87. If e*» + y2 - cos x, find dy/dx.
1.88. If x — a(e — sin e), y = a(l — cos e), find dy/dx.
1.89. If y = (Sx + !)/(! - 2x), find d*y/dx2 at * = 2.
1.90. If y = (cj sin x + c2 cos x)/^fx, show that xy" + 2y' + xy = 0 where clt c2 are any constants.
1.91. Find d2y/dx2 for the function defined in Problem 1.88.
1.92. Find the equation of the tangent line to the curve xy2 + y — Sx — l at the point (1,1).
1.93. Find the differentials of (a) y = x2 In x + Sx, (b) y = (2x - l)/(x + 2).
1.94. (a) Show that if x > 0 and A« is numerically small compared with x, then is approximately equal to (6) Use this result to find the approximate values o f a n d
•>
(c) By obtaining a similar result for V* + A*, find an approximate value for and
1.95. Prove the law of the mean, page 8, and illustrate by an example.
INTEGRALS
1.96. Find the integrals (a) (b) (e) (d) (e) (/) (g) (*) 1.97. Prove the integration formulas (a) 7, (b) 11, (c) 12, (d) 13, (e) 14 on page 6.
1.98. Find ( a (e)
1.99. Evaluate (a) (b) (c)
1.100. Prove the integration formulas (a) 15, (6) 16, (c) 17, (d) 18, (e) 19, (/) 20, (g) 21, (h) 22 on page 6.
1.101. Find
1.102. Find
(6) (d)
1.103. (a) Find the area bounded by y = 4x — x2 and the x axis.
(b) Find the area bounded by the curves y — x2 and y = x.
1.104. Using the trapezoidal rule, find the approximate value of by dividing the interval from 0 to 1 into (a) 5, (b) 10 equal parts, (c) Show that the integral has the exact value ir/4 and compare with the approximate values.
1.105. Prove the trapezoidal rule which gives the approximate value of
where h = (b — a)/n.
1.106. Assume that a function f(x) is approximated by the parabolic function c0 + Cj» + c2x2 in the interval a S x S a + 2h [see Fig. 1-10]. Show that we have approximately
The result is often called Simpson's rule.
1.107. (a) Show how Simpson's rule [Problem 1.106] can be used to find the approximate value of
and (6) use the result to find the approximate values of the integrals in Problems 1.28 and 1.104. Compare the accuracy of the results with those of the
trapezoidal rule. Fig. I.JQ
(b) (c) (d)
(a) (b)
(c) (d)
(e)
(a) (c) (e)
SEQUENCES AND SERIES
1.108. Find the limit of the sequence whose nth term is un = (n- l)/(2w +1), justifying your answer.
1.109. Show that the series converges and find its sum.
1.110. Prove that and (6) if then
1.111. Show that .181818 ... = .18 + .0018 + .000018 + • • • = 2/11.
1.112. Investigate t h e convergence o f t h e serie (e)
1.113. Prove that the series converges for
1.114. Prove (a) Theorem 1-3, (6) Theorem 1-4, (c) Theorem 1-5, page 6.
1.115. (a) Consider the series wt — u2 + u3 — M4 + • • • where uk > 0. Prove that if un+l£un and then the series converges. (6) Thus show that is convergent but not absolutely convergent.
1.116. Find the interval of convergence of (a) (b)
1.117. Find the interval of convergence of (a) (6) UNIFORM CONVERGENCE
1.118. Prove that the series *(1 — x) + x2(l — x) + xs(l — x) + • • • converges for — 1 < x a 1 but is not uniformly convergent in this interval. Is it uniformly convergent in —1/2 S x ë 1/2? Explain.
1.119. Investigate the uniform convergence of the series (a) in (6) for 1.120. Prove that a power series c0 + c^x — a) + c2(x — a)2 + • • • converges uniformly in any interval
which lies entirely within its interval of convergence. [Hint: Use the Weierstrass M test.]
1.121. (a) Prove that if Sn(x) converges uniformly to S(x) in o = x a 6, then
(6) Show that
and supply a possible explanation.
1.122. Prove Theorem 1-11, page 7.
1.123. (a) By using analogy with series, give a definition for uniform convergence and absolute conver- gence of the integral
(b) Prove a test for uniform [and absolute] convergence of the integral in (a) analogous to the Weierstrass M test for series.
(c) Prove theorems for integrals analogous to Theorems 1-9, 1-10 and 1-11 on page 7.
(b) (c) (d)
(f)
TAYLOR SERIES
1.124. Obtain the formal Taylor series (a) 1, (6) 3, (c) 4, (d) 5 given on page 8.
1.125. Expand sin * in a Taylor series about a = W6 and use the result to find sin 31° approximately.
Compare with the result from a table of sines.
1.126. Find an approximate value for (a)
1.127. Show that (a) sinh x = , (b) for PARTIAL DERIVATIVES
1.128. If f ( x , y ) = x sin (y/x), find (a) f(2,v/2), (b) fx(2,ir/2), (e) fy(2,w/2), (d) f^(2,v/2), (e) f^(2, v/2), (/) /Sz(2,T/2), (O) fm(2,v/2).
1.129. Verify that /„, = fyx for the case where /(*, y) = (x - y)/(x + y) if * + y ¥> 0.
1.130. If f ( x , y ) = xz tan-1 (y/x), show t x and 2/?
1.131. If V(x,y,z) = (*2 + 3/2 + «2)-1/2, show that if x,y,z are not all zero.
1.132. If f ( x , y ) = ycos(x-2y), find df if 1.133. If f(x,y,z) = x*z - yz^ + x*, find df.
1.134. If z = x2f(y/x) where / is differentiate, show t Problem 1.130.
1.135. If prove that if U is a twice differentiable function of * and y,
TAYLOR SERIES FOR FUNCTIONS OF TWO OR MORE VARIABLES
1.136. Expand f(x, y) = 2xy + »2 + ya in a Taylor series about a = 1, 6 = 2 and verify the expansion.
1.137. Write Taylor's series for f(x, y, z) expanded about x — a, y = b, z = c and illustrate by expanding f(x, y, z) = xz + y2 about x = 1, y = —1, 2 = 2.
LINEAR EQUATIONS AND DETERMINANTS
1.138. Solve the systems (a) (6) 1.139. Determine whether the system of equations
has non-trivial solutions. If so give two such solutions. If not replace the first equation by kx — y + z — 0 and determine fc so that the system has non-trivial solutions and give two such solutions.
1.140. Show that if two rows (or columns) of a second or third order determinant are interchanged, the sign of the determinant is changed.
1.141. Show that if the elements in two rows (or columns) of a second or third order determinant are equal or proportional, the value of the determinant is zero.
Are there any excweptional values for
nd compare with
1.142. Discuss the system of equations from a geometrical viewpoint.
MAXIMA AND MINIMA. THE METHOD OF LAGRANGE MULTIPLIERS.
1.143. Find the relative maxima and minima of f(x) = 12 + 8«2 — x* and use the results to graph the function.
1.144. Find the relative maxima and minima of (a) x2 In x, (b) a sec 6 + b esc e.
1.145. Find the area of the largest rectangle which can be inscribed in a semicircle of radius a with base on its diameter.
1.146. If /'(a) = 0 and /"(a) = 0, is it possible that f(x) has a maximum or minimum at x = a?
Explain. [Hint: Consider f(x) = x* or f(x) = —x*.]
1.147. A box having only five rectangular sides is required to have a given volume V. Determine its dimensions so that the surface area will be a minimum.
1.148. Find the shortest and largest distance from the origin to the curve x2 + xy + y2 = 16 and give a geometric interpretation. [Hint: Use the method of Lagrange multipliers to find the maximum of
«2 + y2.]
1.149. (a) Find the relative maximum and minimum of x2 + y2 + z2, given that 4a;2 + 9y2 + 16z2 = 576 and (6) interpret geometrically.
1.150. Generalize the method of Lagrange multipliers to the case where /(*, y, z) is to be made an extremum [i.e. maximum or minimum] subject to two constraint conditions <t>i(x, y, z) — 0, <f>2(x, 2/>z) = 0.
[Hint: Let h(x, y, z) = f ( x , y, z) + Xj^z, y, z) + X202(&> V>z) and prove that dh/dx — 0, dh/dy = 0, dh/dz — 0 where \i, X2 are two Lagrange multipliers.]
LEIBNITZ'S RULE 1.151. I f
rule.
1.152. Find dl/da if (a) (b)
1.153. Prove Leibnitz's rule. [Hint: C o n s i d e r a n d write the result as the sum of three integrals. Then let
COMPLEX NUMBERS. POLAR FORM
1.154.Perform each of the indicated operations: (a) 2(5 - 3i) - 3(-2 + i) + 5(i - 3), (b) (3 - 2i)3, (c) (/)
1.155. If Zj and z2 are complex numbers, prove (a) (6) giving any restrictions.
1.156. Prove (a) (b) (c) 1.157. Find all solutions of 2x* - 3x3 - 7a;2 - 8x + 6 - 0.
1.158. Let zl and z2 be represented by points Pl and P2 in the Argand diagram. Construct lines OP1 and OP2, where O is the origin. Show that zl + z2 can be represented by the point P3, where OP3 is the diagonal of a parallelogram having sides OPj and OP2. This is called the parallelogram law of addition of complex numbers. Because of this and other properties, complex numbers can be con- sidered as vectors in two dimensions. The general subject of vectors is given in Chapter 5.
1.159. Interpret geometrically the inequalities of Problem 1.156.
d dl/da by (a) integrating first and (b) using Leibnitz's
1.160. Express in polar form (o)3A/3 + 3¿, (b) -2-2i, (c) (d) 5, (e) -5i.
1.161. Evaluate (a) (6) 1.162. Determine all the indicated roots and represent them graphically:
(a) (b) (c)
1.163. If and prove (a) : Interpret geometrically.
1.164. Show that De Moivre's theorem is equivalent to (e**)n = ein<*.