Exercises

PRELIMINARY CALCULUS
Find the volume of a cone enclosed by the surface formed by rotating about the x-axis
the line y = 2x between x = 0 and x = h.
Using (2.46), the volume is given by
h
π(2x)2 dx =
V =
0
=
4
3
πx3
h
0
h
4πx2 dx
0
= 43 π(h3 − 0) = 43 πh3 . As before, it is also possible to form a volume of revolution by rotating a curve
about the y-axis. In this case the volume enclosed between y = a and y = b is
b
πx2 dy.
(2.47)
V =
a
2.3 Exercises
2.1
Obtain the following derivatives from first principles:
(a) the first derivative of 3x + 4;
(b) the first, second and third derivatives of x2 + x;
(c) the first derivative of sin x.
2.2
2.3
Find from first principles the first derivative of (x + 3)2 and compare your answer
with that obtained using the chain rule.
Find the first derivatives of
(a) x2 exp x, (b) 2 sin x cos x, (c) sin 2x, (d) x sin ax,
(e) (exp ax)(sin ax) tan−1 ax, (f) ln(xa + x−a ),
(g) ln(ax + a−x ), (h) xx .
2.4
Find the first derivatives of
(a) x/(a + x)2 , (b) x/(1 − x)1/2 , (c) tan x, as sin x/ cos x,
(d) (3x2 + 2x + 1)/(8x2 − 4x + 2).
2.5
Use result (2.12) to find the first derivatives of
(a) (2x + 3)−3 , (b) sec2 x, (c) cosech3 3x, (d) 1/ ln x, (e) 1/[sin−1 (x/a)].
2.6
2.7
2.8
2.9
Show that the function y(x) = exp(−|x|) defined by


for x < 0,
exp x
y(x) = 1
for x = 0,

exp(−x) for x > 0,
is not differentiable at x = 0. Consider the limiting process for both ∆x > 0 and
∆x < 0.
Find dy/dx if x = (t − 2)/(t + 2) and y = 2t/(t + 1) for −∞ < t < ∞. Show that
it is always non-negative, and make use of this result in sketching the curve of y
as a function of x.
If 2y + sin y + 5 = x4 + 4x3 + 2π, show that dy/dx = 16 when x = 1.
Find the second derivative of y(x) = cos[(π/2) − ax]. Now set a = 1 and verify
that the result is the same as that obtained by first setting a = 1 and simplifying
y(x) before differentiating.
76
2.3 EXERCISES
2.10
The function y(x) is defined by y(x) = (1 + xm )n .
(a) Use the chain rule to show that the first derivative of y is nmxm−1 (1 + xm )n−1 .
(b) The binomial expansion (see section 1.5) of (1 + z)n is
(1 + z)n = 1 + nz +
n(n − 1) 2
n(n − 1) · · · (n − r + 1) r
z + ···+
z + ··· .
2!
r!
Keeping only the terms of zeroth and first order in dx, apply this result twice
to derive result (a) from first principles.
(c) Expand y in a series of powers of x before differentiating term by term.
Show that the result is the series obtained by expanding the answer given
for dy/dx in (a).
2.11
Show by differentiation and substitution that the differential equation
4x2
2.12
dy
d2 y
− 4x
+ (4x2 + 3)y = 0
dx2
dx
has a solution of the form y(x) = xn sin x, and find the value of n.
Find the positions and natures of the stationary points of the following functions:
(a) x3 − 3x + 3; (b) x3 − 3x2 + 3x; (c) x3 + 3x + 3;
(d) sin ax with a = 0; (e) x5 + x3 ; (f) x5 − x3 .
2.13
2.14
Show that the lowest value taken by the function 3x4 + 4x3 − 12x2 + 6 is −26.
By finding their stationary points and examining their general forms, determine
the range of values that each of the following functions y(x) can take. In each
case make a sketch-graph incorporating the features you have identified.
(a) y(x) = (x − 1)/(x2 + 2x + 6).
(b) y(x) = 1/(4 + 3x − x2 ).
(c) y(x) = (8 sin x)/(15 + 8 tan2 x).
2.15
2.16
Show √
that y(x) = xa√2x exp x2 has no stationary points other than x = 0, if
exp(− 2) < a < exp( 2).
The curve 4y 3 = a2 (x + 3y) can be parameterised as x = a cos 3θ, y = a cos θ.
(a) Obtain expressions for dy/dx (i) by implicit differentiation and (ii) in parameterised form. Verify that they are equivalent.
(b) Show that the only point of inflection occurs at the origin. Is it a stationary
point of inflection?
(c) Use the information gained in (a) and (b) to sketch the curve, paying
particular attention to its shape near the points (−a, a/2) and (a, −a/2) and
to its slope at the ‘end points’ (a, a) and (−a, −a).
2.17
The parametric equations for the motion of a charged particle released from rest
in electric and magnetic fields at right angles to each other take the forms
x = a(θ − sin θ),
2.18
2.19
y = a(1 − cos θ).
Show that the tangent to the curve has slope cot(θ/2). Use this result at a few
calculated values of x and y to sketch the form of the particle’s trajectory.
Show that the maximum curvature on the catenary y(x) = a cosh(x/a) is 1/a. You
will need some of the results about hyperbolic functions stated in subsection 3.7.6.
The curve whose equation is x2/3 + y 2/3 = a2/3 for positive x and y and which
is completed by its symmetric reflections in both axes is known as an astroid.
Sketch it and show that its radius of curvature in the first quadrant is 3(axy)1/3 .
77
PRELIMINARY CALCULUS
C
ρ
c
ρ
r + ∆r
O
Q
r
p + ∆p
P
p
Figure 2.13 The coordinate system described in exercise 2.20.
2.20
2.21
A two-dimensional coordinate system useful for orbit problems is the tangentialpolar coordinate system (figure 2.13). In this system a curve is defined by r, the
distance from a fixed point O to a general point P of the curve, and p, the
perpendicular distance from O to the tangent to the curve at P . By proceeding
as indicated below, show that the radius of curvature, ρ, at P can be written in
the form ρ = r dr/dp.
Consider two neighbouring points, P and Q, on the curve. The normals to the
curve through those points meet at C, with (in the limit Q → P ) CP = CQ = ρ.
Apply the cosine rule to triangles OP C and OQC to obtain two expressions for
c2 , one in terms of r and p and the other in terms of r + ∆r and p + ∆p. By
equating them and letting Q → P deduce the stated result.
Use Leibnitz’ theorem to find
(a) the second derivative of cos x sin 2x,
(b) the third derivative of sin x ln x,
(c) the fourth derivative of (2x3 + 3x2 + x + 2) exp 2x.
2.22
If y = exp(−x2 ), show that dy/dx = −2xy and hence, by applying Leibnitz’
theorem, prove that for n ≥ 1
y (n+1) + 2xy (n) + 2ny (n−1) = 0.
2.23
Use the properties of functions at their turning points to do the following:
(a) By considering its properties near x = 1, show that f(x) = 5x4 − 11x3 +
26x2 − 44x + 24 takes negative values for some range of x.
(b) Show that f(x) = tan x − x cannot be negative for 0 ≤ x < π/2, and deduce
that g(x) = x−1 sin x decreases monotonically in the same range.
2.24
2.25
Determine what can be learned from applying Rolle’s theorem to the following
functions f(x): (a) ex ; (b) x2 + 6x; (c) 2x2 + 3x + 1; (d) 2x2 + 3x + 2; (e)
2x3 − 21x2 + 60x + k. (f) If k = −45 in (e), show that x = 3 is one root of
f(x) = 0, find the other roots, and verify that the conclusions from (e) are
satisfied.
By applying Rolle’s theorem to xn sin nx, where n is an arbitrary positive integer,
show that tan nx + x = 0 has a solution α1 with 0 < α1 < π/n. Apply the
theorem a second time to obtain the nonsensical result that there is a real α2 in
0 < α2 < π/n, such that cos2 (nα2 ) = −n. Explain why this incorrect result arises.
78
2.3 EXERCISES
2.26
Use the mean value theorem to establish bounds in the following cases.
(a) For − ln(1 − y), by considering ln x in the range 0 < 1 − y < x < 1.
(b) For ey − 1, by considering ex − 1 in the range 0 < x < y.
2.27
For the function y(x) = x2 exp(−x) obtain a simple relationship between y and
dy/dx and then, by applying Leibnitz’ theorem, prove that
xy (n+1) + (n + x − 2)y (n) + ny (n−1) = 0.
2.28
Use Rolle’s theorem to deduce that, if the equation f(x) = 0 has a repeated root
x1 , then x1 is also a root of the equation f (x) = 0.
(a) Apply this result to the ‘standard’ quadratic equation ax2 + bx + c = 0, to
show that a necessary condition for equal roots is b2 = 4ac.
(b) Find all the roots of f(x) = x3 + 4x2 − 3x − 18 = 0, given that one of them
is a repeated root.
(c) The equation f(x) = x4 + 4x3 + 7x2 + 6x + 2 = 0 has a repeated integer root.
How many real roots does it have altogether?
2.29
2.30
2.31
Show that the curve x3 + y 3 − 12x − 8y − 16 = 0 touches the x-axis.
Find the following indefinite integrals:
(a) (4 + x2 )−1 dx; (b) (8 + 2x − x2 )−1/2 dx for 2 ≤ x ≤ 4;
√
(c) (1 + sin θ)−1 dθ; (d) (x 1 − x)−1 dx for 0 < x ≤ 1.
Find the indefinite integrals J of the following ratios of polynomials:
(a)
(b)
(c)
(d)
(x + 3)/(x2 + x − 2);
(x3 + 5x2 + 8x + 12)/(2x2 + 10x + 12);
(3x2 + 20x + 28)/(x2 + 6x + 9);
x3 /(a8 + x8 ).
2.32
Express x2 (ax + b)−1 as the sum of powers of x and another integrable term, and
hence evaluate
b/a
x2
dx.
ax
+b
0
2.33
Find the integral J of (ax2 + bx + c)−1 , with a = 0, distinguishing between the
cases (i) b2 > 4ac, (ii) b2 < 4ac and (iii) b2 = 4ac.
Use logarithmic integration to find the indefinite integrals J of the following:
2.34
(a)
(b)
(c)
(d)
2.35
2.36
Find the derivative of f(x) = (1 + sin x)/ cos x and hence determine the indefinite
integral J of sec x.
Find the indefinite integrals, J, of the following functions involving sinusoids:
(a)
(b)
(c)
(d)
2.37
sin 2x/(1 + 4 sin2 x);
ex /(ex − e−x );
(1 + x ln x)/(x ln x);
[x(xn + an )]−1 .
cos5 x − cos3 x;
(1 − cos x)/(1 + cos x);
cos x sin x/(1 + cos x);
sec2 x/(1 − tan2 x).
By making the substitution x = a cos2 θ + b sin2 θ, evaluate the definite integrals
J between limits a and b (> a) of the following functions:
(a) [(x − a)(b − x)]−1/2 ;
(b) [(x − a)(b − x)]1/2 ;
79
PRELIMINARY CALCULUS
(c) [(x − a)/(b − x)]1/2 .
2.38
2.39
Determine whether the following integrals exist and, where they do, evaluate
them: ∞
∞
x
(a)
exp(−λx) dx; (b)
dx;
2 + a2 )2
(x
0
−∞
1
∞
1
1
dx;
dx;
(d)
(c)
2
01 x
1 π/2x + 1
x
cot θ dθ;
(f)
dx.
(e)
2 1/2
0
0 (1 − x )
Use integration
by parts to evaluate
the following:
y
y
x2 sin x dx;
(a)
0y
(c)
x ln x dx;
(b)
1 y
sin−1 x dx;
2.40
ln(a2 + x2 )/x2 dx.
(d)
0
1
Show, using the following methods, that the indefinite integral of x3 /(x + 1)1/2 is
J=
2
(5x3
35
− 6x2 + 8x − 16)(x + 1)1/2 + c.
(a) Repeated integration by parts.
(b) Setting x + 1 = u2 and determining dJ/du as (dJ/dx)(dx/du).
2.41
The gamma function Γ(n) is defined for all n > −1 by
∞
xn e−x dx.
Γ(n + 1) =
0
Find a recurrence relation connecting Γ(n + 1) and Γ(n).
(a) Deduce (i) thevalue
of Γ(n + 1) when
√n is a non-negative integer, and (ii)
the value of Γ 72 , given that Γ 12 = π.
(b) Now,
3 taking factorial m for any m to be defined by m! = Γ(m + 1), evaluate
− 2 !.
2.42
Define J(m, n), for non-negative integers m and n, by the integral
π/2
cosm θ sinn θ dθ.
J(m, n) =
0
(a) Evaluate J(0, 0), J(0, 1), J(1, 0), J(1, 1), J(m, 1), J(1, n).
(b) Using integration by parts, prove that, for m and n both > 1,
J(m, n) =
m−1
n−1
J(m − 2, n) and J(m, n) =
J(m, n − 2).
m+n
m+n
(c) Evaluate (i) J(5, 3), (ii) J(6, 5) and (iii) J(4, 8).
2.43
By integrating by parts twice, prove that In as defined in the first equality below
for positive integers n has the value given in the second equality:
π/2
n − sin(nπ/2)
sin nθ cos θ dθ =
In =
.
n2 − 1
0
2.44
Evaluate the following definite integrals:
1
∞
(a) 0 xe−x dx; (b) 0 (x3 + 1)/(x4 + 4x + 1) dx;
π/2
∞
(c) 0 [a + (a − 1) cos θ]−1 dθ with a > 12 ; (d) −∞ (x2 + 6x + 18)−1 dx.
80