CHALLENGE PROBLEMS
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CHALLENGE PROBLEMS:
CHAPTER 6
A Click here for answers.
Click here for solutions.
S
; 1. Three mathematics students have ordered a 14-inch pizza. Instead of slicing it in the traditional
way, they decide to slice it by parallel cuts, as shown in the figure. Being mathematics majors, they
are able to determine where to slice so that each gets the same amount of pizza. Where are the cuts
made?
1
dx.
x7 ⫺ x
The straightforward approach would be to start with partial fractions, but that would be brutal. Try
a substitution.
2. Evaluate y
1
3
7
3. Evaluate y (s
1 ⫺ x7 ⫺ s
1 ⫺ x 3 ) dx.
14 in
0
FIGURE FOR PROBLEM 1
4. A man initially standing at the point O walks along a pier pulling a rowboat by a rope of length L.
The man keeps the rope straight and taut. The path followed by the boat is a curve called a tractrix
and it has the property that the rope is always tangent to the curve (see the figure).
(a) Show that if the path followed by the boat is the graph of the function y 苷 f 共x兲, then
y
pier
f ⬘共x兲 苷
L
(x, y )
(b) Determine the function y 苷 f 共x兲.
(L, 0)
O
dy
⫺sL 2 ⫺ x 2
苷
dx
x
x
5. A function f is defined by
f 共x兲 苷
FIGURE FOR PROBLEM 4
y
␲
0
cos t cos共x ⫺ t兲 dt
0 艋 x 艋 2␲
Find the minimum value of f .
6. If n is a positive integer, prove that
y
1
0
共ln x兲n dx 苷 共⫺1兲n n!
7. Show that
y
1
0
共1 ⫺ x 2 兲n dx 苷
2 2n 共n!兲2
共2n ⫹ 1兲!
Hint: Start by showing that if In denotes the integral, then
Ik⫹1 苷
2k ⫹ 2
Ik
2k ⫹ 3
; 8. Suppose that f is a positive function such that f ⬘ is continuous.
(a) How is the graph of y 苷 f 共x兲 sin nx related to the graph of y 苷 f 共x兲? What happens
as n l ⬁?
(b) Make a guess as to the value of the limit
lim
Thomson Brooks-Cole copyright 2007
nl⬁
y
1
0
f 共x兲 sin nx dx
based on graphs of the integrand.
(c) Using integration by parts, confirm the guess that you made in part (b). [Use the fact that, since
f ⬘ is continuous, there is a constant M such that f ⬘共x兲 艋 M for 0 艋 x 艋 1.]
9. If 0 ⬍ a ⬍ b, find lim
tl0
再y
1
0
冎
ⱍ
1兾t
关bx ⫹ a共1 ⫺ x兲兴 t dx
.
ⱍ
2
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CHALLENGE PROBLEMS
t⫹1
x
; 10. Graph f 共x兲 苷 sin共e 兲 and use the graph to estimate the value of t such that xt f 共x兲 dx is a maxi-
mum. Then find the exact value of t that maximizes this integral.
ⱍ ⱍ
11. The circle with radius 1 shown in the figure touches the curve y 苷 2x twice. Find the area of the
y
region that lies between the two curves.
12. A rocket is fired straight up, burning fuel at the constant rate of b kilograms per second. Let
v 苷 v共t兲 be the velocity of the rocket at time t and suppose that the velocity u of the exhaust gas is
constant. Let M 苷 M共t兲 be the mass of the rocket at time t and note that M decreases as the fuel
burns. If we neglect air resistance, it follows from Newton’s Second Law that
y=| 2x |
0
FIGURE FOR PROBLEM 11
F苷M
x
dv
⫺ ub
dt
where the force F 苷 ⫺Mt. Thus
1
M
dv
⫺ ub 苷 ⫺Mt
dt
Let M1 be the mass of the rocket without fuel, M2 the initial mass of the fuel, and M0 苷 M1 ⫹ M2 .
Then, until the fuel runs out at time t 苷 M2 b, the mass is M 苷 M0 ⫺ bt.
(a) Substitute M 苷 M0 ⫺ bt into Equation 1 and solve the resulting equation for v. Use the initial
condition v 共0兲 苷 0 to evaluate the constant.
(b) Determine the velocity of the rocket at time t 苷 M2 兾b. This is called the burnout velocity.
(c) Determine the height of the rocket y 苷 y共t兲 at the burnout time.
(d) Find the height of the rocket at any time t.
13. Use integration by parts to show that, for all x ⬎ 0,
0⬍y
⬁
0
sin t
2
dt ⬍
ln共1 ⫹ x ⫹ t兲
ln共1 ⫹ x兲
; 14. The Chebyshev polynomials Tn are defined by
Tn共x兲 苷 cos共n arccos x兲
n 苷 0, 1, 2, 3, . . .
(a) What are the domain and range of these functions?
(b) We know that T0共x兲 苷 1 and T1共x兲 苷 x. Express T2 explicitly as a quadratic polynomial and T3
as a cubic polynomial.
(c) Show that, for n 艌 1,
Tn⫹1共x兲 苷 2x Tn共x兲 ⫺ Tn⫺1共x兲
(d)
(e)
(f)
(g)
(h)
(i)
Use part (c) to show that Tn is a polynomial of degree n.
Use parts (b) and (c) to express T4 , T5 , T6 , and T7 explicitly as polynomials.
What are the zeros of Tn ? At what numbers does Tn have local maximum and minimum values?
Graph T2 , T3 , T4 , and T5 on a common screen.
Graph T5 , T6 , and T7 on a common screen.
Based on your observations from parts (g) and (h), how are the zeros of Tn related to the zeros
of Tn⫹1 ? What about the x-coordinates of the maximum and minimum values?
Thomson Brooks-Cole copyright 2007
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(j) Based on your graphs in parts (g) and (h), what can you say about x⫺1
Tn共x兲 dx when n is odd
and when n is even?
(k) Use the substitution u 苷 arccos x to evaluate the integral in part (j).
(l) The family of functions f 共x兲 苷 cos共c arccos x兲 are defined even when c is not an integer (but
then f is not a polynomial). Describe how the graph of f changes as c increases.
CHALLENGE PROBLEMS
ANSWERS
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Solutions
1. About 1.85 inches from the center
3. 0
5. f 共␲兲 苷 ⫺␲兾2
9. 共b ba⫺a 兲1兾共b⫺a兲e⫺1
11. 2 ⫺ sin⫺1 (2兾s5 )
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CHALLENGE PROBLEMS
SOLUTIONS
E
Exercises
1.
By symmetry, the problem can be reduced to finding the line x = c such that the shaded area is one-third of the area
√
c√
of the quarter-circle. The equation of the circle is y = 49 − x2 , so we require that 0 49 − x2 dx = 13 · 14 π(7)2
1
x
2
⇔
√
49 − x2 +
49
2
c
sin−1 (x/7)
=
0
[by Formula 30] ⇔
49
π
12
1
c
2
√
49 − c2 +
49
2
sin−1 (c/7) =
49
π.
12
This equation would be difficult to solve exactly, so we plot the left-hand side as a function of c, and find that the
equation holds for c ≈ 1.85. So the cuts should be made at distances of about 1.85 inches from the center of the
pizza.
3. The given integral represents the difference of the shaded areas, which appears
to be 0. It can be calculated by integrating with respect to either x or y, so we
√
find x in terms of y for each curve: y = 3 1 − x7 ⇒ x = 7 1 − y 3 and
√
y = 7 1 − x3 ⇒ x = 3 1 − y 7 , so
1
0
3
1 − y7 −
7
√
√
7
1 − x3 − 3 1 − x7 dx. But this
1
0
1 − y 3 dy =
√
√
3
1 − x7 − 7 1 − x3 dx = 0.
1
0
equation is of the form z = −z. So
5. Recall that cos A cos B = 12 [cos(A + B) + cos(A − B)]. So
π
0
f (x) =
=
1
2
=
π
2
cos t cos(x − t) dt =
t cos x +
cos x +
1
4
1
2
sin(2t − x)
sin(−x) −
π
0
1
2
1
4
π
0
[cos(t + x − t) + cos(t − x + t)] dt =
=
π
2
cos x +
sin(−x) =
π
2
1
4
sin(2π − x) −
1
4
1
2
π
0
[cos x + cos(2t − x)] dt
sin(−x)
cos x
The minimum of cos x on this domain is −1, so the minimum value of f (x) is f (π) = − π2 .
1
0
7. In accordance with the hint, we let Ik =
integrate Ik+1 by parts with u = 1 − x2
1 − x2
k+1
k
dx, and we find an expression for Ik+1 in terms of Ik . We
⇒ du = (k + 1) 1 − x2
k
(−2x), dv = dx ⇒ v = x, and
Thomson Brooks-Cole copyright 2007
then split the remaining integral into identifiable quantities:
Ik+1 = x(1 − x2 )k+1
= (2k + 2)
1
0
1
+ 2(k + 1)
0
1 − x2
k
1
0
x2(1 − x2 )k dx
1 − 1 − x2
dx = (2k + 2)(Ik − Ik+1 )
CHALLENGE PROBLEMS
So Ik+1 [1 + (2k + 2)] = (2k + 2)Ik
I0 = 1 =
⇒ Ik+1 =
2k + 2
Ik . Now to complete the proof, we use induction:
2k + 3
20 (0!)2
, so the formula holds for n = 0. Now suppose it holds for n = k. Then
1!
Ik+1 =
=
2(k + 1)22k (k!)2
2(k + 1) 2(k + 1)22k (k!)2
2k + 2
2k + 2 22k (k!)2
Ik =
=
=
·
2k + 3
2k + 3 (2k + 1)!
(2k + 3)(2k + 1)!
2k + 2 (2k + 3)(2k + 1)!
[2(k + 1)]2 22k (k!)2
22(k+1) [(k + 1)!]2
=
(2k + 3)(2k + 2)(2k + 1)!
[2(k + 1) + 1]!
So by induction, the formula holds for all integers n ≥ 0.
9. 0 < a < b. Now
1
0
[bx + a(1 − x)]t dx =
Now let y = lim
t→0
b ut
a (b−a)
bt+1 − at+1
(t + 1)(b − a)
du [put u = bx + a(1 − x)] =
1/t
. Then ln y = lim
t→0
ut+1
(t + 1)(b − a)
b
=
a
bt+1 − at+1
.
(t + 1)(b − a)
bt+1 − at+1
1
ln
. This limit is of the form 0/0,
t
(t + 1)(b − a)
so we can apply l’Hospital’s Rule to get
bt+1 ln b − at+1 ln a
b ln b − a ln a
b ln b
a ln a
bb/(b−a)
1
=
−
1
=
−
−
ln
e
=
ln
−
.
bt+1 − at+1
t+1
b−a
b−a
b−a
eaa/(b−a)
ln y = lim
t→0
Therefore, y = e−1
11.
bb
aa
1/(b−a)
.
An equation of the circle with center (0, c) and radius 1 is
x2 + (y − c)2 = 12 , so an equation of the lower semicircle is
√
y = c − 1 − x2 . At the points of tangency, the slopes of the line
and semicircle must be equal. For x ≥ 0, we must have
√
x
y0 = 2 ⇒ √
= 2 ⇒ x = 2 1 − x2 ⇒
2
1−x
x2 = 4(1 − x2 ) ⇒ 5x2 = 4 ⇒ x2 =
4
5
⇒ x=
2
5
√
5
√
√
5 = 45 5. The slope of the perpendicular line segment is − 12 , so an equation of the line
√
√
√
√
√
√
segment is y − 45 5 = − 12 x − 25 5
⇔ y = − 12 x + 15 5 + 45 5 ⇔ y = − 12 x + 5, so c = 5 and
√
√
an equation of the lower semicircle is y = 5 − 1 − x2 . Thus, the shaded area is
and so y = 2
2
5
√
(2/5) 5
2
Thomson Brooks-Cole copyright 2007
0
√
(2/5) 5
√
√
√
x√
1
30
5 − 1 − x2 − 2x dx = 2 5 x −
1 − x2 − sin−1 x − x2
2
2
0
√
4
5 1
2
1
−
· √ − sin−1 √
− 2(0)
=2 2−
5
2
5
5
5
2
2
1
= 2 − sin−1 √
= 2 1 − sin−1 √
2
5
5
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5
6
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CHALLENGE PROBLEMS
13. We integrate by parts with u =
−1
1
, dv = sin t dt, so du =
and
ln(1 + x + t)
(1 + x + t)[ln(1 + x + t)]2
v = − cos t. The integral becomes
∞
I=
0
= lim
b→∞
⎛
− cos t
sin t dt
= lim ⎝
ln(1 + x + t) b→∞
ln(1 + x + t)
− cos b
1
+
+
ln(1 + x + b)
ln(1 + x)
∞
0
where J =
∞
0
Thomson Brooks-Cole copyright 2007
0
−
0
⎞
cos t dt
⎠
(1 + x + t)[ln(1 + x + t)]2
− cos t dt
1
+J
=
ln(1 + x)
(1 + x + t)[ln(1 + x + t)]2
− cos t dt
. Now −1 ≤ − cos t ≤ 1 for all t; in fact, the inequality is strict
(1 + x + t)[ln(1 + x + t)]2
except at isolated points. So −
−
b
b
1
1
<J <
ln(1 + x)
ln(1 + x)
∞
0
dt
<J <
(1 + x + t)[ln(1 + x + t)]2
⇔ 0<I <
2
.
ln(1 + x)
∞
0
dt
(1 + x + t)[ln(1 + x + t)]2
⇔