Department of Mathematics, K.T.H.M. College, Nashik F.Y.B.Sc. Calculus Practical (Academic Year 2016-17) Practical 1: Graphs of Elementary Functions 1. a) Graph of y 1 = -f (x) ≡ mirror image of Graph of y = f (x) about X axis b) Graph of y 2 = f (-x) ≡mirror image of Graph of y = f (x) about Y axis c) Graph of y 3 = f (x - a) ≡ Graph of y = f (x) displaced along X axis by amount a d) Graph of y 4 = f (x)+ b ≡ Graph of y = f (x) displaced along Y axis by amount b 2. Graph of f -1 (x) ≡ mirror image of Graph of f (x) about the line y = x F.Y.B.Sc. Calculus Practical, Page: 1
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Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
Practical 1: Graphs of Elementary Functions
1. a) Graph of y1 = −f(x) ≡ mirror image of Graph of y = f(x) about X axisb) Graph of y2 = f(−x) ≡mirror image of Graph of y = f(x) about Y axisc) Graph of y3 = f(x− a) ≡ Graph of y = f(x) displaced along X axis by amount ad) Graph of y4 = f(x) + b ≡ Graph of y = f(x) displaced along Y axis by amount b
2. Graph of f−1(x) ≡ mirror image of Graph of f(x) about the line y = x
F.Y.B.Sc. Calculus Practical, Page: 1
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
F.Y.B.Sc. Calculus Practical, Page: 2
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
F.Y.B.Sc. Calculus Practical, Page: 3
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
F.Y.B.Sc. Calculus Practical, Page: 4
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
F.Y.B.Sc. Calculus Practical, Page: 5
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
Practical 2: Real Numbers
1. If 0 < a < b, show thata) a <
√ab < b b) 1
b< 1
a
Ans:
a) 0 < a < b=⇒ a.a < a.b and a.b < b.b=⇒ a2 < ab and ab < b2
=⇒ a <√ab and
√ab < b
=⇒ a <√ab < b
b) 0 < a < b=⇒ 1
aba < 1
abb (∵ ab > 0)
=⇒ 1b< 1
a
2. Find all x ∈ R satisfying the following inequalities.a) |x− 1| > |x+ 1| b) |x|+ |x+ 1| < 2
Ans:
a) |x− 1| > |x+ 1|⇐⇒ |x− 1| − |x+ 1| > 0 . . . (I)End points are −1 and 1.I Case: x ≤ −1In this case x− 1 ≤ −2 < 0 and x+ 1 ≤ 0. So from (I),−(x− 1)− (−(x+ 1)) > 0=⇒ 2 > 0This is true. So all x ≤ −1 satisfy (I).The solution set in I case is (−∞,−1]II Case: −1 < x ≤ 1In this case, x− 1 < 0 and x+ 1 > 0. So from (I),−(x− 1)− (x+ 1) > 0=⇒ −2x > 0=⇒ x < 0The solution set in II case is (−1, 0)III Case: x ≥ 1In this case x− 1 > 0 and x+ 1 > 2 > 0. So from (I),(x− 1)− (x+ 1) > 0=⇒ −2 > 0. This is impossible.The solution set in III case is φ
Solution Set = (−∞,−1] ∪ (−1, 0) ∪ φ = (−∞, 0)
b) |x|+ |x+ 1| < 2 . . . (II)End points are 0 and −1.I Case: x < −1
F.Y.B.Sc. Calculus Practical, Page: 6
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
In this case x < 0 and x+ 1 ≤ 0. So from (II),−x− (x+ 1) < 2=⇒ −2x− 1 < 2=⇒ −2x < 1=⇒ x > −1
2
This is impossible as x < −1The solution set in I case is φ.II Case: −1 ≤ x < 0In this case x < 0 and x+ 1 ≥ 0. So from (II),−x+ (x+ 1) < 2=⇒ 1 < 2This is true.The solution set in II case is [−1, 0).III Case: x ≥ 0In this case x ≥ 0 and x+ 1 > 0. So from (II),x+ (x+ 1) < 2=⇒ 2x+ 1 < 2=⇒ x < 1
2
The solution set in III case is [0, 12).
Solution Set = φ ∪ [−1, 0) ∪ [0, 12) = [−1, 1
2)
3. Prove that the following are not rational numbers.a)√
3 b)√
3 +√
5
Ans:
a) Suppose on contrary that√
3 is rational. Take√
3 = pq, where p and q are integers
and p, q do not have any common factor. Consider√3 = p
q
=⇒ 3 = p2
q2
=⇒ p2 = 3q2
=⇒ 3 divides p (∵ 3 is prime)=⇒ p = 3k (k is some integer)=⇒ q2 = 3k2
=⇒ 3 divides q (∵ 3 is prime)Thus 3 is a common factor of p and q. This contradicts the fact that p, q do not haveany common factor. Hence
√3 must be irrational.
b) Consider√3 +√
5 is rational√3−√
5 is rational (∵√
3−√
5 = −2√3+√5)
=⇒ (√
3 +√
5) + (√
3−√
5) is rational=⇒ 2
√3 is rational
=⇒√
3 is rationalThis is a contradiction. Hence
√3 +√
5 is irrational.
4. Let K = {s+ t√
2 : s, t ∈ Q}. Show that K satisfies the following conditions.
F.Y.B.Sc. Calculus Practical, Page: 7
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
a) If x1, x2 ∈ K, then x1 + x2 ∈ Kb) If x 6= 0 is in K, then 1
0 < |x− 0| < δ =⇒ |f(x)− f(0)| < ε is not possible for any δ > 0 .
This proves that limx→0
sin( 1x) does not exist.
2. If exists, evaluate the following limits.
a) limx→−1
x+52x+3
b) limx→0
|x|+2x3x−5|x| c) lim
x→0
sin( 1x)
1x
d) limx→1
√x−1x−1
Ans:
a) limx→−1
x+52x+3
= −1+52(−1)+3
= 41
= 4
F.Y.B.Sc. Calculus Practical, Page: 9
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
b) limx→0+
|x|+2x3x−5|x| = lim
x→0+
x+2x3x−5x = lim
x→0+
3x−2x = −3
2
and
limx→0−
|x|+2x3x−5|x| = lim
x→0−−x+2x3x+5x
= limx→0+
x8x
= 18
So limx→0+
|x|+2x3x−5|x| 6= lim
x→0−|x|+2x3x−5|x| and lim
x→0
|x|+2x3x−5|x| does not exist.
c) Consider
−1 ≤ sin 1x≤ 1
=⇒ −x ≤ x sin 1x≤ x if x ≥ 0 and −x ≥ x sin 1
x≥ x if x ≤ 0
=⇒ limx→0+
−x ≤ limx→0+
x sin 1x≤ lim
x→0+x and lim
x→0−−x ≥ lim
x→0−x sin 1
x≥ lim
x→0−x
=⇒ 0 ≤ limx→0+
x sin 1x≤ 0 and 0 ≥ lim
x→0−x sin 1
x≥ 0
=⇒ limx→0+
x sin 1x
= limx→0−
x sin 1x
= 0
=⇒ limx→0
x sin 1x
= 0
=⇒ limx→0
sin 1x
1x
= 0
d) Consider
limx→1
√x−1x−1 = lim
x→1
√x−1x−1 ×
√x+1√x+1
= limx→1
x−1(x−1)(
√x+1)
= limx→1
1√x+1
= 1√1+1
= 12
3. a) Show that limx→0
e1x−1e1x+1
does not exist.
b) Discuss the continuity of the function χ : R→ R defined as
χ(x) =
{1 if x ∈ Q0 if x 6∈ Q
Ans:
a) limx→0+
e1x−1e1x+1
= limx→0+
e1x
(1− 1
e1x
)e1x
(1+ 1
e1x
) = limx→0+
(1− 1
e1x
)(1+ 1
e1x
) = limx→0+
1−01+0
= 1
limx→0−
e1x−1e1x+1
= 0−10+1
= −1
So limx→0+
e1x−1e1x+16= lim
x→0−e1x−1e1x+1
and limx→0
e1x−1e1x+1
does not exist.
b) Let c ∈ R. Take ε = 12> 0 and δ be any positive real number. Take a rational r
and irrational s in the interval (c− δ, c+ δ)− {c} (i.e. in deleted δ neighborhood ofc. Then|χ(r)− χ(s)| = |1− 0| > 1
2= ε.
So0 < |x− c| < δ =⇒ |χ(x)− χ(c)| < ε is not possible for any δ > 0.
F.Y.B.Sc. Calculus Practical, Page: 10
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
This proves that limx→c
χ(x) does not exist. Therefore χ is discontinuous at c. Since c
was arbitrary point of R, χ is discontinuous at all points of R.
4. Discuss the continuity of the following functions.
a) f(x) =
2x− 1 if x ≤ 1x2 if 1 < x < 23x− 4 if 2 ≤ x < 4
x32 if x ≥ 4
b) f(x) =
e1x2
1−e1x2
if x 6= 0
1 if x = 0
c) f(x) =√
x−1x+3
for all x ∈ R
Ans:
a) Possible points of discontinuity are 1,2 and 4.i) c = 1 :
limx→1−
f(x) = limx→1−
(2x− 1) = 2(1)− 1 = 1
limx→1+
f(x) = limx→1+
(x2) = 1
So limx→1−
f(x) = limx→1+
f(x) = limx→1
f(x) = 1 = f(1).
f is continuous at 1.
ii) c = 2 :
limx→2−
f(x) = limx→2−
(x2) = 22 = 4
limx→2+
f(x) = limx→2+
(3x− 4) = 3(2)− 4 = 2
So limx→2−
f(x) 6= limx→2+
f(x)
limx→2
f(x) does not exist.
f is discontinuous at 2.
iii) c = 4 :
limx→4−
f(x) = limx→4−
(3x− 4) = 3(4)− 4 = 8
limx→4+
f(x) = limx→4+
(x32 ) = 4
32 = 8
So limx→4−
f(x) = limx→4+
f(x) = limx→4
f(x) = 8 = f(4).
f is continuous at 4.
f is continuous at all points of R except at c = 2.
b) Possible point of discontinuity: 0.
limx→0
e1x2
1−e1x2
= 01−0 = 0 6= 1 = f(1)
F.Y.B.Sc. Calculus Practical, Page: 11
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
So f is discontinuous at c =0.
f is continuous at all points of R except at c = 0.
c) By the algebra of continuous functions, f(x) will be continuous at all points atwhich it is defined. It is defined only when x+ 3 6= 0 and x−1
x+3≥ 0.
I Condition: x+ 3 6= 0In this case x 6= −3
II Conditiion x−1x+3≥ 0
In this case,(x− 1) ≥ 0 and x+ 3 ≥ 0 OR x− 1 <≤ 0 and x+ 3 ≤ 0=⇒ x ≥ 1 and x ≥ −3 OR x ≤ 1 and x ≤ −3=⇒ x ≥ 1 OR x ≤ −3
The function f(x) is continuous on (−∞,−3) ∪ [1,∞)
5. Determine the set of points of discontinuity of the following functions.a) f(x) = x2−2
x2+1, b) f(x) = x2+2x+3
x+2
Ans:
a) f(x) = x2−2x2+1
is a rational function (ratio of two polynomials). It is discontinuousonly where denominator vanishes. SoPoints of discontinuity of f(x) = {x : x2 + 1 = 0} = φ.The function f(x) is continuous at all points of R.
b) f(x) = x2+2x+3x+2
is a rational function (ratio of two polynomials). It is discontinuousonly where denominator vanishes. SoPoints of discontinuity of f(x) = {x : x+ 2 = 0} = {−2}.The function f(x) is continuous at all points of R except at x = −2.
6. If function f(x) is continuous on [−2, 2] where
f(x) =
sin axx
+ 2 if − 2 ≤ x < 03x+ 5 if 0 ≤ x ≤ 1√x2 + 8− b if 1 < x ≤ 2
Show that a+ b+ 2 = 0.
Ans: f(x) is continuous at x = 0
=⇒ limx→0−
f(x) = f(0)
=⇒ limx→0
(sin axx
+ 2)
= 3(0) + 5
=⇒ limx→0
(sin axax× a)
+ 2 = 5
=⇒ 1× a+ 2 = 5
=⇒ a+ 2 = 5
F.Y.B.Sc. Calculus Practical, Page: 12
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
=⇒ a = 3
Also
f(x) is continuous at x = 1
=⇒ limx→1+
f(x) = f(1)
=⇒√
12 + 8− b = 3(1) + 5
=⇒√
9− b = 8
=⇒ 3− b = 8
=⇒ b = −5
Thus a = 3 and b = −5. Therefore a+ b+ 2 = 3 + (−5) + 2 = 0.
7. Show that the equation 2xx− 1 = 0 has at least one root in (0, 1).
Ans: Take f(x) = 2xx− 1 = 0 on [0, 1]. Then
f(0) = 20(0)− 1 = −1 < 0 and
f(1) = 21(1)− 1 = 2− 1 = 1 > 0
Therefore by Bolzano’s Theorem, there is at least one c ∈ (0, 1) such that f(c) = 0.This implies that
2cc− 1 = 0
=⇒ c is root of 2xx− 1 = 0 in (0, 1).
8. Show that the equation cos x = x has a solution in the interval [0, π2].
Ans: Take f(x) = cos x− x on [0, π2]. Then
f(0) = cos 0− 0 = 1− 0 = 1 > 0 and
f(π2) = cos(π
2)− π
2= 0− π
2= −π
2< 0
Therefore by Bolzano’s Theorem, there is at least one c ∈ (0, π2) such that f(c) = 0.
This implies that
cos c− c = 0
=⇒ c is solution of cosx = x in [0, π2].
F.Y.B.Sc. Calculus Practical, Page: 13
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
Practical 4: Differentiation
1. Use the definition of derivative, to find derivative of the following functions.a) f(x) = x3 b) f(x) = 1
x
Ans: a) f ′(x) = limh→0
f(x+h)−f(x)h
= limh→0
(x+h)3−x3h
= limh→0
(x3+3x2h+3xh2+h3)−x3h
= limh→0
3x2h+3xh2+h3
h
= limh→0
(3x2 + 3xh+ h2)
= 3x2
f ′(x) = 3x2
Ans: b) f ′(x) = limh→0
f(x+h)−f(x)h
= limh→0
1x+h− 1x
h
= limh→0
x−(x+h)x(x+h)h
= limh→0
−hx2h+h2
= limh→0
−1x2+h
= −1x2
f ′(x) = −1x2
2. a) If f(x) = |x|, show that f ′(x) = 1 when x > 0 and f(x) = −1 when x < 0. Alsoshow that the function f(x) is not differentiable at x = 0 .b) If f(x) = x|x| (∀x ∈ R), show that f ′′(x) = 2 when x > 0 and f”(x) = −2 whenx < 0. Also show that the function f(x) is differentiable at x = 0.
c) If f(x) =
{1− x if x ≤ 1(1− x)2 if x > 1
,
show that f ′+(1) = 0 and f ′−(1) = −1
Ans:
a) f(x) = |x| = x if x > 0 implying that f ′(x) = 1 when x > 0f(x) = |x| = −x if x < 0 implying that f ′(x) = −1 when x < 0
limh→0+
f(0+h)−f(0)h
= limh→0+
h−0h
= limh→0+
1 = 1
andlimh→0−
f(0+h)−f(0)h
= limh→0−
−h−0h
= limh→0−
−1 = −1
So limh→0−
f(0+h)−f(0)h
6= limh→0+
f(0+h)−f(0)h
and limh→0
f(0+h)−f(0)h
does not exist. This proves
that f(x) is not differentiable at x = 0.
b) f(x) = x|x| = xx = x2 if x > 0 implying that f ′(x) = 2x and f”(x) = 2 whenx > 0f(x) = x|x| = x(−x) = −x2 if x > 0 implying that f ′(x) = −2x and f”(x) = −2when x < 0
limh→0+
f(0+h)−f(0)h
= limh→0+
h2−0h
= limh→0+
h = 0
limh→0−
f(0+h)−f(0)h
= limh→0−
−h2−0h
= limh→0+
−h = 0
So limh→0−
f(0+h)−f(0)h
= limh→0+
f(0+h)−f(0)h
= 0 and limh→0
f(0+h)−f(0)h
exists and equals to
0. This proves that f(x) is differentiable at x = 0 and f ′(0) = 0
c) f ′+(1) = limh→0+
f(1+h)−f(1)h
= limh→0+
[1−(1+h)]2−0h
= limh→0+
h2
h= lim
h→0+h = 0
F.Y.B.Sc. Calculus Practical, Page: 14
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
andf ′−(1) = lim
h→0−f(1+h)−f(1)
h= lim
h→0−[1−(1+h)]−0
h= lim
h→0−−hh
= limh→0−
−1 = −1
3. Test whether the conditions of Rolles Mean Value Theorem hold for given functionson the interval and if so find the value of c.a) f(x) = log
(x2+ab(a+b)x
)on [a, b] b) f(x) = (x− a)m(x− b)n on [a, b]
Ans:
a) f(x) is continuous on [a, b] and f(x) is differentiable in (a, b). Also
f(a) = log(a2+ab(a+b)a
)= log
(a2+ab
(a2+ab)x
)= log 1 = 0
and
f(a) = log(b2+ab(a+b)b
)= log
(b2+ab(a+b2)x
)= log 1 = 0
The hypothesis of Rolle’s Mean Value theorem is satisfied. To find c, consider
f ′(c) = 0
=⇒ 1(c2+ab)(a+b)c
(2c) = 0
=⇒ c2 + ab = 0
=⇒ c =√ab
b) f(x) is continuous on [a, b] and f(x) is differentiable in (a, b). Also
f(a) = (a− a)m(a− b)n = 0
and
f(a) = (b− a)m(b− b)n = 0
The hypothesis of Rolle’s Mean Value theorem is satisfied. To find c, consider
f ′(c) = 0
=⇒ m(c− a)m−1(c− b)n + n(c− a)m(c− b)n−1 = 0
=⇒ (c− a)m−1(c− b)n−1(mc−mb+ nc− na) = 0
=⇒ mc−mb+ nc− na = 0
=⇒ c(m+ n)− (mb+ na) = 0
F.Y.B.Sc. Calculus Practical, Page: 15
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
=⇒ c = mb+nam+n
4. Test whether the conditions of Lagranges Mean Value Theorem hold for given func-tions on the interval and if so find the value of c.
a) f(x) = (x− 1)(x− 2)(x− 3) on [0, 4] b) f(x) = tan−1 x on [0, 1]
Ans:
a) f(x) is continuous on [0, 4] and f(x) is differentiable in (0, 4). So the hypothesisof Lagrange’s Mean Value Theorem is satisfied. To find value of c, consider
b) f(x) is continuous on [0, 1] and f(x) is differentiable in (0, 1). So the hypothesisof Lagrange’s Mean Value Theorem is satisfied. To find value of c, consider
f ′(c) = f(1)−f(0)1−0
=⇒ 11+c2
=π4−01
=⇒ 11+c2
= π4
=⇒ 1 + c2 = 4π
=⇒ c = ±√
4π− 1
=⇒ c =√
4π− 1 (∵ c ∈ (0, 1))
5. a) Show that the value of c when Cauchy’s Mean Value Theorem is applied to thefunctions f(x) = sinx and g(x) = cos x on [a, b] is the arithmetic mean of a and b.b) Show that the value of c when Cauchy’s Mean Value Theorem is applied to the
functions f(x) =√x and g(x) =
√1x
on [a, b] is the geometric mean of a and b.
Ans:
a) The functions f(x) = sin x and g(x) = cosx are continuous on [a, b] and differen-tiable in (a, b). So the hypothesis of Cauchy’s Mean Value Theorem is satisfied. To
F.Y.B.Sc. Calculus Practical, Page: 16
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
find c, consider
f ′(c)g′(c)
= f(b)−f(a)g(b)−g(a)
=⇒ cos c− sin c
= sin b−sin acos b−cos a
=⇒ − cot c =2 sin( b−a
2) cos( b+a
2)
−2 sin( b−a2
) sin( b+a2
)
=⇒ − cot c = − cot (b+a)2
=⇒ c = b+a2
=⇒ c is arithmetic mean of a and b.
b) The functions f(x) =√x and g(x) =
√1x
are continuous on [a, b] and differentiable
in (a, b). So the hypothesis of Cauchy’s Mean Value Theorem is satisfied. To find c,consider
f ′(c)g′(c)
= f(b)−f(a)g(b)−g(a)
=⇒12c−
12
− 12c−
32
=√b−√a√
1b−√
1a
=⇒ −c = −√ba(√
b−√a√
a−√b
)=⇒ c =
√ab
=⇒ c is geometric mean of a and b
6. Show that the function f(x) = x3 − 3x2 + 3x− 100 is increasing on R.
Ans: f(x) = x3 − 3x2 + 3x− 100
=⇒ f ′(x) = 3x2 − 6x+ 3
=⇒ f ′(x) = 3(x− 1)2
=⇒ f ′(x) > 0 for all x ∈ R except for x = 1.
=⇒ f(x) is increasing on R.
7. Find values of x for which the function f(x) = 2x3 − 9x2 + 12x+ 2 is decreasing.
Ans: f(x) = 2x3 − 9x2 + 12x+ 2
=⇒ f ′(x) = 6x2 − 18x+ 12
F.Y.B.Sc. Calculus Practical, Page: 17
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
=⇒ f ′(x) = 6(x− 1)(x− 2)
Therefore f ′(x) < 0
=⇒ 6(x− 1)(x− 2) < 0
=⇒ x− 1 < 0 and x− 2 > 0 OR x− 1 > 0 and x− 2 < 0
=⇒ x < 1 and x > 2 OR x > 1 and x < 2
=⇒ x > 1 and x < 2
=⇒ 1 < x < 2
=⇒ x ∈ (1, 2)
So f(x) is decreasing for x ∈ (1, 2).
8. Differentiate log(1 + x2) with respect to tan−1 x
Ans: Take u = log(1 + x2) and v = tan−1 x. Then
dudx
= 11+x2× 2x = 2x
1+x2
and
dvdx
= 11+x2
So,
dudv
= dudx× dx
dv= 2x
1+x2× 1+x2
1= 2x
9. Differentiate tan−1(cosx+sinxcosx−sinx
)w.r.t x
Ans: Take y = tan−1(cosx+sinxcosx−sinx
). Then
y = tan−1(1+tanx1−tanx
)=⇒ y = tan−1
(tan(π
4+ x)
)=⇒ y = π
4+ x
=⇒ dydx
= 1
F.Y.B.Sc. Calculus Practical, Page: 18
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
Ans: y sin( yx) + x = x sin( y
x) dydx
y sin( yx)+x
x sin( yx)= dy
dx
yx
+ 1sin( y
x)
= dydx
Put y = vx.
dydx
= v + x dydx
v + 1sin v
= v + x dvdx
1sin v
= x dvdx
Using variable seperable form
1xdx = sin vdv
=⇒∫
1xdx =
∫sin vdv
=⇒ log x = − cos v + c
Replacing v by yx,
log x = − cos( yx) + c
3. x dydx− y =
√x2 − y2
Ans: x dydx− y =
√x2 − y2
=⇒ x dydx
= y +√x2 − y2
=⇒ dydx
=y+√x2−y2x
Put y = vx
dydx
= v + x dvdx
=⇒ v + x dvdx
= vx+√x2−v2x2x
=⇒ v + x dvdx
= v +√
1− v2
=⇒ x dvdx
=√
1− v2
Using variable seperable form,
1√1−v2dv = 1
xdx
F.Y.B.Sc. Calculus Practical, Page: 29
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
By taking integration of both sides∫1√
1−v2 dv =∫
1xdx
sin−1 v = log x+ c
By replacing v = yx
sin−1 yx
= log x+ c
4. Show that the differential equations is (2x3 − xy2 − 2y + 3)dx − (x2y + 2x)dy = 0exact and solve it.
Ans: M = 2x3 − xy2 − 2y + 3 and N = −x2y − 2x
=⇒ ∂M∂y
= −2xy − 2 and ∂N∂x
= −2xy − 2
=⇒ ∂M∂y
= ∂N∂x
Therefore the given differential equation is exact. The solution is given by∫y const
Mdx+∫(terms in N free from x) dy = c
=⇒∫
y const
(2x3 − xy2 − 2y + 3)dx+∫
0dy = c
=⇒ 2x4
4− x2y2
2− 2xy + 3x = c
=⇒ x4
2− x2y2
2− 2xy + 3x = c
=⇒ x4 − x2y2 − 4xy + 6x = c
5. Find order and degree of the following differential equations.
a) dydx
+xy = x3 b) d2ydx2
+ ( dydx
)2 + 5y = 0 c)(1 + ( dy
dx)2) 3
2 = d2ydx2
Ans: a) dydx
+ xy = x3
The order is 1 and degree is 1.
b) d2ydx2
+ ( dydx
)2 + 5y = 0
The order is 2 and degree is 1.
c)(1 + ( dy
dx)2) 3
2 = d2ydx2
The order is 2 and degree is 2.
F.Y.B.Sc. Calculus Practical, Page: 30
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
6. Which of the following functions are homogeneous?
a) f(x, y) = x2+y2 b) g(x, y) =√x2 + y2 c) h(x, y) = x2+xy2
Ans: a) f(tx, ty) = t2x2 + t2y2 = t2(x2 +y2) = t2f(x, y). Therefore f is homogeneousfunction of degree 2.
b) g(tx, ty) =√t2x2 + t2y2 =
√t2(x2 + y2) = t
√x2 + y2 = tg(x, y). Therefore g
is homogeneous function of degree 1.
c) h(tx, ty) = t2x2 + txt2y2 = t2(x2 + txy2) 6= t2h(x, y). Therfore this is not ho-mogeneous differential equation.
7. Solve the differential equation dydx
= x+yx+2y−3
Ans: Compairing given differential equation with dydx
= a1x+b1y+c1a2x+b2y+c2
We get a1 = 1, b1 =1, a2 = 1, b2 = 2, c1 = 0, c2 = 1
buta1a2
= 116= b1
b2= 1
2Therefore put x = X + h, andy = Y + k =⇒ dx
dX= 1
and dydY
= 1 =⇒ dydx
= dYdX
Therefore given differential becomes dYdX
= X+h+Y+kX+h+2(Y+k)−3
=⇒ dYdX
= X+Y+(h+k)X+2Y+(h+2k−3)
Put h + k = 0 and h + 2k − 3 = 0. We obtain by solving h = −3 and k = 3.Therefore dY
dX= X+Y
X+2Y
Put Y = V X. Then dYdX
= V +X dVdX
V +X dVdX
= X+V XX+2V X
= !+V1+2V
X dVdX
= 1+V1+2V
= −V = 1+V−V−2V 2
1+2V
=⇒ X dVdX
= 1−2V 2
1+2V
=⇒ 1+2V1−2V 2dV = 1
XdX
=⇒ 11−2V 2dV + 2V
1−2V 2dV = 1XdX
=⇒ 12( 1
2−V 2)
dV − (−2V )1−2V 2dV = 1
XdX
F.Y.B.Sc. Calculus Practical, Page: 31
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
=⇒ 12( 1
2−V 2)
dV − 12(−4V )1−2V 2dV = 1
XdX
By taking integration of both sides, we get
12
∫1
2( 12−V 2)
dV − 12
∫ (−4V )1−2V 2 dV =
∫1X
dX
12× 1
2
[log(
V− 1√2
V+√2)]− 1
2log(1− 2V 2) = logX
Multiplying by 4 on both sides, we get[log(
V− 1√2
V+√2)]− 2 log(1− 2V 2) = 4 logX
=⇒[log(
V− 1√2
V+√2× 1
(1−2V 2)2)]
= logX4 + logC
=⇒V− 1√
2
V+√2× 1
(1−2V 2)2= X4C
=⇒√2v−1√2v+1× 1
(1−√2V )2(1+
√2V )2
= X4C
=⇒√2v−1√2v+1× 1
(√2V−1)2(1+
√2V )2
= X4C
=⇒ 1(√2V+1)(
√2V−1)(1+
√2V )2
= X4C
=⇒ 1(2V 2−1)(1+
√2V )2
= X4C
Put V = YX
1
(2 Y2
X2−1)(1+√2 YX)2
= X4C
On Simplifying, we get
(2Y 2 −X2)(X +√
2Y )2 = 1C
Replace X by x+ 3 and Y by y − 3
(2(y − 3)2 − (x+ 3)2)((x+ 3) +√
2(y − 3))2 = 1C
= d
On Simplifying we get, (2y2 − 12y − x2 − 6x+ 9)(x+√
2y + 3(1−√
2))2 = d
8. Solve (1 + x2) dydx
+ 2xy − 1 = 0
Ans: (1 + x2) dydx
+ 2xy − 1 = 0
=⇒ dydx
+ 2x1+x2
y = 11+x2
Compairing given differential equation with dydx
+ P (x)y = Q(x)
F.Y.B.Sc. Calculus Practical, Page: 32
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
We get P (x) = 2x1+x2
and Q(x) = 11+x2
Therefore integrating factor is
I.F. = e∫P (x)dx = e
∫2x
1+x2dx
= elog(1+x2) = (1 + x2)
Therefore solution is y.I.F. =∫(I.F.)Q(x)dx
=⇒ y.(1 + x2) =∫(1 + x2) 1
1+x2dx
=⇒ y.(1 + x2) =∫
dx
=⇒ y.(1 + x2) = x+ c
=⇒ y = x+c1+x2
9. Solve (1 + y2)dx = (tan−1 y − x)dy
Ans: (1 + y2) = (tan−1 y − x) dydx
(1 + y2)dxdy
= (tan−1 y − x)
dxdy
+ 11+y2
x = tan−1 y1+y2
−
Comparing given differential equation with dxdy
+ P (y)x = Q(y)
We get P (y) = 11+y2
and Q(y) = tan−1 y1+y2
Therefore integrating factor is
I.F. = e∫P (y)dy = e
∫1
1+y2dy
= etan−1 y
Therefore solution is x.I.F. =∫(I.F.)Q(y)dy
x.I.F. =∫etan
−1 y(tan−1 y1+y2
)dy
Put tan−1 = y =⇒ sec2 udu = dy
Therefore xetan−1 y =
∫etan
−1 tanu(tan−1 tanu1+tan2 u
)sec2 udu
=⇒ xetan−1 y =
∫euudu
Using product rule for integration we get
=⇒ xetan−1 y = u
∫eudu−
∫∂u∂u
(∫eudu)du
F.Y.B.Sc. Calculus Practical, Page: 33
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
=⇒ xetan−1 y = u
∫eudu−
∫(∫eudu)du
=⇒ xetan−1 y = ueu − eu + C
By repacing u by tan−1 y we get
=⇒ xetan−1 y = tan−1 yetan
−1 y − etan−1 y + C
=⇒ x = tan−1 y + C
etan−1 y− 1
=⇒ x = tan−1 y + Ce− tan−1 y − 1
10. Solve (x+ 2y3) dydx
= y
Ans: (x+ 2y3) dydx
= y
=⇒ y dxdy
= x+ 2y3
=⇒ dxdy− x
y= 2y2 . . . (I)
The above equation is of the form dydx
+ P.x = Q where P and Q are functionsof y.
Therefore
I.F. = e∫Pdy = e
∫ −1ydy=e− log y= 1
y
Multiplying equation (I) by I.F.,
1y(dxdy− x
y) = 2y2 1
y
=⇒ ddy
(x 1y) = 2y
Integrating w.r.t. y
x 1y
=∫
2ydy + c
=⇒ xy
= y2 + c
=⇒ x = y(c+ y2)
This is the general solution of given differential equation.
11. Find orthogonal trajectories of the family x2 + y2 = a2
Ans: Differentiate the family x2 + y2 = a2 with respect to x
=⇒ 2x+ 2y dydx
= 0
F.Y.B.Sc. Calculus Practical, Page: 34
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
=⇒ x+ y dydx
= 0
=⇒ dydx
= −xy
By replacing dydx
by −dxdy
we get
=⇒ −dxdy
= −xy
=⇒ dxdy
= xy
By using variable separable form
=⇒ 1ydy = 1
xdx
Taking integration on both sides∫1ydy =
∫1xdx
log y = log x+ log c
=⇒ log y = log cx
=⇒ y = cx
Therefore orthogonal trajectories of the family x2 + y2 = a2 is the family of straightline pasiing through origine i.e. y = cx where c is a slope of the line
F.Y.B.Sc. Calculus Practical, Page: 35
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
Practical 7: Miscellaneous
1. By knowing the graph of f(x) = ex, draw the graphs of a) f(x) = −ex b) f(x) = e−x
Ans: a) Graph of −f(x) ≡ mirror image of Graph of f(x) about X axis
=⇒ Graph of −ex ≡ mirror image of Graph of ex about X axis
b) Graph of f(−x) ≡ mirror image of Graph of f(x) about Y axis
=⇒ Graph of e−x ≡ mirror image of Graph of ex about Y axis
F.Y.B.Sc. Calculus Practical, Page: 36
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
2. By knowing the graph of f(x) = sin x, draw the graphs of a) f(x) = sin(x) + 2 b)f(x) = sin(x)− 2 c) f(x) = sin(x− 2) d) f(x) = sin(x+ 2)
Ans: a) Graph of f(x) + b ≡ Graph of f(x) displaced along Y axis by amount b
=⇒ Graph of sin(x) + 2 ≡ Graph of sinx displaced along Y axis by amount 2
b) Graph of f(x) + b ≡ Graph of f(x) displaced along Y axis by amount b
=⇒ Graph of sin(x)− 2 ≡ Graph of sinx displaced along Y axis by amount −2
c) Graph of f(x− a) ≡ Graph of f(x) displaced along X axis by amount a
F.Y.B.Sc. Calculus Practical, Page: 37
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
=⇒ Graph of sin(x− 2) ≡ Graph of sinx displaced along X axis by amount 2
d) Graph of f(x− a) ≡ Graph of f(x) displaced along X axis by amount a
=⇒ Graph of sin(x+ 2) ≡ Graph of sinx displaced along X axis by amount −2
3. Evaluate the integral∫
x2+3(x−1)(x2+4)
dx
Ans: Partial fractions are
x2+3(x−1)(x2+4)
= Ax−1 + Bx+c
x2+4. . . (I)
=⇒ x2 + 3 = A(x2 + 4) + (Bx+ C)(x− 1)
=⇒ x2 + 3 = (A+B)x2 + (C −B)x+ (4A− C)
F.Y.B.Sc. Calculus Practical, Page: 38
Department of Mathematics, K.T.H.M. College, NashikF.Y.B.Sc. Calculus Practical (Academic Year 2016-17)
=⇒ A+B = 1, C −B = 0, 4A− C = 3
=⇒ B = C,A+ C = 1, 4A− C = 3
=⇒ 5A = 4, C = 15
= B
=⇒ A = 45, B = 1
5, C = 1
5
So from (I),∫x2+3
(x−1)(x2+4)dx
=∫( 4
5
x−2 +15(x+1)
x2+4
)dx
= 45
∫1
x−1dx+ 15
∫x+1x2−4dx
= 45
log |x− 1|+ 110
∫( 2xx2+4
+ 2x2+4
)dx
= 45
log |x− 1|+ 110
∫2xx2+4
dx+ 15
∫1
x2+4dx
= 45
log |x− 1|+ 110
log |x2 + 4|+ 110
tan−1 x2
+ c
4. If y = (sin−1 x)2, show that (1− x2)yn+2 − (2n+ 1)xyn+1 − n2yn = 0
Ans: y = (sin−1 x)2
=⇒ y1 = 2 sin−1 x 1√1−x2
=⇒ (1− x2)y21 − 4 sin−1 x = 0
=⇒ (1− x2)y21 − 4y = 0
=⇒ (1− x2)2y1y2 − 2xy21 − 4y1 = 0
=⇒ 2y1[(1− x2)y2 − xy1 − 2] = 0
=⇒ (1− x2)y2 − xy1 − 2 = 0
Differentiating n times by using Leibnitz’s Theorem, we get