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ASSIGNMENT
Differential Calculus I
Q. 1. Discuss the continuity of
Q. 2. If
Q. 3. Differentiate the following function w.r.t. x….
Q. 4. If then show that
Q. 5.
Q. 6. Differentiate with respect to
Q. 7. If show that
Q. 8. Find the value of a and b such that the function defined by
Differential Calculus II
Q. 1. Discuss the continuity of
Q. 2. Find the relationship between a and b so that the function f defined by
Q. 3. If
then show that
Q. 4. Q. 5.If cos y = x Cos (a + y), with
prove that
Q. 6. Differentiate w.r.to x.
Q. 7.
Q. 8. If for some c > 0, prove that
is a constant independent of a and b.
Differential Calculus III
Q. 1. Discuss the continuity of
Q. 2. For what value of is the function defined by
continuous at x = 0? What about continuity at x = 1?
Q. 3.
Q. 4.
Q. 5.
Q. 6.
Q. 7.
Q. 8.
Continuity & Differentiation
Q. 1. Find the values of a and b such that the function defined by f(x) = ( 5, if x ≤ 2 ax + b if 2<x<10 21, if x 10 ) is a continuous function.
Q. 2. Find of sin2y + cos (xy) = p
Q. 3. Differentiate w.r.t. x ( x cosx)x + (x sinx)1/x
Q. 4. If x = , = show that
Q. 5. If y = (tan-1x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2.
Q. 6. Differentiate sin-1 w.r.t. x
Q. 7. If x for -1<x<1, show that
Q. 8. Find if y = a t + 1/t , x = ( t + 1/t)a
Q. 9. Discuss the continuity of the function given by :-
Q. 10. If the function f(x) is given by f(x) =
is continuous at x = 1, find the values of a and b.
Q. 11. If y = [x + ]n, then prove that
Q. 12. Prove :
Q. 13. Find when y = sec-1
Q. 1. Find the value of the following :
i.
ii.
iii.
Q. 2. Prove That
i.
ii.
iii.
Q. 3. Solve
i.
ii.
Q. 4. Simplify :
i.
ii.
Q. 5.
Q. 6.
Q. 7.
Q. 8.
Q. 14. If ex + ey = ex+y, prove that
Q. 15. Given that cos
prove that
Q. 16. If x=a(q + sinq), y= a(1+ cosq), prove that
Q. 17.
Q. 18. Find the value of ‘k’ if
is continuous at x =
Q. 19. If
Q. 20. If Cos y = x Cos2 ( a + y ) , with Cos a ≠ 1, prove that
Question 3 The function f is defined as {x ²+ax+b ,0≤x<23 x+2 ,2≤x ≤4
2ax+5b ,4<x ≤8 If f(x)
is continuous on [0,8], find the values of a and b. Answer [a=3,b=-2]
Rate of Change of Quantities.
Q.1. A point source of light along a straight road is at a height of ‘a’ metres. A boy ‘b’ metres in height is walking along the road. How fast is his shadow increasing if he is walking away from the light at the rate of c metres per minute?
Solution :
Fig.
Let lamp-post be AB and CD be the boy whose distance from lamp-post at any time t be x m, let CE = y m be its shadow. Then dx/dt = c m/m.
As, ∆ BAE ~ ∆ DCE, AB/CD = AE/CE => a/b = (x + y)/y => ay = b(x + y) => (a – b) y = bx => (a – b)dy/dt = b dx/dt = bc Therefore, dy/dt = bc/(a – b). [Ans.]
Q.2. The two equal sides of an isosceles triangle with fixed base b cm are decreasing at the rate of 3 cm/sec. How fast is the area decreasing when the two equal sides are equal to the base?
Solution :
Fig.
Q.3. The volume of a cube is increasing at the rate of 7 cubic centimeters per second. How fast is the surface area of the cube increasing when the length of an edge is 12 centimeters?
6.2. Increasing and Decreasing Function.
Q.1. Find the intervals in which the function f(x) = x3 – 12 x2 + 36 x + 17 is
i. increasing,ii. decreasing.
i. x ] – ∞, 2[ U ] 6, ∞ [ . [ε Ans.]
(ii) x ]2, 6[ [ε Ans.]
Q.2. Find the intervals in which the function f(x) = 2x3 – 9x2 + 12x + 15 is (i) increasing and (ii) decreasing.
Solution :
Therefore, disjoint intervals on real number line are (– ∞, 1), (1, 2), (2, ∞)
Intervals Test Value
Nature of f’(x) f’(x) = 6(x – 2)(x – 1)
f(x)
( – ∞, 1) x = 0 ( + ) (– ) (– ) = ( + ) > 0
↑
(1, 2) x = 1.5 ( + )( – )( – ) = ( – ) < 0
↓
(2, ∞) x = 3 ( + )( + )( + ) = ( + ) > 0
↑
Therefore, f(x) is increasing in ( – ∞, 1), (2, ∞) and decreasing in (1, 2). [Ans.]
Tangents and Normals.
Q.1. If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t), show that [dy/dx]at t= /4 = b/a.π
Q.2. If x = a(cos + log tan /2) and y = a sin , find the value of dy/dx θ θ θat = /4.θ π
Q.3. Find the slope of the tangent to the curve y = 3x4 – 4x at x = 1.
Q.4. For the curve y = 3x2 + 4x, find the slope of the tangent to the curve at the point whose x-coordinate is – 2.
Q.5. Find the equation of the tangent and the normal to the curve y = x3 at the point P(1, 1).
Q.6. Find the equation of the tangent to the curve: x = + sin , y = 1 + θ θcos at = /4.θ θ π
Q.7. Find the equation of the tangent to the curve x = sin 3t, y = cos 2t, at t = /4.π
2√2 x – 3y – 2 = 0. [Ans.]
Q.8. At what points will the tangent to the curve y = 2x3 – 15x2 + 36x – 21 be parallel to x-axis? Also, find the equations of tangents to the curve at those points.
6.4. Approximation.
Q.1. If f(x) = 3x2 + 15x + 5, then find the approximate value of f(3.02), using differentials.
77.66. [Ans.]
Q. 1. An open box, with a square base, is to be made out of a given quantity of metal sheet of area C2. Show that the maximum volume of the box is C3/6√3.
Q.2. A window is in the form of a rectangle surmounted by a semi-circle. If the total perimeter of the window is 30 m, find the dimensions of the window so that maximum light is admitted.
Solution :
AB = 30/( + 4) m and BC = 30/( + 4) m. [π π Ans.]
Q.3. Find the point on the curve y2 = 4x which is nearest to the point (2, –8).
the nearest point is (4, – 4) [Ans.]
Q.4. Find the largest possible area of the right-angled triangle whose hypotenuse is 5 cm.
Solution :
.
= 25/4 sq. units. [Ans.]
Q.5. Prove that the radius of the right circular cylinder of the greatest curved surface that can be inscribed in a given cone is half of the radius of the cone.
Solution :
Q.6. A right-angled triangle with constant area S is given. Prove that the hypotenuse of the triangle is least when the triangle is isosceles.
Solution :
Q.7. Three sides of a trapezium are equal, each being 10 cm. Find the area of the trapezium when it is maximum.
Solution :
the maximum area of the trapezium is 75√3. [Ans.]
Q.8. Show that the semi-vertical angle of the right circular cone of given total surface area and maximum volume is sin –11/3.
Solution :
Q.9. Show that a rectangle of maximum perimeter which can be inscribed in a circle of radius r is a square of side √2 r.
Solution :
Fig.
Let ABCD be the rectangle inscribed in a circle of radius r and centre O. BD is the diameter = 2r. Let LOBA = , 0 < < /2.θ θ π Now, AB = 2r cos and AD = 2r sin .θ θ Perimeter of the rectangle, p = 2(AB + CD) = 2(2r cos + 2r sin )θ θ = 4r (cos + sin )θ θ Therefore, dp/d = 4r (– sin + cos )θ θ θ and d2p/d 2 = 4r (– cos – sin ) = – 4r(cos + sin ).θ θ θ θ θ
Now, dp/d = 0 => 4r (– sin + cos ) = 0θ θ θ Or, tan = 1 => = /4. [As, 0 < < /2]θ θ π θ π Also [d2p/d 2] = /4 = – 4r ( sin /4 + cos /4 )θ θ π π π = – 4r (1/√2 + 1/√2 ) = – 4r.2/√2 = – 4√2 r < 0. Therefore, p is maximum when = /4.θ π That is when BC = 2r sin /4 = 2r. 1/√2 = √2 r and AB = 2r cos /4 = π π2r. 1/√2 = √2 r.AB and BC are adjacent sides, hence ABCD is a square. Hence, perimeter of ABCD is maximum when it is a square. [Proved.]
Q.10. Show that the rectangle of maximum area that can be inscribed in a circle is a square.
Q.12. Show that the height of a cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/√3.
Or,
Prove that the height of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/√3. Also find the maximum volume. Solution :
Q.13. Find the altitude of a right circular cone of maximum curved surface which can be inscribed in a sphere of radius r.
Solution :
Q.14. A wire of length 20 m is available to fence off a flower bed in the form of a sector of a circle. What must be the radius of the circle, if we wish to have a flower bed with the greatest possible area?
Solution :
Q.15. Show that the height of a cylinder of maximum volume that can be inscribed in a cone of height h is 1/3h.
Solution :
Q.16. Show that the volume of the greatest cylinder that can be inscribed in a cone of height h and semi-vertical is 4/27 h3tan2 .α π α
Solution :
Q.17. Show that the semi-vertical angle of a cone of maximum volume and of given slant height is tan –1(√2).
Solution :
Q.18. Find the volume of the largest cone that can be inscribed in a sphere of radius R.
Solution :
Fig.
Let
Q.19. Prove that the area of right-angled triangle of a given hypotenuse is maximum when the triangle is isosceles.
Solution :
Fig.
Q.20. A closed circular cylinder has a volume of 2156 c.c. What will be the radius of its base so that its total surface area is minimum. Find the height of the cylinder when its total surface area is minimum.
Or
Show that the height of the closed right circular cylinder, of given volume and minimum total surface area, is equal to its diameter.
Q.21. Three numbers are given whose sum is 180 and the ratio between first two of them is 1:2. if the product of the number is greatest, find the numbers.
[ numbers are 40 , 80 , 60 . [Ans.]
Q.22. ABC is a right-angled triangle of given area S. Find the sides of the triangle for which the area of the circumscribed circle is least.
Solution :
Q.24. A box is to be constructed from a square metal sheet of side 60 cm by cutting out identical squares from the four corners and turning up the sides. Find the length of the side of the square to be cut out so that the box has maximum volume.
Solution :
Q.25. Find the shortest distance of the point C (0.c) from the parabola y = x2, c > 1/2.
Solution :
Let P(x, y) be any point on the given parabola y = x2, then | CP | = √{(x – 0)2 + (y – c)2} = √{y + (y – c)2} [ writing y for x2 as, y = x2] =√{y2 – (2c – 1)y + c2}. Or, | CP |2 = y2 – (2c – 1)y + c2 Now, | CP | is the shortest if and only if | CP | 2 is the shortest. Writing, | CP |2 as f(y), we get f(y) = y2 – (2c – 1)y + c2 ------------ (i) f’(y) = 2y – (2c – 1) and f”(y) = 2. Now, f’(y) = 0 => 2y – (2c – 1) = 0 Or, y = (2c – 1)/2. Hence, f”{(2c – 1)/2} = 2 > 0. Therefore, f(y) is minimum when y = (2c – 1)/2 i.e. | CP | is minimum when y = (2c – 1)/2 and the minimum value of | CP | = √[{(2c – 1)/2} + {(2c – 1)/2 – c}2] = √[(2c – 1)/2 + 1/4] = √[(4c – 1)/2]. [Ans.]
Q.26. An enemy vehicle is moving along the curve y = x2 + 2. Find the shortest distance between the vehicle and our artillery located at (3, 2). Also find the co-ordinates of the vehicle when the distance is shortest.
Solution :
[when x = 1, y = 12 + 2 = 3. Thus the co-ordinates of the vehicle when the distance is the shortest are (1, 3). [Ans.]
Q.27. Given the sum of the perimeters of a square and a circle, that the sum of their areas is least when the side of the square is equal to the diameter of the circle.
ASSIGNMENT(continuity & differentiability) (XII)
**Question 1 Determine a and b so that the function f given by
f(x) = 1−sin ² x3 cos ² x , x<п/2
=a, x=п/2
= b(1−sinx)(п−2x ) ² , x>п/2
Is continuous at x=п/2.
Answer [a = 1/3 , b = 8/3]
**Question 2 Find k such that following functions are continuous at indicated point
(i) f(x) ={1−cos4 x8x ²
, x ≠0
k , x=0 at x=0
(ii) f(x) = (2x+2 - 16)/(4x – 16) , x≠2
= k, x = 0 at x=2.
Answer [ (i) k=1,(ii) k=1/2]
**Question 3 The function f is defined as {x ²+ax+b ,0≤x<23 x+2 ,2≤x ≤4
2ax+5b ,4<x ≤8
If f(x) is continuous on [0,8], find the values of a and b.
Answer [a=3,b=-2]
** Question 4 If f(x) = {√1+ px−√1−pxx
,−1≤x<0
2 x+1x−1
,0≤ x≤1 is continuous in the [-
1,1], find p.
Answer [p=-1]
**Question 5 Find the value of a and b such that the f(x) defined as
f(x) = { x+a√2 sinx ,0≤x<п /42 xcotx+b ,п/ 4≤ x ≤п/2
acos2 x−bsinx , п2< x≤п is continuous for all values of x in
[0,п].
ANSWER [a=п/6 , b=-п/12]
** Question 6 Prove that limx→π /4
tan3 x−tanx
cos (x+ √π4
) = -4
[ Hint: Nr. Can be written as tanx(tanx-1)(tanx+1) =- [tanx(cosx-sinx)(tanx+1)]/cosx Cosx-sinx = √ 2 cos¿) ]
**Question 7 Prove that (i) limx→ 1
√ 2
x−cos (sin−1 x )1−tan (sin−1 x ) = −1
√2 [ Hint: put x=
sin ]Ѳ
(ii) limx→∞ x ¿¿ ) = -3/2. [Hint: π4 = tan−11 & use formula of tan−1 x−tan−1 y
]
Question 8 f(x) = a x2+bx2+1
, limx→0f (x ) =1 & limx→∞ f (x) =1, then p.t. f(-
2)=f(2)=1. [ Hint: limx→∞
1x2 =0]
Question 9 limx→0
e x−1√1−cosx [Dr. = √2|sinx/2| &lim
x→0
e x−1x =1
|sinx/2| =+ve & -ve as x→0+ & x→0- , ⇨ limit does not exist]
Question 10 Show that the function
f(x)¿{sin 3 xtan 3 x
, x<0
32, x=0
log(1+3 x)e2x−1
, x>0
is continuous at x=0.
[Hint: use limx→0
e x−1x =1 , lim
x→0
log (1+x)x =1]
Question11 Show that f(x) = |x-3|,x∊R is cts. But not diff. at x=3.
[Hint:show L.H.lt=R.H.lt by |x-3| = x-3, if x ≥3 and –x+3, if x<3, L.hd=-1≠1(R.h.d)
QUESTION 12 Discuss the continuity of the fn. f(x) = |x+1|+|x+2|,
at x = -1 & -2 [Hint:f(x) = {−2x−3 ,when x←21 ,when−2≤ x←1
2 x+3 ,when x ≥−1 yes cts. At x=-1,-2
Question 13 Find the values of p and q so that f(x) ={x ²+3x+ p , if x≤12x+2 , if x>1 is
diff. at x = 1. [ answer is p=3 , q=5]
Question 14 For what choice of a, b, c if any , does the function
F(x) = {ax ²+bx+c ,0≤ x≤1bx−c ,1<x ≤2c , x>2
becomes diff at x=1,2 & show that a=b=c=0.
Question15For what values a,b f(x)={ e2 x−1 ,when x≤0
ax+ bx ²2,when x>0 is diff.at x=0
[Hint: L.H.d= 2by using limx→0
ex−1x =1& R.H.d=a, since f‘(x)=0exists, a=2,b∊R]
Q. 16 Discuss the diff. Of f(x) = | x-1| + |x-2|
[ Hint: we have f(x)= {−2x+3 , x<11 ,1≤ x<2
2x−3 , x ≥2 to be examined not diff. At x=1,2]
ASSIGMENT OF DIFFERENTITION
Question 1 Show that y = aex and y = be –x cut at right angles ab=1 [ by equating , we get ex = √ ba ⇨ x= ½ log ( b/a) , find slopes(dy/dx) at pt. of intersection is (½ log ( b/a , √ab).
Question 2 (i) If y√1−x ² + x√1− y ² = 1, prove that
dy/dx= (-1)√ 1− y ²1−x ²
[Hint: put y=sinѲ & x= sinφ , use formula of sin(θ+φ ¿¿
(ii) If cos-1¿ ) = tan-1a , find dy/dx. [let cos(tan-1a )= k(constant), then assume c= 1-k/1+k , dy/dx= y/x] (iii) If y x = e y−x, prove that dy/dx = (1+ logy ) ²
logy (iv) If xm.yn = (x+y)m+n, then find dy/dx. [ y/x]Question 3 Differentiate w.r.t. x : **(i) Using logarithmic differentiation, differentiate:
Solution:
(ii) x tanx + √ x ²+1x (iii) (iogx)x + xlogx
Question 4 (i) If y x = e y−x, prove that dy/dx = (1+ logy ) ²logy
(ii) If f(1)= 4,f’(1)=2,find d/dx{logf(ex)} at the point x =0.[1/2](iii)If y = √ x+√x+√ x+…….∞ ,show that (2y – 1)dy/dx =1.(iv) If x = (t+1/t)a , y= a(t+1/t) where a>0,a≠1,t≠0, find dy/dx.
[Hint: take dy/dt & dx/dt , then find dy/dx = ylogy/ax. ]Question5(i)differentiate: Sec-1(1/(2x2 – 1)),w.r.t.sin-1(3x –4x3).[Hint: let u=1st fn. & v= 2nd fn. , find du/dv = 1](ii)differentiate: tan-1 (√1+x ²−1
x ),w.r.t. sin-1 ( 2x1+ x ² ) if -
1<x<1;x≠0[ du/dv= ¼, put x=tan ⇨ u= /2, v=2 , u&v as assumedѲ Ѳ Ѳ above](iii) If y = e(msin-1x) , show that (1-x2)y2 – xy1 – m2y= 0.Question 6 Water is driping out from a conical funnel, at the uniform rate of 2cm3/sec. through a tiny hole at the vertex at the bottom. When the slant height of the water is 4cm.,find the rate of decrease of the slant height of the water given that the vertical angle of the funnel is 1200 .[Hint: Let l is slant height ,V = 1/3.π .l(√ 3/2)2.l/2=πl3/8(vertical angle will be 600 (half cone), take dv/dt=-2cm3/sec. l=-1/3⇨ π cm/s.]**Question 7(i) Let f be differentiable for all x. If f(1)=-2 and if f `(x) ≥2 ∀ x∊[1, 6], then prove f(6) ≥8.[ use L.M.V.Thm.,f`(c)≥2,c∊[1, 6]](ii) If the function f(x)= x3 – 6x2+ax+b defined on [1, 3] satisfies the rolle’s theorem for c = (2√ 3 +i)/√ 3 , then p.t. a = 11 & b∊R.[Hint: Take f(1)=f(3) , use rolle’s thm. f`(c)=0⇨ a=11]Question 8 (i) Show that f(x)= x/sinx is increasing in (0, п/2)[HINT: f’(x)>0 , tanx >x](ii) Find the intervals of increase and decrease for f(x) = x3 + 2x2 – 1.[Answer is increasing in (-∞, -4/3)U(0, ∞) & decreasing in (-4/3, 0)](iii) Find the interval of increase & decrease for f(x) =log(1+x)-(x/1+x) ORProve that x/1+x < log(1+x) < x for x > 0.[ Hint: f(x)strictly ↑ in [0, ∞) , x>0 ⇨f(x)>f(0), let g(x)=x-log(1+x)g(x)>0 ↑ in [0,∞) & f(x) ↓ in (-∞, 0].](iv) For which value of a , f(x)=a(x+sinx)+a is increasing. [Hint: f’(x) a(1+cosx) ≥0 ⇨ a>0 ∵ -1≤cosx≤1]
**Question 9 Problem: Using differentials, approximate the expression
Solution: We let
Hence, x = 0.05 and y = /4.
Differentiating, we obtain
Substituting, we get
Question 10 For the curve y = 4x3 − 2x5, find all the points at which the tangents passes through the origin. [Hint: eqn. Of tangent at (x0,y0) , put x,y=0,(x0,y0)lies on given curve]Question 11 Find the stationary points of the function f(x) = 3x4 – 8x3+6x2 and distinguish b/w them. Also find the local max. And local mini. Values, if they exist.[ f’(x)=0⇨ x=0,1 f has local mini. At x=0∵f’’>0 & f’’(1)=0, f has point of inflexion at x=1,f(1)=1] Question 12 Show that the semi – vertical angle of right circular cone of given total surface area and max. Volume is sin-1 1/3.[Hint: take S=Пr(l+r) ⇨ l= S/пr – r , take derivative of V² OR can use trigonometric functions for l & h]
Question 13 A window has the shape of a rectangle surmounted by an equilateral ∆. If the perimeter of the window is 12 m., find the dimensions of the rectangle so that it may produce the largest area of the window.[Hint: let x=length, y=breadth, then y=6 – 3y/2, A= xy+√3x2 /4, take derivative of A & it is max. ,x=4(6+√3)/11 ,y=6(5−√3)/11]
Application of Derivatives
Q. 1. The volume of a cube is increasing aa constant rate. Prove that the increase in surface area varies inversely as the length of the edge of the cube.
Q. 2. Use differentials to find the approximate value of
Q. 3. It is given that for the function f(x) = x3 – 6x2 + ax + b on [1, 3],
Rolle’s theorem holds with c = 2+ . Find the values of a and b if f(1)= f(3) = 0
Q. 4. Find a point on the curve y = (x – 3)2, where the tangent is parallel to the line joining (4, 1) and (3, 0).
Q. 5. Find the intervals in which the function f(x) = x4 – 8x3 + 22x2 – 24x + 21 is decreasing or increasing.
Q. 6. Find the local maximum or local minimum of the
function.
Q. 7. Find the point on the curve y2 = 4x which is nearest to the point (2, 1).
Q. 8. A figure consists of a semi-circle with a rectangle on its diameter. Given the perimeter of the figure, find its dimensions in order that the area may be maximum.
Q. 9. A balloon which always remain spherical has a variable
diameter . Find the rate of change of its volume with respect to x.
Q. 10. Find the intervals in which f(x) = (x+1)3 (x – 3)3 is strictly increasing or decreasing.
Q. 11. Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1
Q. 12. Using differentials, find the approximate value of (26.57)1/3
Q. 13. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Q. 14. Find the equation of the tangent and normal to the hyperbola
at the point (x0,y0)
Q. 15. Find the intervals of the function is strictly increasing or strictly decreasing.
Q. 16. An open topped box is to be constructed by removing equal squares from each corner of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the sides. Find the volume of the largest such box.
Q. 17. Prove that the volume of the largest cone that can be inscribed
in a sphere of radius R is of the volume of the sphere.
Q. 18. Show that the right circular cylinder of given surface area and maximum volume is such that its height is equal to the diameter of the base.
Q. 19. The sum of the perimeter of a circle, and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.
Q. 20. A window is in the form of a rectangle surmounded by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
Q. 21. Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?
Q. 22. For the curve y = 4 x3– 2 x5 , find all the points at which the tangent passes through the origin.
Q. 23. An Apache helicopter of enemy is flying along the curve given by y = x2+ 7. A soldier, placed at (3, 7), wants to shoot down the helicopter when it is nearest to him. Find the nearest distance.
Q. 24. A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum ?
Q. 25. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
Q. 26. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank
Q. 1. If y = x4 - 10 and if x changes from 2 to 1.99, what is the approximate change in y.
Q. 2. A circular plate expands under heating so that its radius increases by 2%. Find the approximate increase in the area of the plate if the radius of the plate before heating is 10 cm.
Q. 3. Find the approximate value of f(3.02) when f(x) = 3x2+5x+3.
Using differentials find the approximate value of
Q. 4.
Q. 5.
Q. 6. tan46o. given 1o = 0.01745 radians.
Answers
1. 5.682. 4p3. 45.464. 5.025. 0.19256. 1.03490
Q. 1. Find intervals in which the function given
by is
(a) strictly increasing (b) strictly decreasing.
Q. 2. Show that the function f given by f (x) = tan–1(sin x + cos x), x > 0 is
always an strictly increasing function in
Find the intervals in which the following functions are increasing or decreasing
Q. 1. f(x) = -x2 - 2x +15.
Q. 2. f(x) = 2x3 +9x2 +12x +20
Q. 3. (x+1)3(x-3)3.
Q. 4. x4 - .
Q. 5. f(x) sin3x.
Q. 6. f(x) = sinx + cosx.
Q. 7. f(x) sin4x + cos4 x on [0, p/2]
Q. 8. f(x) = log(1+x) - .
Answers
1.
2.
3.
4.
5.
6.
7.
8.
Q. 1. Find the maximum slope of the curve y = -x3 + 3x2+ 2x - 27. and what point is it
Q. 2. A right circular cone of maximum volume is inscribed in a sphere of radius r. find its altitude. Also show that the maximum volume of the cone is 8/27 times the volume of the sphere.
Q. 3. A point on the hypotenuse of a triangle is at a distance a and b from the sides of the triangle. Show that the maximum length of the
hypotenuse is
Q. 4. From a piece of tin 20cm. in square, a simple box without top is made by cutting a square from each corner and folding up the
remaining rectangular tips to form the sides of the box. What is the dimension of the squares is cut in order that the volume of the box is maximum.
Q. 5. If length of three sides of a trapezium other than base are equal to 10cm, then find the area of the trapezium when it is maximum.
Q. 6. Find the shortest distance of the point (o,c) from the parabola y = x2, where 0 £ x £ 5.
Q. 7. A window consists of a rectangle surmounted by a semicircle. If the perimeter of the window is p centimetres, show that the window will allow the maximum possible light when the radius of the semi
circles cm.
Q. 8. Show that the semi vertical angle of the cone of given surface
area and maximum volume is .
Q. 9. A wire of length a is cut into two parts which are bent respectively in the form of a square and a circle. Show that the least
value of the areas so formed is .
Q. 10. Show that the volume of the greatest cylinder which can be
inscribed in a cone of height h and semi vertical angle a is
Q. 11. An open tank with square base and vertical sides is to be constructed from metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half the width.
Q. 12. Find the area of the greatest isosceles triangle that can be inscribed in a given ellipse
having its vertex coincident with one end of the major axis.
Q. 13. Find the maximum and minimum points for the following:
i.
ii.
iii.
Q. 14. The section of a window consists of a rectangle surmounted by an equilateral triangle. If the perimeters be given as 16m. find the dimensions of the window in order that the maximum amount of light may be admitted.
Q. 15. A square tank of capacity 250 cu.m has to be dug out. The cost of land is Rs. 50.per sq.m. The cost of digging increases with the depth and for the whole tank is 400(depth)2 rupees. Find the dimensions of the tank for the least total cost.
Q. 16. Find the dimensions of the rectangle of greatest area that can be inscribed in a semi circle of radius r.
Q. 17. A running track of 440 ft is to be laid out enclosing a football field, the shape of which is rectangle with semicircle at each end. If the area of the rectangular portion is to be maximum find the length of the sides.
Q. 18. Find the maximum and minimum values of y = |4-x2|, -3 £ x £ 3. Also determine the greatest and least values.
Q. 1. Whether
.
If so find the point of contact.
Q. 2. Find points at which the tangent to the curve y = x3 – 3x2– 9x + 7 is parallel to the x-axis.
Q. 3. Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 andx = – 2 are parallel.
Q. 4. Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
Q. 5. Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.
Q. 6. Find the equation of the tangent to the curve which is parallel to the line 4x - 2y + 5 = 0
Q. 7. Find the equation of the tangent to
.
Q. 8. Find the equation of the tangent and normal to
Q. 9. Find the tangent and normal to
Q. 10. Show that the normal at q to x = acosq+aqsinq and y = asinq-aqcosq is at constant distance from the origin.
Q. 11. Find the equation of the normal to x3 +y3 = 8xy where it meet y2 = 4x other than the origin.
Q. 12. Show that touches at the point where the curve crosses y-axis.
Q. 13. Find the angle of intersection of the curves xy = a2
and
Q. 14. Find the equation(s) of normal(s) to the curve 3x2 - y2 = 8 which is (are) parallel to the line x+3y = 4.
Ans: x+3y-8 = 0 and x+3y+8 = 0}
Q. 15. For the curve y = 4x3 -2x5, find all the points at which the tangent passes through the origin.
Ans: (0,0),(1,2),(-1,-2)
Q. 16. Prove that the sum of intercepts of the tangent to the
curve with the co-ordinate axes is constant.
ASSIGNMENT OF INTEGRATION
Question 1 Evaluate: (i)** Integrate .[ Use the power substitution
Put ]
** (iii) Integrate . [ Use the power substitution Put ]
(iii) ∫0
π /4
secx tan3 x dx [answer is (2 - √2)/3 ]
(iv) ∫ dx [multiply÷ by sin(a-b)] (v)∫ √ 1−√ x
1+√x dx
[multiply & divide by √1−√ x ] (Vi)∫ xx3−1
dx [by partial fraction]
(v)∫ (x−4)ex
(x−2)3 dx [ use ∫ex(f(x)+f’(x))dx] (vi)∫o
π /2 dx3+2 sinx+cosx dx [put sinx=
2 tanx /21+ tan ² x /2, cosx =1−tan ² x /2
1+tan ² x /2 , then put t=tanx/2. Answer is tan−12 – п/2]
(vii) ∫0
3 /2
¿ xcosπx∨¿¿ dx [∫0
1 /2
¿ xcosπx∨¿¿+ ∫1 /2
3 /2
¿ xcosπx∨¿¿ = ∫+ve dx+∫ -ve dx ,
answer is 5/2п- 1/п2] (viii) [ write sin2x = 1-cos2x answer is
п/6] (ix) ∫0
π /2
√ tanx + √cotx dx [ answer is √2π] (x) ∫0
a
sin−1 √ xa+x dx [ put x=atan2Ѳ
, answer is a/2( -2) ] п (xi) ∫0
π x1+sin ² x dx [ use property ∫
0
a
f ( x )dx = ∫0
a
f (a− x )dx , ∫0
2a
f ( x )dx =2∫0
a
f ( x )dx ∵f(2a-x) = f(x) , then put t=tanx, answer is /2√2 ] п² (xii) ∫
−5
0
f ( x ) dx , where f(x) =|x|+|x+2|+|x+5|. [∫−5
−2
(−x+3)dx + ∫−2
0
( x+7) dx , answer is 31.5 ] (xiii) Evaluate ∫ ex (1−x ) ²
( x ²+1 ) ² dx
[use ∫ ex(f(x)+f’(x))dx
Question 2 Using integration, find the area of the regions: (i) { (x,y): |x-1| ≤y ≤√5−x ² } (ii) {(x,y):0≤y≤x2+3; 0≤y≤2x+3; 0≤x≤3}[(i) A= ∫
−1
2
√5−x ²dx- ∫−1
1
(−x+1 )dx - ∫1
2
(x−1) dx = 5/2 [ sin−1( 2√5
) +sin−1( 1√5
)
] – ] [(ii) ½ A=∫0
2
(x ²+3¿)¿ dx +∫2
3
(2x+3)dx , answer is 50/3](iii) Find the area bounded by the curve x 2 = 4y & the line x = 4y – 2.[A = ∫
−1
2 x+24 dx - ∫
−1
2 x ²4 dx = 9/8 sq. Unit.]
**(iv) Sketch the graph of f(x) = {|x−2|+2 , x≤2x ²−2 , x>2 ,evaluate∫
0
4
f (x ) dx[hint: ∫
0
4
f (x )dx = ∫0
2
(4−x )dx + ∫2
4
(x ²−2) dx = 62/3.]**Question 3 evaluate ∫ √1+ x
√ x dx [ mult. & divide by √1+x , put 1+x =A.(d/dx)(x2+x)+B ,find A=B=1/2, integrate]Definite integral as the limit of a sum , use formula : ∫
a
b
f ( x )dx limh→ 0
∑r=1
n
f (a+rh),
where nh=b-a & n→∞ Question 4 Evaluate (i )∫0
4
¿¿) dx (ii) ∫0
3
(x ²−2 x+2) dx
[ use limh→0
eh−1h = 1 for part (i) , use formulas of special sequences, answer is 6]
Some special case :
(1) Evaluate: ∫ dx(x−3)√ x+1 [ put x+1=t²] (2) ∫ dx
(x ²−4)√ x+1 [ put x+1 = t² ]
(3) Evaluate: ∫ dx(x+1)√x ²−1 (4) Evaluate: ∫ dx
x ²√x ²+1 [ put x=1/t for both]
(5) Evaluate: ∫ (x ²+1)dxx4+1
[ divide Nr. & Dr. By x2 , then write x²+1/x²=(x-1/x)²
+2 according to Nr. , let x-1/x=t]
(6) Evaluate ∫ x √1+x−x ² dx [ let x=A(d/dx) ( 1+x-x²) +B]
(7) Integrating by parts evaluate ∫ x ²( xsinx+cosx ) ² = ∫ ( xsecx ) . xcosx
(xsinx+cosx ) ²
(8) Evaluate ∫ 11+cotxdx =∫ sinx
sinx+cosxdx [ put sinx=Ad/dx(sinx+cosx)
+B(sinx+cosx)+C
If Nr. Is constant term then use formulas of sinx,cosx as Ques. No. 1 (vi) part]
The important discussions in Differential Equations are as follows:
ASSESSMENT OF DIFFERENTIAL EQUATIONS FOR
CLASS—XII Level--1 Q.1 Find the order and degree of the following differential equations. State also whether they are linear or non-linear.
(i) X2( d ² ydx ² )3 + y ( dydx )
4 y4 =0. (ii) d ² ydx ² = 3√1+(
dydx ¿) ² ¿ .
Q.2 Form the differential equation corresponding to y2 = a (b – x)(b+ x) by eliminating parameters a and b. Q.3 Solve the differential equation (1+e2x) dy + (1+y2) ex dx = 0, when x= 0, y =1.
Q.4 Solve the differential equation: dydx = 1−cosx1+cosx .
Q.5 Verify that y = A cosx – Bsinx is a solution of the differential
equation d ² ydx ² + y = 0.
Answers of Level—11. (i) order =2 , degree=3 , non linear( because degree is more than 1 ) , (ii) order 2 , degree 3 , non-linear .
2. y2 = a(b – x)(b+ x) = a (b2 – x2), 2y dydx =-2ax ⇨ ydydx = -ax, again
differentiate Y d ² ydx ² + ¿ )2 = -a , by using the value of a from above
step , we will get , x{ Y d ² ydx ² + ¿ )2 } = ydydx .
3. dy1+ y ² = - exdx
1+e2 x , Integrating both sides, we get
tan−1 y = - ∫ exdx
1+e2 x , put ex = t⇨ tan−1 y = - tan−1t +c
Using x=0, y=1, we have y = 1/ex.
4. y = 2 tan(x/2) – x +c , put tan(x/2) = 1−cosx1+cosx
5. dydx = - A sinx – B cosx , d ² ydx ² = - A cosx + B sinx = -y.
Level---2 Q.1 Solve: y dx + x log ( yx ) dy – 2x dy = 0 .
Q. 2 Which of following transformations reduce the differential
equation dzdx + zx log z = zx ² ( log z ) ² into the form
dudx + P(x) u = Q(x) ?
(i) u = log x (ii) u = ex (iii) u = ¿-1 (iv) u = ( log z )2
Q.3 Solve: dydx +xy = xy3
Q.4 Solve: dydx = cos (x+ y) + sin(x+ y )
Q.5 Solve: dydx + x sin 2 y = x3 cos2y
Answers of Level ---2 1. put x = vy , answer = 1+ log ( yx ) = ky .
2. (iii) differentiate w.r.t. x dudx = - 1¿¿ .1
z dzdx , put the value of dzdx in
the given differential equation. 3. put 1y ² = t, answer is 1
y ² = 1 + c
ex ². 4. put x+y = v, answer is log(1+ tan (x+ y )2
) = x + c.
5. put tan y = v, I.F. = ex ² , also use sin 2 y = 2siny cosy.
Order of a Differential Equation.
Q.1. Write the order and degree of the following differential equation :
(dy/dx)4 + 3yd2y/dx2 = 0.
Solution :
We have, (dy/dx)4 + 3yd2y/dx2 = 0 Order of the differential equation = order of highest derivative = 2. [Ans.]
Degree of the differential equation = degree of highest derivative = 1. [Ans.]
Q.2. Write the order and degree of the differential equation :
(d2y/dx2)2 + (dy/dx)3 + 2y = 0
Solution :
We have, (d2y/dx2)2 + (dy/dx)3+ 2y = 0 Order = 2, Degree = 2. [Ans.]
Q.1. (i)Verify that y = A cos x – B sin x is a solution of the differential equationd2y/dx2 + y = 0.
(ii) Form the differential equation of the family of curves y = a sin (x + b), where a and b are arbitrary constants.
Q.2. Form the differential equation of the family of circles having centre on x-axis and passing through the origin.
Q.3. Verify that y = 3 cos (log x) + 4 sin (log x) is a solution of the differential equation,
x2 d2y/dx2 + x dy/dx + y = 0.
[Hence, y = 3 cos (log x) + 4 sin (log x) is a solution of x2 d2y/dx2 + x dy/dx + y = 0. [Proved.]
9.4. Differential Equations with Variables Separable.
Q.1. Solve the following differential equation :
dy/dx = log (x + 1).
[ (x + 1) log (x + 1) – x + c. [Ans.]
Q.2. Solve the following differential equation :
dy/dx = ex+yx+y + x2.ey.
[– ye–y = ex + x3/3 + c => ex + ey + x3/3 + c = 0 [Ans.]
Q.3. Solve the following differential equation :
x(1 + y2)dx – y(1 + x2)dy = 0, given that y = 0 when x = 1.
[ x2 – 2y2 – 1 = 0 [Ans.]
Q.4. Solve the following differential equation : (1 + e2x)dy + (1 + y2)ex dx = 0.tan-1y + tan -1ex = c. [Ans.]
9.5. Homogeneous Differential Equations.
Q.1. Solve the following differential equation : (y2 – x2) dy = 3xy dx.
[ – 1/4log(y/x) – 3/8 log |4 – y2/x2| = log x + c. [Ans.]
Q.2. Solve the following differential equation : 2xy dx + (x2 + 2y2) dy = 0. [3x2y + 2y3 = c. [Ans.]
Q.3. Solve the following differential equation : x dy/dx – y + x tan(y/x) = 0. [ x sin v = c => x sin(y/x) = c. [Ans.]
Q.4. Solve the following differential equation : (x2 – y2)dx + 2 xydy = 0, given that y = 1 when x = 1.
Q.6. Solve the following differential equation : x2 dy/dx = y2 + 2xy. Given that y = 1 when x = 1. [ y = x2/(2 – x). [Ans.]
Linear Differential Equations.
Q.1. Solve the following differential equation : sin x dy/dx + cos x.y = cos x.sin2x
[y = (1/3) sin2 x + c cosec x. [Ans.]
Q.2. Solve the following differential equation : dy/dx – y/x = 2x2.[Thus y = x3 + cx. [Ans.]
Q.3. Solve the following differential equation : dy/dx + (sec x).y = tan x.
[y = sec x + tan x – x + c. [Ans.]
Q.4. Solve the following differential equation : dy/dx + 2tan x .y = sin x.
[ y = cos x + c cos2 x. Ans. ]
Q.6. Solve the differential equation : (1 – x2)dy/dx + xy = ax.
[ y = a + c√(1 – x2). [Ans.]
Q.7. Solve the differential equation : dy/dx + 2y tan x = sin x, given that y = 0 if x = /3.π
[ y = cos x – 2 cos2 x. [Ans.]
Q.8. Solve the following differential equation : cos2 x dy/dx + y = tan x.
[ y = tan x – 1 + ce-tan x . [Ans.]
Q.9. Solve the following differential equation : (x2 + 1) dy/dx + 2xy = √(x2 + 4).[ x/2√(x2 + 4) + 2log|x + √(x2 + 4)| + c. [Ans.
Area of The Region Bounded by a Curve and a Line.
Q.1. Find the area of the region bounded by the parabola x2 = 4y and the line x = 4y – 2.
Solution :
= 9/8 sq. units. [Ans.]
Q.2. Find the area of the region bounded by y2 = 4x, x = 1, x = 4 and x-axis in the first quadrant.
Solution :
= 28/3 sq. units. [Ans.]
Area of triangle.
Q.1. Using integration, find the area of the triangle ABC, the coordinates of whose vertices are A(2, 0), B(4, 5) and C(6, 3).
Solution :
= 7 sq. units. [Ans.]
Q.2. Using integration find the area of the triangular region whose vertices are (1, 0), (2, 2) and (3, 1).
Solution : [Ans. = 3/2]
Q. 1. Evaluate as a limit of a sum.
Q. 2. Evaluate .
Q. 3. Evaluate .
Q. 4. Evaluate the integral .
Q. 5. Evaluate .
Q. 6. Evaluate
Q. 7. Evaluate
Q. 8. Evaluate
Q. 9. Prove that .
Q. 10. Evaluate .
Q. 1. Find the area lying above the x-axis and included between the circle x2+y2=8x and the parabola y2 =4x.
Q. 2. Find the area lying above the x-axis and included between the circle x2 +y2=16a2 and the parabola y2 =6ax.
Q. 3. Find the area of the smaller region bounded by the
ellipse and the line
Q. 4. Find the area of the region included between x2 =4y , y = 2 , y = 4 and the y-axis in the first quadrant.
Q. 5. Find the area between the parabolas 4ay = x2 and y2 = 4ax.
Q. 6. Find the area bounded by the curve y2 = 4ax and the line y = 2a and y-axis.
Q. 7. Find the area bounded by the parabola y2 = 8x and its latus rectum
Q. 8. Find the area of the circle x2 + y2 = 16, which is exterior to the parabola y2 = 6x.
Q. 9. Sketch the region common to the circle x2 + y2 = 8 and the parabola x2 = 4y. Also find the area of the common region using integration.
Q. 10. Draw the rough sketch of the region and find the area enclosed by the region using method of integration.
Q. 11. Using integration, find the area of the triangle ABC whose vertices are A(2,3), B(2,8) and C(6,5).
Q. 12. Using integration, find the area of the triangle ABC whose vertices are A(2,5), B(4,7) and C(6,2).
Q. 13. Using integration, find the area of the triangle ABC whose vertices are A(-1,1), B(0.5) and C(3,2).
Q. 14. Compute the area bounded by the lines x+2y = 2, y-x=1 and 2x+y = 7.
Q. 15. Compute the area bounded by the lines y = 4x+5, y = 5-x, and 4y = x+5.
Q. 16. Compute the area bounded by the lines 2x+y = 4, 3x-2y = 6, and x-3y+5=0.
Q. 17. Using integration, find the area of the region bounded by x-7y+19=0, and y =çxú.
Q. 18. Using integration, find the area of the region bounded by the line
i. 2y= -x+8, x-axis and the lines x = 2 and x = 4.ii. y -1 = x, x-axis and the lines x = -2 and x = 3
iii. y = , line y = x and the positive x- axis.
Q. 19. Find the area of the region enclosed between the two circles x2 + y2 = 1 and (x-4)2 + y2 =16.
Q. 20. Find the area of the region in the first quadrant enclosed by the x-axis, the line y = 4x and the circle x2 + y2 = 32
Q. 21. Find the area of the smaller part of the circle x2 + y2 =a2 cut off
by the line .
Q. 22. Find the area of the region
Q. 23. Sketch the graph of the curve y = and evaluate
Q. 24. Sketch the graph of the curve and find the area bounded by y =
, x=-2, x=3, y=0.
Q. 25. Find the area bounded by the line y = sin2x and y = cos2x between x = 0 and x=p/4
Q. 26. Prove that the curves y2 = 4x and x2 = 4y divide the area of the square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.
