calculus

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iii) f (x) = (3x + 4 tan x) / x x ≠ 0. k x = 0 ... 5ax – 2b x > 1. Determine the ..... If y = x + tan x , prove that cos2x (d2y / dx2) – 2y + 2x = 0 (1992, 2001 C). If y = A cos ...


1. CALCULUS CONTINUITY

Definitions and formula : 1. A function ‘f’ is said to be continuous at x = a if = f (a)

i.e = = f (a)

2. = (putting x = a – h)

= (putting x = a + h)

3. = 1 θ is measured in radian

4. = n a n-1

5. = e

Continuity : 1. Examine the continuity of the function given below at x = 3 f (x) = (2x2 – 7x + 3) / (x – 3) x ≠ 3 4 x = 3

2. Find the constant ‘k’ f (x) becomes continuous at x = 0 f (x) = (x4 + x3 + 2x) / (ex – 1) , x ≠ 0 k x = 0

3. Determine ‘k’ so that the following functions are continuous at the indicated points. i) f (x) = x2 / (ex – cos mx) x ≠ 0 k x = 0 at x = 1

ii) f (x) = (1 – x) tan (Πx / 2) x ≠ 1 k x = 1 at x = 0

iii) f (x) = (3x + 4 tan x) / x x ≠ 0 k x = 0 at x = 0

iv) f (x) = x2 sin (1 / x) + 2x2 + 3 x ≠ 0 k x = 0 at x = 0

v) f (x) = [√(1 + x) – √ (1 + x)] / x x ≠ 0 k x = 0 at x = 0

vi) f (x) = (1 – cos x) / sin2x x ≠ 0 k x = 0 at x = 0

vii) f (x) = sin 3x / sin 2x x ≠ 0 k x = 0 at x = 0

viii) f (x) = x sin (1/x) x ≠ 0 k x = 0 at x = 0

ix) f (x) = [log (1 + x)] / sin x x ≠ 0 k x = 0

x) f (x) = 1 if x ≤ 3 ax + b if 3 < x < 5 7 if 5 ≤ x Determine the values of a & b so that f(x) is a continuous function.

xi) f (x) = ax2 + b x < 2 2 x = 2 2ax – b x > 2 Determine the values of a and b so that f(x) is continuous at x = 2

xii) If f(x) = 3ax + b x < 1 11 x = 1 5ax – 2b x > 1 Determine the value of a and b so that f(x) is continuous at x = 1

xiii) If f (x) = (1 + 3x)1/x x ≠ 0 e3 x = 0 Show that f(x) continuous at x = 0

xiv) Find the relationship between ‘a’ and ‘b’ so that the function ‘f’ defined by f (x) = ax + 1 if x ≤ 3 bx + 3 if x > 3 is continuous at x = 3

4. if f(x) = | x | / x when x ≠ 0 1 when x = 0 Find whether f(x) is continuous at x = 0

5. Show that the function f(x) = ( | x | - x) / x x ≠ 0 k x = 0 is discontinuous at x = 0 whatever the value of k may be

6. If f(x) = (sin(a + 1)x) + (sin x)) / x if x < 0 = c if x = 0 = (√(x + bx2) – √x) / (bx3/2) if x > 0 is continuous at x = 0. Find the value of a, b and c.

7. Let f(x) = (1 – cos 4x) / x2 x < 0 a x = 0 √x / (√(16 + (√x)) – 4) x > 0 Determine the value of a so that f(x) is continuous at x = 0

8. If f(x) = (sin 3x) / x when x ≠ 0 1 when x = 0 Find whether f is continuous at x = 0 (CBSE 1992)

9. If f(x) = [ x ] / x when x ≠ 0 1 when x = 0 Find whether f is continuous at x = 0 (1992)

10. Discuss the continuity of the function f(x) at the point x = ½ where f(x) = x 0 ≤ x < ½ ½ x = ½ 1 – x ½ ≤ x ≤ 1

11. Discuss the continuity of the function f(x) = 2 – x x < 0 2 + x x ≥ 0 at x = 2

12. f (x) = (x2 – 9) / (x – 3) x ≠ 3 k x = 3 Find k is f(x) is continuous at x = 3 (1994 C)

13. Discuss continuity of the function f(x) at the origin, where f(x) = x / | x | x ≠ 0 1 x = 0 (1995 C)

14. For what value of k is the function f(x) = (Sin 2x) / x x ≠ 0 k x = 0 continuous at x = 0 ( 1996)

15. If f(x) = x / (sin 3x) when x ≠ 0 3 when x = 0 Find whether the function f(x) is continuous at x = 0 (1997)

16. If f(x) = (Sin 2x) / (Sin 3x) x ≠ 0 2 x = 0 ( 1997)

17. For what value of k is the following function continuous at x = 0 ? f(x) = (1 – cos 4x) / 8x2 x ≠ 0 2 x = 0 ( 2000 C)

18. Evaluate the left hand limit and right hand limit of the following function at x = 1 f(x) = 1 + x2 if 0 ≤ x ≤ 1 2 - x if x > 1 Does exist ? (2001 C)

19. Evaluate the left hand & right hand limits of the following functions at x = 1 f(x) = 5x – 4 if 0 < x ≤ 1 4x2 – 3x if 1 < x < 2 Does exist ? (2001 C) 20. Show that the function f(x) = 2x - | x | is continuous at x = 0 (2002)

21. If the function f(x) = 3ax + b x > 1 11 x = 1 5ax – 2b x < 1 (2002) Is continuous at x = 1, find the value of a and b

22. Discuss the continuity of the function f(x) at x = 0 If f(x) = 2x – 1 x < 0 2x + 1 x ≥ 0 (2002) Continuity Questions asked in CBSE sample papers 1. Find all the points of discontinuity of the function f defined by f(x) = x + 2 x ≤ 1 x – 2 1 < x < 2 0 x ≥ 2

2. Show that the function f(x) = | x + 2 | is continuous at every x Є IR but fails to be differentiable at x = - 2

3. If f(x) = [(x – 5) / (| x – 5 |)] + a if x < 5 a + b if x = 5 [(x – 5) / (| x – 5 |)] + b if x > 5 is continuous function. Find a and b.

4. 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 continuous function

5. Find the value of k so that the function f is continuous at the indicated point. f(x) = kx + 1 if x ≤ Π cos x if x > Π at x = Π

DERIVATIVES

Definitions and Formula 1. dy/dx is derivative of first order and is also denoted by y′ or y1

2. d/dx (constant) = 0

3. If u and v are both differentiable functions of x then a. d / dx ( u ± v ) = (du / dx) ± (dv / dx) b. d / dx ( uv) = u (dv / dx) + v (du / dx) c. d / dx ( u / v ) = [v (du / dx) – u (dv / dx)] / v2 d. d / dx k U = k (du / dx) , k Є IR

4. Derivative of [ f(x) ] is found by applying logarithm differentiation.

5. Marginal cost , MC = d / dx { C(x) }

6. Marginal revenue , MR = d / dx { R(x) }

7. a. d/dx xn = n xn-1 b. d/dx (Sin x) = cos x c. d/dx (Cos x) = - sin x d. d/dx (tan x) = sec2x e. d/dx (sec x) = sec x tan x f. d/dx (cosec x) = - cosec x . cot x g. d/dx (Cot x) = - cosec2x h. d/dx ax = ax log a i. d/dx ex = ex j. d/dx √x = 1 / 2√x k. d/dx (log x) = 1 / x l. d/dx (sin-1x) = 1 / √(1 – x2) m. d/dx (Cos-1x) = - 1/ √(1 – x2) n. d/dx (tan-1x) = 1 / (1 + x2) o. d/dx (Cot-1x) = - 1 / (1 + x2) p. d/dx (sec-1x) = 1 / (x √(x2 – 1)) q. d/dx (Cosec-1x) = - 1 / (x √(x2 – 1))



Differentiate each of the following w.r.t. x

1. (sin x) (1992)

2. x (1992, 2000 C)

3. (sin x + x2) / cos 2x (1993)

4. (ex + log x) / sin 3x (1993)

5. sin x sin 2x (1994 C)

6. tan-1(cos x / (1 + sin x)) (1998)

7. tan-1((1 – cos x) / sin x) (1998)

8. 5x /( √(1 – x2 )) + sin2(2x + 3) (2000 C)

9. e (2006)

10. log (x + √ (1 – x2)) (2003)

11. log (sin √(x2 + 1)) (2003)

12. e + (tan x)x (2003)

13. tan-1 [(√(1 + x) – √(1 – x)) / (√(1 + x) + √(1 – x))] (2003)

14. If x = at2 and y = 2at , find d2y / dx2 (1992)

15. If y = x + tan x , prove that cos2x (d2y / dx2) – 2y + 2x = 0 (1992, 2001 C)

16. If y = A cos nx + B sin nx , prove that (d2y / dx2) + n2y = 0 (1992, 2001 C)

17. Find dy/dx when x = a(1 – cos θ) , y = a( θ + sin θ) (1995, 1996)

18. Differentiate sin x2 w.r.t x2 (1995)

19. Find dy/dx when y = a(θ + sin θ) & x = a(1 + cos θ) (1995)

20. Differentiate log x w.r.t (1/x) (1995 C)

21. Find dy/dx when x = at2 , y = 2at (1995 C)

22. Differentiate tan-1(2x / (1 – x2)) w.r.t tan-1 x

23. Differentiate sin-1(2x / (1 + x2)) w.r.t tan-1 x (1996)

24. If y = sec x – tan x , show that cos x (d2y/dx2) = y2 (1999)

25. Differentiate cos-1θ w.r.t log(1 + θ)

26. Differentiate log(1 + θ) w.r.t sin-1 θ

27. If y = log √((1 – cos x) / (1 + cos x)) , then show that dy/dx = cosec x (2003)

4 marks Questions : 28. If y = tan x + sec x , then prove that (d2y/dx2 ) = cos x / (1 – sin x)2 (1992 C)

29. Differentiate tan-1 ((√1 + x2 + 1) / x ) w.r.t x (1993)

30. Differentiate tan-1 (√(1 + x2 ) + x) w.r.t x (1993)

31. Prove that d/dx (cos-1 (√(1 + x) / 2)) = - 1 / (2 √(1 – x2)) (1993 C)

32. Differentiate sin-1(3x – 4x3) w.r.t x (1994 C)

33. Differentiate sin-1(1 / √(1 + x2)) w.r.t x (1994)

34. If y = tan-1 x , then show that (1 + x2) (d2y/dx2 ) + 2x (dy/dx) = 0 (1996 C, 1999)

35. Find dy/dx when y = x + [(x2 – 1) / (x2 + 1)] (1997)

36. Find dy/dx when y = x + [(2x2 – 3) / (x2 + x + 2)] (1997)

37. If y = tan-1 x + sec x , prove that d2y/dx2 = cos x / (1 – sin x)2 (1997 C)

38. If xp . yq = (x + y)p+q , prove that dy/dx = y / x (2000)

39. Find d2y/dx2 when y = log(x2 / ex) (2000)

40. If y = ex(sin x + cos x) , prove that (d2y/dx2 ) – 2 (dy/dx) + 2y = 0 (2002)

41. If y = √((1 – sin 2x) / (1 + sin 2x)) , show that (dy/dx) + sec2((Π / 2) – x) = 0 (2002)

Ans : 1. (– 3 sin 3x) / (2 √cos 3x)

2. 1 / √(2x + 3)

3. (cos x) / (2 √sin x)

4. (– sin x) / (2 √cos x)

5. – 3 / (4x7/ 4)

6. (cos 2x) / (√sin 2x)

7. e√2x / √2x

8. sec2√x / 2√x

9. – cosec2 √x / 2√x

10. – sin √x / 2√x

11. x2 – 1 / x2

12. 1 / 1 + x2

13. – ½ cot x √ cosec x

14. ½ tan x √sec x

15. – 3 sin(3x + 2)

16. 2x cos(x2 + 1)

17. – 2x sin (x2 + 1)

18. 2x cos x2

19. 2x sec2x2

20. – sin 2x

21. (sin x) (Cos-1 x . cot x . (log sin x) / √(1 – x2 ))

22. x ((sin-1 x / x) + ((log x) / √(1 – x2)))

23. tan 2x (cos x + 2x) + 2 sec22x (x2 + sin x )

24. (sin 3x (ex + (1 / x)) – 3 (ex + log x) cos 3x) / sin23x

25. – ½ sin x + (3/2) sin 3x

26. – ½

27. ½

28. [(15 – 5x2) / (3 (1 – x2) 4/3 ] + 2 sin (4x + 6)

29. 2xe

30. 1 / √ (1 + x2)

31. [x / √ (x2 + 1)] . cot (√ (x2 + 1))

32. cos x esin x + (tan x)x [ x sec x . cosec x + log tan x ]

33. 1 / 2√(1 – x2)

34. – 1 / (2at3)

37. cot (θ / 2)

38. cos x2

39. – cot (θ / 2)

40. – x

41. 1 / t

42. 2

43. 2

45. – [√((1 + θ) / (1 – θ))]

46. √((1 – θ) / (1 + θ))

49. – 1 / (2(1 + x2))

50. 1 / (2(1 + x2))

52. 3 / (√(1 – x2))

53. – 1 / (1 + x2)

55. x sin x – cos x [((cos x + sin x) log x) + ((sin x – cos x) / x) ] + (4x / (x2 + 1)2)

56. x cot x [((cot x / x) – (cosec2x . log x))] + [(2x2 + 14x + 3) / (x2 + x + 2 )2]

59. – 2 / x2

DERIVATIVES

1. Differentiate y = xsin x + (sin x)cos x w.r.t x

2. Differentiate tan-1 (2x / (1 – x2)) w.r.t sin-1 (2x / (1 + x2))

3. Differentiate y = (sin x)cos x + 2 tan x

4. Differentiate tan-1 (2x / (1 – x2)) w.r.t cos-1 ((1 – x2) / (1 + x2))

5. If y = ex (a cos x + b sin x) , then show that (d2y / dx2) - 2 (dy/dx) + 2y = 0

6. If y = (sin x)tan x + (cos x)sec x , find dy/dx

7. Find dy/dx when x = sin3t / √cos 2t , y = cos3t / √ cos 2t

8. If y = 3 cos (log x) + 4 sin (log x) , show that x2y2 + xy1 + y = 0

9. Differentiate tan-1 { (√(1+x2) – √(1 – x2)) / (√(1 + x2 ) + √(1 – x2 ))} w.r.t x

10. Find dy/dx if x = √ a , y = √ a

11. Prove d/dx [ (x / 2) √(a2 – x2) + (a2 / 2) sin-1 (x / a) ] = √ (a2 – x2)

12. If (cos x )y = (cos y)x , find dy/dx

13. If y = √ sin x + (√ sin x + (√ sin x + …. to ∞ )) , prove that dy/dx = (cos x / (2y – 1))

14. Find dy/dx if y = tan-1 [(√(1+x2) + √(1 – x2)) / (√(1 + x2 ) - √(1 – x2 ))] , 0 < | x | < 1

15. If x = a sin pt and y = b cos pt , find the value of d2y / dx2 at t = 0

16. If xy + yx = log a , find dy/dx

17. Differentiate sin-1 (2x + 1 / (1 + 4x)) w.r.t. x

18. If x = a (θ – sin θ) , y = a (1 + cos θ) , find d2y / dx2 at θ = Π / 2

19. Differentiate cot-1 [ (√(1+sin x) + √(1 – sin x)) / (√(1 + sin x) - √(1 – sin x)) ] , 0 < x < Π / 2 w.r.t. x

20. If (x √(1 + y)) + (y √(1 + x)) = 0 , for - 1 < x < 1 , prove that dy/dx = - 1 / (1 + x)2

21. If xy = ex – y , show that dy/dx = log x / [(1 + log x)2]

22. Find dy/dx if yx + xy + xx = ab

23. If x = sin t , y = sin mt , prove that (1 – x2) (d2y/dx2 ) – x (dy/dx) + m2y = 0

24. Find d2y/dx2 if y = (3at) / (1 + t) , x = (2at2) / (1 + t)

25. If y = (cos x)log x + (log x)x , find dy/dx

26. If x = a (Cos t + log tan (t / 2)) , y = a sin t , find d2y / dx2 at t = Π / 4

27. If y = b tan-1 ((x / a) + tan-1 (y / x)) find dy/dx

28. If √(1 – x2) + √(1 – y2) = a (x – y) , show that dy /dx = √((1 – y2) / (1 – x2))

APPLICATIONS OF DERIVATIVES 1) A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the rectangle that will produce the largest area of the window.

2) A point on the hypotenuse of a right triangle is at a distance of ‘a’ and ‘b’ from the sides of the triangle. Show that the minimum length of the hypotenuse if ( a[pic]+ b [pic])[pic]

3) Find the maximum area of an isosceles triangle inscribed in ellipse [pic]+[pic]=1, with one end of major axis as its vertex.

4) Find the point on the curve y[pic]=2x which is at a minimum distance from the point(1,4)

5) A right circular cylinder is inscribed in a right circular cone. Show that the curved surface area of the cylinder is maximum when the diameter of cylinder is equal to the radius of the base of the cone.

6) Show that the right triangle of maximum area that can be inscribed in a fixed circle is an isosceles triangle.

7) A window in the shape of a rectangle is surmounted by an equilateral triangle, the perimeter of the window is 100 meters. Find the dimensions if the area is maximum

8) If the length of three sides of a trapezium other that base are equal to 10 cm, then find the area of trapezium when it is maximum.

9) Find the shortest distance between t he line y-=1 and the curve x=y[pic].

10) Show that the height of a cylinder of maximum volume that can be inscribed in a sphere of radius R is [pic]. 11) A wire of length 20 m is to be cut into two pieces one of the pieces will be bent into the shape of a square and the other into the shape of an equilateral triangle. What should be the length of the two pieces so that the sum of the areas of the square and triangle is maximum?

12) Show that the semi-vertical angle of a right circular cone of given surface area and maximum volume is sin-1 [pic]. 13) Show that the rectangle of maximum area that can be inscribed in a circle of radius r is a square of side [pic] r.

14) The sum of the length of the hypotenuse and a side of a right triangle is given,. Show that the area of the triangle is maximum when the angle between them is [pic] 15) A wire of length 36 cm is cut into two pieces. One of the pieces is turned in the form of a square and the other in the form of an equilateral triangle. Find the length of each piece so that the sum of the area of the two be minimum.

16) Find the volume of the largest cylinder that can be inscribed in a sphere of radius r.

17) 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 m[pic]. if building of tank costs Rs. 70 per sq. metre for the base and Rs. 45 per sq. metre for sides, what is the cost of least expensive tank? 18) A manufacturer can sell x items at a price of Rs.[pic]5 - [pic]each. The cost price of x items is Rs. [pic] + 500 . Find the number of items he should sell to earn maximum profit.

19) Show that the total surface area of a closed cuboid with square base ad given volume is minimum, when it is a cube?

20) Show that the height of the cylinder of maximum volume that can be inscribed in a cone of height H is [pic].

INDEFINITE INTEGRALS DEFINITIONS AND FORMULA I) INTEGRATION: It is the process of finding antiderivative (ie primitive) of a function. i.e. If [pic][pic]=[pic], then [pic]

II) 1) [pic], [pic] 2) [pic] 3) [pic]

III) 1) [pic] 2) [pic] 3) [pic][pic]4)[pic]

5) [pic] 6)[pic] 7) [pic]

8) [pic] 9) [pic] 10) [pic] 11) [pic] 12) [pic] 13) [pic]

14) [pic] 15)[pic] 16) [pic] 17) [pic] 18) [pic] 19) [pic] 20) [pic] 21) [pic] 22) [pic] 23) [pic] 24) [pic][pic][pic][pic] 25) [pic]

IV) TYPES OF INTEGRATION AND THEIR WORKING RULE

1) [pic], [pic]

W.R: [pic]

2) [pic] W.R.: put [pic]

3) [pic] W.R: put [pic]

4) [pic] [pic] [pic] [pic] [pic] [pic] [pic]

W.R: write [pic] [pic] [pic][pic] 5) Integration of odd powers of [pic]& [pic] W.R: Write [pic] or [pic] = [pic] or =[pic]

5) [pic] W.R. use : [pic]

6) [pic]

W.R : put [pic] and [pic] where [pic]& [pic]

7) [pic] or [pic] or [pic] W.R: use completing square method.

8) [pic] or [pic] or [pic][pic]

W.R : write [pic]. A & B are constants to be evaluated suitably. 9) [pic] W.R: Divide N and D by [pic]

10) [pic]

W.R: [pic] and split into two integrals of type (10).

11) [pic]

W.R: 1=[pic] and split into two integrals of type (10) 12) [pic] W.R: Divide N & D by [pic] & put [pic]

13) [pic] W.R: use [pic] [pic] and put [pic]

14) [pic]

W.R: put numerator=A[pic](Denominator)+B(Denominator) where A & B are constant obtained by comparing coefficient of [pic] and [pic]

15) [pic] W.R: The order of preference for first function is as these appear in word “ILATE”. I: Inverse trigonometric function. L: Logarithmic function. A: Algebraic function. T: Trignometric function. E: Exponential function.

16) [pic] When [pic] and [pic] are polynomial in [pic] with no common factors and degree of [pic] [pic] degree of [pic], method of resolving into partial fractions. a) When denominator contains linear non-repeated factors e.g [pic]where A, B & C are constants which have to be evaluated by comparing coefficient of like terms. b) When denominator contains linear repeated factors e.g: [pic][pic]

c) When denominator contains quadratic(non-factorizable factors) e.g: [pic]..

d) When denominator contains a mixture of above types of factors: e.g. [pic]



INTEGRATION (4 MARKS QUESTIONS)

1) [pic] 2) [pic] 3) [pic] 4) [pic] [pic][pic]

5) [pic] 6) [pic] 7) [pic] 8) [pic] 9) [pic] 10) [pic][pic] 11) [pic] 12) [pic]

13) [pic][pic] 14) [pic] 15) [pic]

16) [pic] 17) [pic] 18) [pic] 19) [pic]

20) [pic] 21) [pic][pic] 22) [pic] 23) [pic]

24) [pic] 25) [pic] 26) [pic] 27) [pic]

28) [pic] 29) [pic] 30) [pic] 31) [pic]

32) [pic] 33) [pic] 34) [pic] 35) [pic]

36) [pic] 37) [pic] 38) [pic] 39) [pic]

40) [pic] 41) [pic] 42) [pic] 43) [pic]

44) [pic] 45) [pic] 46) [pic] 47)[pic] 48) [pic]

49) [pic] 50) [pic] 51) [pic] 52) [pic][pic]

53) [pic] 54) [pic] 55) [pic] 56) [pic]

57) [pic] 58) [pic] 59) [pic] 60) [pic] [pic]

61) [pic] 62) [pic] 63) [pic] 64) [pic]

65) [pic] 66) [pic] 67) [pic] 68) [pic] 69) [pic]

70) [pic] 71) [pic] 72) [pic] 73) [pic]

74) [pic] 75) [pic] 76) [pic] 77) [pic] 78) [pic]

79) [pic] 80) [pic] 81) [pic]

82) [pic] 83) [pic] 84) [pic] 85) [pic] 86)[pic] 87) [pic] 88) [pic]

89) [pic] 90) [pic] 91) [pic] 92) [pic]

93) [pic] 94) [pic]



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cos-1x

sin-1x

3

x2

sin x

sinx – cos x

cot x

cos-1x

sin-1x

x2

cos-1t

sin-1t

Lim f(x) x( a

Lim f(x) x( a+

Lim f(x) x( a–

Lim f(a -h) h( 0

Lim f(x) x( a–

Lim f(a+h) h( 0

Lim f(x) x( a+

Lim sin θ / θ x( 0

Lim xn – an / x - a x( 0

Lim (1 + x) 1/x x( 0

3

Lim f(x) x( 1

Lim f(x) x( 1

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