Tamil Nadu Board of Secondary EducationHSC Commerce Class 11th

# Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide chapter 5 - Differential Calculus [Latest edition]

#### Chapters ## Chapter 5: Differential Calculus

Exercise 5.1Exercise 5.2Exercise 5.3Exercise 5.4Exercise 5.5Exercise 5.6Exercise 5.7Exercise 5.8Exercise 5.9Exercise 5.10Miscellaneous Problems
Exercise 5.1 [Page 105]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 5 Differential CalculusExercise 5.1 [Page 105]

Exercise 5.1 | Q 1. (i) | Page 105

Determine whether the following function is odd or even?

f(x) = ((a^x - 1)/(a^x + 1))

Exercise 5.1 | Q 1. (ii) | Page 105

Determine whether the following function is odd or even?

f(x) = log (x^2 + sqrt(x^2 + 1))

Exercise 5.1 | Q 1. (iii) | Page 105

Determine whether the following function is odd or even?

f(x) = sin x + cos x

Exercise 5.1 | Q 1. (iv) | Page 105

Determine whether the following function is odd or even?

f(x) = x2 – |x|

Exercise 5.1 | Q 1. (v) | Page 105

Determine whether the following function is odd or even?

f(x) = x + x2

Exercise 5.1 | Q 2 | Page 105

Let f be defined by f(x) = x3 – kx2 + 2x, x ∈ R. Find k, if ‘f’ is an odd function.

Exercise 5.1 | Q 3 | Page 105

If f(x) = x^3 - 1/x^3, then show that "f"(x) + "f"(1/x) = 0

Exercise 5.1 | Q 4 | Page 105

If f(x) = ((x + 1)/(x - 1)), then prove that f(f(x)) = x.

Exercise 5.1 | Q 5 | Page 105

For f(x) = (x - 1)/(3x + 1), write the expressions of "f"(1/x) and 1/("f"(x))

Exercise 5.1 | Q 6. (i) | Page 105

If f(x) = ex and g(x) = logx then find (f + g)(1)

Exercise 5.1 | Q 6. (ii) | Page 105

If f(x) = ex and g(x) = logx then find (fg)(1).

Exercise 5.1 | Q 6. (iii) | Page 105

If f(x) = ex and g(x) = logx then find (3f) (1).

Exercise 5.1 | Q 6. (iv) | Page 105

If f(x) = ex and g(x) = logx then find (5g)(1).

Exercise 5.1 | Q 7. (i) | Page 105

Draw the graph of the following function:

f(x) = 16 – x2

Exercise 5.1 | Q 7. (ii) | Page 105

Draw the graph of the following function:

f(x) = |x – 2|

Exercise 5.1 | Q 7. (iii) | Page 105

Draw the graph of the following function:

f(x) = x |x|

Exercise 5.1 | Q 7. (iv) | Page 105

Draw the graph of the following function:

f(x) = e2x

Exercise 5.1 | Q 7. (v) | Page 105

Draw the graph of the following function:

f(x) = e-2x

Exercise 5.1 | Q 7. (vi) | Page 105

Draw the graph of the following function:

f(x) = |x|/x

Exercise 5.2 [Page 110]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 5 Differential CalculusExercise 5.2 [Page 110]

Exercise 5.2 | Q 1. (i) | Page 110

Evaluate the following:

$\lim_{x->2} \frac{x^3 + 2}{x + 1}$

Exercise 5.2 | Q 1. (ii) | Page 110

Evaluate the following:

$\lim_{x->∞} \frac{2x + 5}{x^2 + 3x + 9}$

Exercise 5.2 | Q 1. (iii) | Page 110

Evaluate the following:

lim_(x->∞) (sum "n")/"n"^2

Exercise 5.2 | Q 1. (iv) | Page 110

Evaluate the following:

lim_(x->0) (sqrt(1+x) - sqrt(1-x))/x

Exercise 5.2 | Q 1. (v) | Page 110

Evaluate the following:

lim_(x->a) (x^(5/8) - a^(5/8))/(x^(2/3) - a^(2/3))

Exercise 5.2 | Q 1. (vi) | Page 110

Evaluate the following:

lim_(x->0) (sin^2 3x)/x^2

Exercise 5.2 | Q 2 | Page 110

If lim_(x->a) (x^9 + "a"^9)/(x + "a") = lim_(x->3) (x + 6), find the value of a.

Exercise 5.2 | Q 3 | Page 110

If lim_(x->2) (x^n - 2^n)/(x-2) = 448, then find the least positive integer n.

Exercise 5.2 | Q 4 | Page 110

If f(x) = (x^7 - 128)/(x^5 - 32), then find lim_(x-> 2) f(x)

Exercise 5.2 | Q 5 | Page 110

Let f(x) = ("a"x + "b")/("x + 1"), if lim_(x->0) f(x) = 2 and lim_(x->∞) f(x) = 1, then show that f(-2) = 0

Exercise 5.3 [Page 112]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 5 Differential CalculusExercise 5.3 [Page 112]

Exercise 5.3 | Q 1. (a) | Page 112

Examine the following function for continuity at the indicated point.

f(x) = {((x^2 - 4)/(x-2) "," if  x ≠ 2),(0 "," if x = 2):} at x = 2

Exercise 5.3 | Q 1. (b) | Page 112

Examine the following function for continuity at the indicated point.

f(x) = {((x^2 - 9)/(x-3) "," if  x ≠ 3),(6 "," if x = 3):} at x = 3

Exercise 5.3 | Q 2 | Page 112

Show that f(x) = |x| is continuous at x = 0.

Exercise 5.4 [Page 115]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 5 Differential CalculusExercise 5.4 [Page 115]

Exercise 5.4 | Q 1. (i) | Page 115

Find the derivative of the following function from the first principle.

x2

Exercise 5.4 | Q 1. (ii) | Page 115

Find the derivative of the following function from the first principle.

ex

Exercise 5.4 | Q 1. (iii) | Page 115

Find the derivative of the following function from the first principle.

log(x + 1)

Exercise 5.5 [Pages 117 - 118]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 5 Differential CalculusExercise 5.5 [Pages 117 - 118]

Exercise 5.5 | Q 1. (i) | Page 117

Differentiate the following with respect to x.

3x4 – 2x3 + x + 8

Exercise 5.5 | Q 1. (ii) | Page 117

Differentiate the following with respect to x.

5/x^4 - 2/x^3 + 5/x

Exercise 5.5 | Q 1. (iii) | Page 118

Differentiate the following with respect to x.

sqrtx + 1/root(3)(x) + e^x

Exercise 5.5 | Q 1. (iv) | Page 118

Differentiate the following with respect to x.

(3 + 2x - x^2)/x

Exercise 5.5 | Q 1. (v) | Page 118

Differentiate the following with respect to x.

x3 ex

Exercise 5.5 | Q 1. (vi) | Page 118

Differentiate the following with respect to x.

(x2 – 3x + 2) (x + 1)

Exercise 5.5 | Q 1. (vii) | Page 118

Differentiate the following with respect to x.

x4 – 3 sin x + cos x

Exercise 5.5 | Q 1. (viii) | Page 118

Differentiate the following with respect to x.

(sqrtx + 1/sqrtx)^2

Exercise 5.5 | Q 2. (i) | Page 118

Differentiate the following with respect to x.

e^x/(1 + x)

Exercise 5.5 | Q 2. (ii) | Page 118

Differentiate the following with respect to x.

(x^2 + x + 1)/(x^2 - x + 1)

Exercise 5.5 | Q 2. (iii) | Page 118

Differentiate the following with respect to x.

e^x/(1 + e^x)

Exercise 5.5 | Q 3. (i) | Page 118

Differentiate the following with respect to x.

x sin x

Exercise 5.5 | Q 3. (ii) | Page 118

Differentiate the following with respect to x.

ex sin x

Exercise 5.5 | Q 3. (iii) | Page 118

Differentiate the following with respect to x.

ex (x + log x)

Exercise 5.5 | Q 3. (iv) | Page 118

Differentiate the following with respect to x.

sin x cos x

Exercise 5.5 | Q 3. (v) | Page 118

Differentiate the following with respect to x.

x3 ex

Exercise 5.5 | Q 4. (i) | Page 118

Differentiate the following with respect to x.

sin2 x

Exercise 5.5 | Q 4. (ii) | Page 118

Differentiate the following with respect to x.

cos2 x

Exercise 5.5 | Q 4. (iii) | Page 118

Differentiate the following with respect to x.

cos3 x

Exercise 5.5 | Q 4. (iv) | Page 118

Differentiate the following with respect to x.

sqrt(1 + x^2)

Exercise 5.5 | Q 4. (v) | Page 118

Differentiate the following with respect to x.

(ax2 + bx + c)n

Exercise 5.5 | Q 4. (vi) | Page 118

Differentiate the following with respect to x.

sin(x2)

Exercise 5.5 | Q 4. (vii) | Page 118

Differentiate the following with respect to x.

1/sqrt(1 + x^2)

Exercise 5.6 [Page 119]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 5 Differential CalculusExercise 5.6 [Page 119]

Exercise 5.6 | Q 1. (i) | Page 119

Find "dy"/"dx" for the following function

xy – tan(xy)

Exercise 5.6 | Q 1. (ii) | Page 119

Find "dy"/"dx" for the following function.

x2 – xy + y2 = 1

Exercise 5.6 | Q 1. (iii) | Page 119

Find "dy"/"dx" for the following function.

x3 + y3 + 3axy = 1

Exercise 5.6 | Q 2 | Page 119

If xsqrt(1 + y) + ysqrt(1 + x) = 0 and x ≠ y, then prove that "dy"/"dx" = - 1/(x + 1)^2

Exercise 5.6 | Q 3 | Page 119

If 4x + 3y = log(4x – 3y), then find "dy"/"dx"

Exercise 5.7 [Page 120]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 5 Differential CalculusExercise 5.7 [Page 120]

Exercise 5.7 | Q 1. (i) | Page 120

Differentiate the following with respect to x.

xsin x

Exercise 5.7 | Q 1. (ii) | Page 120

Differentiate the following with respect to x.

(sin x)x

Exercise 5.7 | Q 1. (iii) | Page 120

Differentiate the following with respect to x.

(sin x)tan x

Exercise 5.7 | Q 1. (iv) | Page 120

Differentiate the following with respect to x.

sqrt(((x - 1)(x - 2))/((x - 3)(x^2 + x + 1)))

Exercise 5.7 | Q 2 | Page 120

If xm . yn = (x + y)m+n, then show that "dy"/"dx" = y/x

Exercise 5.8 [Page 122]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 5 Differential CalculusExercise 5.8 [Page 122]

Exercise 5.8 | Q 1. (i) | Page 122

Find "dy"/"dx" of the following function:

x = ct, y = c/t

Exercise 5.8 | Q 1. (ii) | Page 122

Find "dy"/"dx" of the following function:

x = log t, y = sin t

Exercise 5.8 | Q 1. (iii) | Page 122

Find "dy"/"dx" of the following function:

x = a cos3θ, y = a sin3θ

Exercise 5.8 | Q 1. (iv) | Page 122

Find "dy"/"dx" of the following function:

x = a(θ – sin θ), y = a(1 – cos θ)

Exercise 5.8 | Q 2 | Page 122

Differentiate sin3x with respect to cos3x.

Exercise 5.8 | Q 3 | Page 122

Differentiate sin2x with respect to x2.

Exercise 5.9 [Page 123]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 5 Differential CalculusExercise 5.9 [Page 123]

Exercise 5.9 | Q 1. (i) | Page 123

Find y2 for the following function:

y = e3x+2

Exercise 5.9 | Q 1. (ii) | Page 123

Find y2 for the following function:

y = log x + ax

Exercise 5.9 | Q 1. (iii) | Page 123

Find y2 for the following function:

x = a cosθ, y = a sinθ

Exercise 5.9 | Q 2 | Page 123

If y = 500e7x + 600e-7x, then show that y2 – 49y = 0.

Exercise 5.9 | Q 3 | Page 123

If y = 2 + log x, then show that xy2 + y1 = 0.

Exercise 5.9 | Q 4 | Page 123

If = a cos mx + b sin mx, then show that y2 + m2y = 0.

Exercise 5.9 | Q 5 | Page 123

If y = (x + sqrt(1 + x^2))^m, then show that (1 + x2) y2 + xy1 – m2y = 0

Exercise 5.9 | Q 6 | Page 123

If y = sin(log x), then show that x2y2 + xy1 + y = 0.

Exercise 5.10 [Pages 123 - 125]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 5 Differential CalculusExercise 5.10 [Pages 123 - 125]

Exercise 5.10 | Q 1 | Page 123

If f(x) = x2 – x + 1 then f(x + 1) is:

• x2

• x

• 1

• x2 + x + 1

Exercise 5.10 | Q 2 | Page 123

If f(x) = {(x^2 - 4x if  x >= 2),(x+2 if x < 2):}, then f(5) is

• -1

• 2

• 5

• 7

Exercise 5.10 | Q 3 | Page 123

If f(x) = {(x^2 - 4x if  x >= 2),(x+2 if x < 2):}, then f(0) is

• 2

• 5

• -1

• 0

Exercise 5.10 | Q 4 | Page 123

If f(x) = (1 - x)/(1 + x) then f(-x) is equal to:

• - f(x)

• 1/("f"(x))

• -1/("f"(x))

• f(x)

Exercise 5.10 | Q 5 | Page 123

The graph of the line y = 3 is

• Parallel to x-axis

• Parallel to y-axis

• Passing through the origin

• Perpendicular to x-axis

Exercise 5.10 | Q 6 | Page 123

The graph of y = 2x2 is passing through:

• (0, 0)

• (2, 1)

• (2, 0)

• (0, 2)

Exercise 5.10 | Q 7 | Page 124

The graph of y = ex intersect the y-axis at:

• (0, 0)

• (1, 0)

• (0, 1)

• (1, 1)

Exercise 5.10 | Q 8 | Page 124

The minimum value of the function f(x) = |x| is:

• 0

• - 1

• + 1

Exercise 5.10 | Q 9 | Page 124

Which one of the following functions has the property f(x) = "f"(1/x)?

• f(x) = (x^2 - 1)/x

• f(x) = (1 - x^2)/x

• f(x) = x

• f(x) = (x^2 + 1)/x

Exercise 5.10 | Q 10 | Page 124

If f(x) = 2x and g(x) = 1/2^x then (fg)(x) is:

• 1

• 0

• 4x

• 1/4^x

Exercise 5.10 | Q 11 | Page 124

Which of the following function is neither even nor odd?

• f(x) = x3 + 5

• f(x) = x5

• f(x) = x10

• f(x) = x2

Exercise 5.10 | Q 12 | Page 124

f(x) = -5, for all x ∈ R is a:

• an identity function

• modulus function

• exponential function

• constant function

Exercise 5.10 | Q 13 | Page 124

The range of f(x) = |x|, for all x ∈ R is:

• (0, ∞)

• [0, ∞)

• (-∞, ∞)

• [1, ∞)

Exercise 5.10 | Q 14 | Page 124

The graph of f(x) = ex is identical to that of:

• f(x) = ax, a > 1

• f(x) = ax, a < 1

• f(x) = ax, 0 < a < 1

• y = ax + b, a ≠ 0

Exercise 5.10 | Q 15 | Page 124

If f(x) = x2 and g(x) = 2x + 1 then (fg)(0) is:

• 0

• 2

• 1

• 4

Exercise 5.10 | Q 16 | Page 124

lim_(theta->0) (tan theta)/theta =

• 1

• - ∞

• θ

Exercise 5.10 | Q 17 | Page 124

$\lim_{x->0} \frac{e^x - 1}{x}$=

• e

• nxn-1

• 1

• 0

Exercise 5.10 | Q 18 | Page 124

For what value of x, f(x) = (x+2)/(x-1)  is not continuous?

• -2

• 1

• 2

• -1

Exercise 5.10 | Q 19 | Page 124

A function f(x) is continuous at x = a lim_(x->"a")f(x) is equal to:

• f(-a)

• "f"(1/"a")

• 2f(a)

• f(a)

Exercise 5.10 | Q 20 | Page 124

"d"/"dx" (1/x) is equal to:

• -$\frac{1}{x^2}$

• -$\frac{1}{x}$

• log x

• $\frac{1}{x^2}$

Exercise 5.10 | Q 21 | Page 125

"d"/"dx" (5ex – 2 log x) is equal to:

• 5e^x - 2/x

• 5ex - 2x

• 5e^x - 1/x

• 2 log x

Exercise 5.10 | Q 22 | Page 125

If y = x and z = 1/x then "dy"/"dx" =

• x2

• 1

• -x2

• (-1)/x^2

Exercise 5.10 | Q 23 | Page 125

If y = e2x then ("d"^2"y")/"dx"^2 at x = 0 is:

• 4

• 9

• 2

• 0

Exercise 5.10 | Q 24 | Page 125

If y = log x then y2 =

• 1/x

• - 1/x^2

• - 2/x^2

• e2

Exercise 5.10 | Q 25 | Page 125

"d"/"dx" ("a"^x) =

• 1/(x log_e"a")

• aa

• x loge a

• ax loge a

Miscellaneous Problems [Page 125]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 5 Differential CalculusMiscellaneous Problems [Page 125]

Miscellaneous Problems | Q 1 | Page 125

If f(x) = 1/(2x + 1), x > -1/2, then show that f(f(x)) = (2x + 1)/(2x + 3)

Miscellaneous Problems | Q 2 | Page 125

Draw the graph of y = 9 - x2.

Miscellaneous Problems | Q 3 | Page 125

If f(x)= {((x - |x|)/x if  x ≠ 0),(2 if x = 0):} then show that lim_(x->1)f(x) does not exist.

Miscellaneous Problems | Q 4 | Page 125

Evaluate: lim_(x->1) ((2x - 3)(sqrtx - 1))/(2x^2 + x - 3)

Miscellaneous Problems | Q 5 | Page 125

Show that the function f(x) = 2x - |x| is continuous at x = 0

Miscellaneous Problems | Q 6 | Page 125

Verify the continuity and differentiability of f(x) = {(1 - x if  x < 1),((1 - x)(2 - x) if 1 <= x <= 2),(3 - x if x > 2):} at x = 1 and x = 2.

Miscellaneous Problems | Q 7 | Page 125

If xy . yx , then prove that "dy"/"dx" = y/x((x log y - y)/(y log x - x))

Miscellaneous Problems | Q 8 | Page 125

If xy2 = 1, then prove that 2 "dy"/"dx" + y^3= 0

Miscellaneous Problems | Q 9 | Page 125

If y = tan x, then prove that y2 - 2yy1 = 0.

Miscellaneous Problems | Q 10 | Page 125

If y = 2 sin x + 3 cos x, then show that y2 + y = 0.

## Chapter 5: Differential Calculus

Exercise 5.1Exercise 5.2Exercise 5.3Exercise 5.4Exercise 5.5Exercise 5.6Exercise 5.7Exercise 5.8Exercise 5.9Exercise 5.10Miscellaneous Problems ## Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide chapter 5 - Differential Calculus

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Concepts covered in Class 11th Business Mathematics and Statistics Answers Guide chapter 5 Differential Calculus are Functions and Their Graphs, Limits and Derivatives, Differentiation Techniques.

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