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

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Class 11th Business Mathematics and Statistics Answers Guide - Shaalaa.com
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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
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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]

Choose the correct answer

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
Class 11th Business Mathematics and Statistics Answers Guide - Shaalaa.com

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

Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide chapter 5 (Differential Calculus) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Tamil Nadu Board of Secondary Education Class 11th Business Mathematics and Statistics Answers Guide solutions in a manner that help students grasp basic concepts better and faster.

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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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