#### Chapters

Chapter 2: Algebra

Chapter 3: Analytical Geometry

Chapter 4: Trigonometry

Chapter 5: Differential Calculus

Chapter 6: Applications of Differentiation

Chapter 7: Financial Mathematics

Chapter 8: Descriptive Statistics and Probability

Chapter 9: Correlation and Regression Analysis

Chapter 10: Operations Research

## Chapter 5: Differential Calculus

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

**Determine whether the following function is odd or even?**

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

**Determine whether the following function is odd or even?**

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

**Determine whether the following function is odd or even?**

f(x) = sin x + cos x

**Determine whether the following function is odd or even?**

f(x) = x^{2} – |x|

**Determine whether the following function is odd or even?**

f(x) = x + x^{2}

Let f be defined by f(x) = x^{3} – kx^{2} + 2x, x ∈ R. Find k, if ‘f’ is an odd function.

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

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

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

If f(x) = e^{x} and g(x) = log_{e }x then find (f + g)(1)

If f(x) = e^{x} and g(x) = log_{e }x then find (fg)(1).

If f(x) = e^{x} and g(x) = log_{e }x then find (3f) (1).

If f(x) = e^{x} and g(x) = log_{e }x then find (5g)(1).

**Draw the graph of the following function:**

f(x) = 16 – x^{2}

**Draw the graph of the following function:**

f(x) = |x – 2|

**Draw the graph of the following function:**

f(x) = x |x|

**Draw the graph of the following function:**

f(x) = e^{2x}

**Draw the graph of the following function:**

f(x) = e^{-2x}

**Draw the graph of the following function:**

f(x) = `|x|/x`

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

**Evaluate the following:**

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

**Evaluate the following:**

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

**Evaluate the following:**

`lim_(x->∞) (sum "n")/"n"^2`

**Evaluate the following:**

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

**Evaluate the following:**

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

**Evaluate the following:**

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

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

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

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

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

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 5 Differential Calculus Exercise 5.3 [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

**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

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

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

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

x^{2}

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

e^{x}

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

log(x + 1)

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

**Differentiate the following with respect to x.**

3x^{4} – 2x^{3} + x + 8

**Differentiate the following with respect to x.**

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

**Differentiate the following with respect to x.**

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

**Differentiate the following with respect to x.**

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

**Differentiate the following with respect to x.**

x^{3} e^{x}

**Differentiate the following with respect to x.**

(x^{2} – 3x + 2) (x + 1)

**Differentiate the following with respect to x.**

x^{4} – 3 sin x + cos x

**Differentiate the following with respect to x.**

`(sqrtx + 1/sqrtx)^2`

**Differentiate the following with respect to x.**

`e^x/(1 + x)`

**Differentiate the following with respect to x.**

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

**Differentiate the following with respect to x.**

`e^x/(1 + e^x)`

**Differentiate the following with respect to x.**

x sin x

**Differentiate the following with respect to x.**

e^{x} sin x

**Differentiate the following with respect to x.**

e^{x} (x + log x)

**Differentiate the following with respect to x.**

sin x cos x

**Differentiate the following with respect to x.**

x^{3} e^{x}

**Differentiate the following with respect to x.**

sin^{2} x

**Differentiate the following with respect to x.**

cos^{2} x

**Differentiate the following with respect to x.**

cos^{3} x

**Differentiate the following with respect to x.**

`sqrt(1 + x^2)`

**Differentiate the following with respect to x.**

(ax^{2} + bx + c)^{n}

**Differentiate the following with respect to x.**

sin(x^{2})

**Differentiate the following with respect to x.**

`1/sqrt(1 + x^2)`

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

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

xy – tan(xy)

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

x^{2} – xy + y^{2} = 1

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

x^{3} + y^{3} + 3axy = 1

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

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

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

**Differentiate the following with respect to x.**

x^{sin x}

**Differentiate the following with respect to x.**

(sin x)^{x}

**Differentiate the following with respect to x.**

(sin x)^{tan x}

**Differentiate the following with respect to x.**

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

If x^{m} . y^{n} = (x + y)^{m+n}, then show that `"dy"/"dx" = y/x`

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

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

x = ct, y = `c/t`

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

x = log t, y = sin t

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

x = a cos^{3}θ, y = a sin^{3}θ

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

x = a(θ – sin θ), y = a(1 – cos θ)

Differentiate sin^{3}x with respect to cos^{3}x.

Differentiate sin^{2}x with respect to x^{2}.

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

**Find y _{2} for the following function:**

y = e^{3x+2}

**Find y _{2} for the following function:**

y = log x + a^{x}

**Find y _{2} for the following function:**

x = a cosθ, y = a sinθ

If y = 500e^{7x} + 600e^{-7x}, then show that y_{2} – 49y = 0.

If y = 2 + log x, then show that xy_{2} + y_{1} = 0.

If = a cos mx + b sin mx, then show that y_{2} + m^{2}y = 0.

If y = `(x + sqrt(1 + x^2))^m`, then show that (1 + x^{2}) y_{2} + xy_{1} – m^{2}y = 0

If y = sin(log x), then show that x^{2}y_{2} + xy_{1} + y = 0.

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

#### Choose the correct answer

If f(x) = x^{2} – x + 1 then f(x + 1) is:

x

^{2}x

1

x

^{2}+ x + 1

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

-1

2

5

7

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

2

5

-1

0

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

- f(x)

`1/("f"(x))`

-`1/("f"(x))`

f(x)

The graph of the line y = 3 is

Parallel to x-axis

Parallel to y-axis

Passing through the origin

Perpendicular to x-axis

The graph of y = 2x^{2} is passing through:

(0, 0)

(2, 1)

(2, 0)

(0, 2)

The graph of y = e^{x} intersect the y-axis at:

(0, 0)

(1, 0)

(0, 1)

(1, 1)

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

0

- 1

+ 1

∞

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`

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

1

0

4

^{x}`1/4^x`

Which of the following function is neither even nor odd?

f(x) = x

^{3}+ 5f(x) = x

^{5}f(x) = x

^{10}f(x) = x

^{2}

f(x) = -5, for all x ∈ R is a:

an identity function

modulus function

exponential function

constant function

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

(0, ∞)

[0, ∞)

(-∞, ∞)

[1, ∞)

The graph of f(x) = e^{x} is identical to that of:

f(x) = a

^{x}, a > 1f(x) = a

^{x}, a < 1f(x) = a

^{x}, 0 < a < 1y = ax + b, a ≠ 0

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

0

2

1

4

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

1

∞

- ∞

θ

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

e

nx

^{n-1}1

0

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

-2

1

2

-1

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)

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

-\[\frac{1}{x^2}\]

-\[\frac{1}{x}\]

log x

\[\frac{1}{x^2}\]

`"d"/"dx"` (5e^{x} – 2 log x) is equal to:

`5e^x - 2/x`

5e

^{x}- 2x`5e^x - 1/x`

2 log x

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

x

^{2}1

-x

^{2}`(-1)/x^2`

If y = e^{2x} then `("d"^2"y")/"dx"^2` at x = 0 is:

4

9

2

0

If y = log x then y_{2} =

`1/x`

`- 1/x^2`

`- 2/x^2`

e

^{2}

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

`1/(x log_e"a")`

a

^{a}x log

_{e }aa

^{x}log_{e}a

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

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

Draw the graph of y = 9 - x^{2}.

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

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

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

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.

If x^{y} . y^{x }, then prove that `"dy"/"dx" = y/x((x log y - y)/(y log x - x))`

If xy^{2} = 1, then prove that `2 "dy"/"dx" + y^3`= 0

If y = tan x, then prove that y_{2} - 2yy_{1} = 0.

If y = 2 sin x + 3 cos x, then show that y_{2} + y = 0.

## Chapter 5: Differential Calculus

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

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