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

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) = x² – |x|

Determine whether the following function is odd or even?

f(x) = x + x²

Let f be defined by f(x) = x³ – kx² + 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 – x²

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`

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

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.

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

x²

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)

Differentiate the following with respect to x.

3x⁴ – 2x³ + 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.

x³ e^x

Differentiate the following with respect to x.

(x² – 3x + 2) (x + 1)

Differentiate the following with respect to x.

x⁴ – 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.

x³ e^x

Differentiate the following with respect to x.

sin² x

Differentiate the following with respect to x.

cos² x

Differentiate the following with respect to x.

cos³ x

Differentiate the following with respect to x.

`sqrt(1 + x^2)`

Differentiate the following with respect to x.

(ax² + bx + c)ⁿ

Differentiate the following with respect to x.

sin(x²)

Differentiate the following with respect to x.

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

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

xy – tan(xy)

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

x² – xy + y² = 1

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

x³ + y³ + 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"`

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

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 cos³θ, y = a sin³θ

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

x = a(θ – sin θ), y = a(1 – cos θ)

Differentiate sin³x with respect to cos³x.

Differentiate sin²x with respect to x².

Find y₂ for the following function:

y = e^3x+2

Find y₂ for the following function:

y = log x + a^x

Find y₂ for the following function:

x = a cosθ, y = a sinθ

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

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

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

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

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

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

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

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

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

The graph of the line y = 3 is

The graph of y = 2x² is passing through:

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

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

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

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

Which of the following function is neither even nor odd?

f(x) = -5, for all x ∈ R is a:

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

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

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

`lim_(theta->0) (tan theta)/theta` = 

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

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

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

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

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

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

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

If y = log x then y₂ =

`"d"/"dx" ("a"^x)` = 

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 - x².

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

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

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