11th Business Maths Monthly Test - 3 ( Differential Calculus)

MABS Institution

11th Business Maths Monthly Test - 3 ( Differential Calculus)-Aug 2020

Time : 2.00 Hours

Part - A

Generic 2 - Answer All Question

Evaluate: \(\underset { x\rightarrow 0 }{ lim } \frac { \sin { 3x } }{ \sin { 5x } } \)

Determine whether the following functions are odd or even?
\(f(x)=log({ x }^{ 2 }+\sqrt { { x }^{ 2 } } +1)\)

Evaluate: \(\underset { x\rightarrow 2 }{ lim } \frac { { x }^{ 2 }-4x+6 }{ x+2 } \)

Let f be defined by f(x) = x - 5 and g be defined by g(x) = \(\begin{cases} \frac { { x }^{ 2 }-25 }{ x+5 } if\quad x\neq -5 \\ \lambda \quad if\quad x=-5 \end{cases}\) Find \(\lambda\) such that f(x) = g(x) for all x \(\epsilon \) R

If f(x) = x + \(\frac{1}{x}\), then show that [f(x)]³ + \(3f(\frac{1}{x})\)

If f(x) = log \(\frac{1+x}{1-x}\), show that \(f(\frac{2x}{1+x^2})=2f(x)\)

Find the derivative of the following functions from first principles x²

If f(x) = 2^x, show that f(x) . f(y) = f(x + y)

Find \(\frac{dy}{dx}\) at (1, 1) to the curve 2x² + 3xy + 5y² = 10.

Find the derivative of the following functions from first principles e^-x

Differentiate \({ x }^{ \frac { 2 }{ 3 } }\) from first principles

Evaluate: \(\underset { x\rightarrow 1 }{ lim } \frac { { x }^{ 3 }-1 }{ x-1 } \)

If \(f(x)=\frac { x+1 }{ x-1 } \) ,then prove that f(f(x))=x

Evaluate: \(\underset { x\rightarrow 0 }{ lim } \frac { \sqrt { 2+x } -\sqrt { 2 } }{ x } \)

Part - B

Generic 3 - Answer All Question

Find \(\frac { dy }{ dx } \) for the following function x³+y³+3axy=1

Evaluate the following \(\lim _{ x\rightarrow a }{ \frac { { x }^{ \frac { 5 }{ 8 } }-{ a }^{ \frac { 5 }{ 8 } } }{ { x }^{ \frac { 2 }{ 3 } }-a^{ \frac { 2 }{ 3 } } } } \)

Find y₂ of the following function y = log x + a^x

Draw the graph y = 9 - x²

Find \(\frac{dy}{dx}\) of the following functions: x=acos³θ, y=asin³θ

Differentiate the following with respect to x. \(\frac { { e }^{ x } }{ 1+x } \)

If y = \(\frac{1}{u^2}\) and u = x² - 9, find \(\frac{dy}{dx}\).

Differentiate the following with respect to x.
(sin x )^{tan x}

Differentiate the following with respect to x \(\frac { { e }^{ x } }{ 1+{ e }^{ x } } \)

Differentiate the following with respect to x.
(sin x)^x

Find y₂ of the following functions y = e^3x+2

Differentiate sin² x with respect to x².

If x\(\sqrt { 1+y } +y\sqrt { 1+x } =0\) and x≠y, then prove that \(\frac { dy }{ dx } =\frac { -1 }{ (x+1)^{ 2 } } \).

Verify the existence of the function f(x) = \(\begin{cases} 5x-4\quad if0 at x = 1.

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11th Business Maths Monthly Test - 3 ( Differential Calculus)

MABS Institution

11th Business Maths Monthly Test - 3 ( Differential Calculus)-Aug 2020

Part - A

Generic 2 - Answer All Question

Evaluate: \(\underset { x\rightarrow 0 }{ lim } \frac { \sin { 3x } }{ \sin { 5x } } \)

Determine whether the following functions are odd or even? \(f(x)=log({ x }^{ 2 }+\sqrt { { x }^{ 2 } } +1)\)

Evaluate: \(\underset { x\rightarrow 2 }{ lim } \frac { { x }^{ 2 }-4x+6 }{ x+2 } \)

Let f be defined by f(x) = x - 5 and g be defined by g(x) = \(\begin{cases} \frac { { x }^{ 2 }-25 }{ x+5 } if\quad x\neq -5 \\ \lambda \quad if\quad x=-5 \end{cases}\) Find \(\lambda\) such that f(x) = g(x) for all x \(\epsilon \) R

If f(x) = x + \(\frac{1}{x}\), then show that [f(x)]3 + \(3f(\frac{1}{x})\)

If f(x) = log \(\frac{1+x}{1-x}\), show that \(f(\frac{2x}{1+x^2})=2f(x)\)

Find the derivative of the following functions from first principles x2

If f(x) = 2x, show that f(x) . f(y) = f(x + y)

Find \(\frac{dy}{dx}\) at (1, 1) to the curve 2x2 + 3xy + 5y2 = 10.

Find the derivative of the following functions from first principles e-x

Differentiate \({ x }^{ \frac { 2 }{ 3 } }\) from first principles

Evaluate: \(\underset { x\rightarrow 1 }{ lim } \frac { { x }^{ 3 }-1 }{ x-1 } \)

If \(f(x)=\frac { x+1 }{ x-1 } \) ,then prove that f(f(x))=x

Evaluate: \(\underset { x\rightarrow 0 }{ lim } \frac { \sqrt { 2+x } -\sqrt { 2 } }{ x } \)

Part - B

Generic 3 - Answer All Question

Find \(\frac { dy }{ dx } \) for the following function x3+y3+3axy=1

Evaluate the following \(\lim _{ x\rightarrow a }{ \frac { { x }^{ \frac { 5 }{ 8 } }-{ a }^{ \frac { 5 }{ 8 } } }{ { x }^{ \frac { 2 }{ 3 } }-a^{ \frac { 2 }{ 3 } } } } \)

Find y2 of the following function y = log x + ax

Draw the graph y = 9 - x2

Find \(\frac{dy}{dx}\) of the following functions: x=acos3θ, y=asin3θ

Differentiate the following with respect to x. \(\frac { { e }^{ x } }{ 1+x } \)

If y = \(\frac{1}{u^2}\) and u = x2 - 9, find \(\frac{dy}{dx}\).

Differentiate the following with respect to x. (sin x )tan x

Differentiate the following with respect to x \(\frac { { e }^{ x } }{ 1+{ e }^{ x } } \)

Differentiate the following with respect to x. (sin x)x

Find y2 of the following functions y = e3x+2

Differentiate sin2 x with respect to x2.

If x\(\sqrt { 1+y } +y\sqrt { 1+x } =0\) and x≠y, then prove that \(\frac { dy }{ dx } =\frac { -1 }{ (x+1)^{ 2 } } \).

Verify the existence of the function f(x) = \(\begin{cases} 5x-4\quad if0 at x = 1.

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Determine whether the following functions are odd or even?
\(f(x)=log({ x }^{ 2 }+\sqrt { { x }^{ 2 } } +1)\)

If f(x) = x + \(\frac{1}{x}\), then show that [f(x)]³ + \(3f(\frac{1}{x})\)

Find the derivative of the following functions from first principles x²

If f(x) = 2^x, show that f(x) . f(y) = f(x + y)

Find \(\frac{dy}{dx}\) at (1, 1) to the curve 2x² + 3xy + 5y² = 10.

Find the derivative of the following functions from first principles e^-x

Find \(\frac { dy }{ dx } \) for the following function x³+y³+3axy=1

Find y₂ of the following function y = log x + a^x

Draw the graph y = 9 - x²

Find \(\frac{dy}{dx}\) of the following functions: x=acos³θ, y=asin³θ

If y = \(\frac{1}{u^2}\) and u = x² - 9, find \(\frac{dy}{dx}\).

Differentiate the following with respect to x.
(sin x )^{tan x}

Differentiate the following with respect to x.
(sin x)^x

Find y₂ of the following functions y = e^3x+2

Differentiate sin² x with respect to x².