The value of a for which the function \[f\left( x \right) = \begin{cases}5x - 4 , & \text{ if } 0 < x \leq 1 \\ 4 x^2 + 3ax, & \text{ if } 1 < x < 2\end{cases}\] is continuous at every point of its domain, is
[5] Continuity and Differentiability
Chapter: [5] Continuity and Differentiability
Concept: undefined >> undefined
Find the values of a and b so that the function
\[f\left( x \right)\begin{cases}x^2 + 3x + a, & \text { if } x \leq 1 \\ bx + 2 , &\text { if } x > 1\end{cases}\] is differentiable at each x ∈ R.
[5] Continuity and Differentiability
Chapter: [5] Continuity and Differentiability
Concept: undefined >> undefined
Find the values of a and b, if the function f defined by
\[f\left( x \right) = \begin{cases}x^2 + 3x + a & , & x \leqslant 1 \\ bx + 2 & , & x > 1\end{cases}\] is differentiable at x = 1.
[5] Continuity and Differentiability
Chapter: [5] Continuity and Differentiability
Concept: undefined >> undefined
If f is defined by \[f\left( x \right) = x^2 - 4x + 7\] , show that \[f'\left( 5 \right) = 2f'\left( \frac{7}{2} \right)\]
[5] Continuity and Differentiability
Chapter: [5] Continuity and Differentiability
Concept: undefined >> undefined
If \[f \left( x \right) = \sqrt{x^2 + 9}\] , write the value of
\[\lim_{x \to 4} \frac{f\left( x \right) - f\left( 4 \right)}{x - 4} .\]
[5] Continuity and Differentiability
Chapter: [5] Continuity and Differentiability
Concept: undefined >> undefined
If \[f\left( x \right) = \begin{cases}\frac{\left| x + 2 \right|}{\tan^{- 1} \left( x + 2 \right)} & , x \neq - 2 \\ 2 & , x = - 2\end{cases}\] then f (x) is
[5] Continuity and Differentiability
Chapter: [5] Continuity and Differentiability
Concept: undefined >> undefined
The function f (x) = |cos x| is
[5] Continuity and Differentiability
Chapter: [5] Continuity and Differentiability
Concept: undefined >> undefined
If \[f\left( x \right) = a\left| \sin x \right| + b e^\left| x \right| + c \left| x \right|^3\]
[5] Continuity and Differentiability
Chapter: [5] Continuity and Differentiability
Concept: undefined >> undefined
The function f (x) = x − [x], where [⋅] denotes the greatest integer function is
[5] Continuity and Differentiability
Chapter: [5] Continuity and Differentiability
Concept: undefined >> undefined
Let f (x) = |cos x|. Then,
[5] Continuity and Differentiability
Chapter: [5] Continuity and Differentiability
Concept: undefined >> undefined
The function f (x) = 1 + |cos x| is
[5] Continuity and Differentiability
Chapter: [5] Continuity and Differentiability
Concept: undefined >> undefined
The function \[f\left( x \right) = \frac{\sin \left( \pi\left[ x - \pi \right] \right)}{4 + \left[ x \right]^2}\] , where [⋅] denotes the greatest integer function, is
[5] Continuity and Differentiability
Chapter: [5] Continuity and Differentiability
Concept: undefined >> undefined
Let f (x) = a + b |x| + c |x|4, where a, b, and c are real constants. Then, f (x) is differentiable at x = 0, if
[5] Continuity and Differentiability
Chapter: [5] Continuity and Differentiability
Concept: undefined >> undefined
If \[f\left( x \right) = \begin{cases}\frac{1 - \cos x}{x \sin x}, & x \neq 0 \\ \frac{1}{2} , & x = 0\end{cases}\]
then at x = 0, f (x) is
[5] Continuity and Differentiability
Chapter: [5] Continuity and Differentiability
Concept: undefined >> undefined
\[s^2 \frac{d^2 t}{d s^2} + st\frac{dt}{ds} = s\]
[9] Differential Equations
Chapter: [9] Differential Equations
Concept: undefined >> undefined
\[\frac{d^3 y}{d x^3} + \left( \frac{d^2 y}{d x^2} \right)^3 + \frac{dy}{dx} + 4y = \sin x\]
[9] Differential Equations
Chapter: [9] Differential Equations
Concept: undefined >> undefined
(xy2 + x) dx + (y − x2y) dy = 0
[9] Differential Equations
Chapter: [9] Differential Equations
Concept: undefined >> undefined
\[\sqrt{1 - y^2} dx + \sqrt{1 - x^2} dx = 0\]
[9] Differential Equations
Chapter: [9] Differential Equations
Concept: undefined >> undefined
\[\frac{d^2 y}{d x^2} = \left( \frac{dy}{dx} \right)^{2/3}\]
[9] Differential Equations
Chapter: [9] Differential Equations
Concept: undefined >> undefined
\[2\frac{d^2 y}{d x^2} + 3\sqrt{1 - \left( \frac{dy}{dx} \right)^2 - y} = 0\]
[9] Differential Equations
Chapter: [9] Differential Equations
Concept: undefined >> undefined
