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Continuity and Differentiability

Short Answer Type

601. Verify Roll's Theorem for the function :

straight f left parenthesis straight x right parenthesis equals space 4 space sin space straight x space in space left square bracket 0 comma space straight pi right square bracket.

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602.

If space space straight f open parentheses straight x close parentheses equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin open parentheses straight a plus 1 close parentheses plus 2 sinx over denominator straight x end fraction comma space straight x less than 0 end cell row cell 2 space space space space space space space space space space space space space space space space space space space space space space comma space x equals 0 end cell row cell fraction numerator square root of 1 plus b x end root minus 1 over denominator straight x end fraction space space space space space space comma space straight x greater than 0 end cell end table close

is continuous at x = 0, then find the values of a and b.

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603.

if space cos space left parenthesis straight a plus straight y right parenthesis space equals space cos space straight y space then space prove space that space dy over dx space equals fraction numerator cos squared left parenthesis straight a plus straight y right parenthesis over denominator sin space straight a end fraction Hence space show space that space sin space straight a space fraction numerator straight d squared straight y over denominator dx squared end fraction plus sin space 2 space left parenthesis straight a plus straight y right parenthesis dy over dx equals 0

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604.

Find space dy over dx space if space straight y equals sin to the power of negative 1 end exponent open square brackets fraction numerator 6 straight x minus 4 square root of 1 minus 4 straight x squared end root over denominator 5 end fraction close square brackets

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605.

Show that the function $straight f left parenthesis straight x right parenthesis space equals space open vertical bar straight x minus 3 close vertical bar comma space straight x element of bold R bold comma$ is continuous but not differentiable at x=3.

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606.

Determine the value of the constant ‘k’ so that the function is continuous at x = 0.

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Long Answer Type

607.

For what value of k is the following function continuous at x = 2?

$f (x) = \begin{matrix} 2 x + 1; x < 2 \\ k; x = 2 \\ 3 x - 1; x > 1 \end{matrix}$

608.

Differentiate the following function w.r.t. x:

y = ${(sinx)}^{x} + \sin^{- 1} \sqrt{x}$

609.

Find $\frac{dy}{dx}$ if (x² + y²)² = xy.

610.
If y =3 cos ( log x ) + 4 sin ( log x ), then show that $x^{2} \frac{d^{2} y}{{dx}^{2}} + x \frac{dy}{dx} + y = 0$

y = 3 cos ( log x ) + 4 sin ( log x )

Differentiating the above function with respect to x, we have,

$\frac{dy}{dx} = \frac{- 3 \cos (\log x)}{x} + \frac{4 \cos (\log x)}{x} x \frac{dy}{dx} = - 3 \cos (\log x) + 4 \cos (\log x)$

Again differentiating the above function with respect to x, we have,

$x \frac{d^{2} y}{{dx}^{2}} + \frac{dy}{dx} = \frac{- 3 \cos (\log x)}{x} - \frac{4 \sin (\log x)}{x} \Rightarrow x^{2} \frac{d^{2} y}{{dx}^{2}} + x \frac{dy}{dx} = - (3 \cos (\log x) + 4 \sin (\log x)) \Rightarrow x^{2} \frac{d^{2} y}{{dx}^{2}} + x \frac{dy}{dx} = - y \Rightarrow x^{2} \frac{d^{2} y}{{dx}^{2}} + x \frac{dy}{dx} + y = 0$