Recent questions tagged ch5

Given the function $ f(x)=\large\frac{1}{x+2}.$Find the points of discontinuity of the composite function $y=f(f(x)).$

asked Dec 31, 2012 by sreemathi.v

1 answer

Find the values of $a$ and $b$ such that the function $f$ defined by $f(x)=\left \{\begin{array}{1 1}\large\frac{x-4}{|x-4|}\normalsize+a, & if\;x<4\\a+b, & if\;x=4\\\large\frac{x-4}{|x-4|}\normalsize +b, & if\;x>4\end{array}\right.$ is a continuous function at $x=4.$

asked Dec 31, 2012 by sreemathi.v

1 answer

Prove that the function f defined by $ f(x)=\left \{\begin{array}{1 1}\large\frac{x}{\mid x\mid+2x^2}, & x\neq 0\\k, &x=0\end{array}\right.$remains discontinuous at x=0,regardless the choice of k.

asked Dec 31, 2012 by sreemathi.v

1 answer

Find the value of $k$ in the following so that the function $f$ is continuous at the indicated point $f(x)=\left \{\begin{array}{1 1}\large\frac{1-\cos kx}{x\sin x}, & if\;x\neq 0\\\large\frac{1}{2}, & if\;x=0\end{array}\right.\;at\;x=0$

asked Dec 31, 2012 by sreemathi.v

1 answer

Find the value of $k$ in the following so that the function $f$ is continuous at the indicated point $f(x)=\left \{\begin{array}{1 1}\large\frac{\sqrt {1+kx}-\sqrt{1-kx}}{x}, & if\;-1\leq x< 0\\\large\frac{2x+1}{x-2}, & if\;0\leq x\leq 1\end{array}\right.\; at\; x=0$

asked Dec 31, 2012 by sreemathi.v

1 answer

Find the value of k in the following so that the function $f$ is continuous at the indicated point: $f(x)=\left \{\begin{array}{1 1}\frac{\large 2^{x+2}-16}{4^{\large x}-16}, & if\;x\neq 2\\k, & if\;x=2\end{array}\right.$ at $x=2$

asked Dec 31, 2012 by sreemathi.v

1 answer

Find the value of k in the following so that the function f is continuous at the indicated point $f(x)=\left \{\begin{array}{1 1}3x-8, & if\;x\neq 5\\2k, & if\;x=5\end{array}\right.\;at\;x=5$

asked Dec 31, 2012 by sreemathi.v

1 answer

Find whether f(x) is continuous or discontinuous at the indicated point $f(x)= x ^2+x-1\;at\;x=1.$

asked Dec 31, 2012 by sreemathi.v

1 answer

Find whether $f(x)$ is continuous or discontinuous at the indicated point $f(x)=\left \{\begin{array}{1 1}\large\frac{x^2}{2}, & if\;0\leq x\leq 1\\2x^2-3x+\large\frac{3}{2}\normalsize,& if\;0< x\leq 2\end{array}\right.$ at $x=1$.

asked Dec 31, 2012 by sreemathi.v

1 answer

Find whether f(x) is continuous or discontinuous at the indicated point $f(x)=\left \{\begin{array}{1 1}\large\frac{e^ {\Large\frac{1}{x}}}{1+e^ {\Large\frac{1}{x}}},\normalsize & if\;x\neq 0\\0, & if\;x=0\end{array}\right.$ at $x=0$.

asked Dec 31, 2012 by sreemathi.v

1 answer

Find whether $f(x)$ is continuous or discontinuous at the indicated point $f(x)=\left \{\begin{array}{1 1}( x-a)\sin\large\frac{1}{x-a},\normalsize & if\;x\neq 0\\0, & if\;x=0\end{array}\right.$ at $x=a$.

asked Dec 31, 2012 by sreemathi.v

1 answer

Find whether f(x) is continuous or discontinuous at the indicated point $f(x)=\left \{\begin{array}{1 1}\mid x\mid\cos\large\frac{1}{x}, \normalsize& if\;x\neq 0\\0, & if\;x=0\end{array}\right.$at x=0.

asked Dec 31, 2012 by sreemathi.v

1 answer

Find whether f(x) is continuous or discontinuous at the indicated point $f(x)=\left \{\begin{array}{1 1}x-4, & if\;x\neq 4\\2(x-4)\\0, & if\;x=4\end{array}\right.$at x=4.

asked Dec 31, 2012 by sreemathi.v

1 answer

Examine the continuity of f(x) at the indicated point $f(x)=\left \{\begin{array}{1 1}\large\frac{2x^2-3x-2}{x-2} &\normalsize if\;x\neq 2\\5 & if\;x=2\end{array}\right.$ at x=2.

asked Dec 31, 2012 by sreemathi.v

1 answer

Examine the continuity of f(x) at the indicated point $f(x)=\left \{\begin{array}{1 1}\large\frac{1-\cos 2x}{x^2} & \normalsize if\;x\neq 0\\5 & if\;x=0\end{array}\right.$at x=0.

asked Dec 31, 2012 by sreemathi.v

1 answer

Examine the continuity of f(x) at the indicated point $f(x)=\left \{\begin{array}{1 1}3x+5 & if\;x\geq 2\\x^2 & if\;x<2\end{array}\right.$ at $x=2.$

asked Dec 31, 2012 by sreemathi.v

1 answer

Examine the continuity of the function $f(x)=x^3+2x^2-1$ at $x=1$

asked Dec 31, 2012 by sreemathi.v

1 answer

If $ y = e^{\large a \: cos^{-1}x}, -1 \leq x \leq 1$, show that $(1- x^2)\large\frac{d^2y}{dx^2}-\normalsize x\large\frac{dy}{dx} -\normalsize a^2y \; = \; 0.$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

If $y = \begin{vmatrix} f(x) & g(x) & h(x) \\l & m &n \\ a & b & c \end{vmatrix}$, prove that $\large\frac{dy}{dx}\normalsize = \begin{vmatrix} f'(x) & g'(x) & h'(x) \\l & m &n \\ a & b & c \end{vmatrix}$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.

asked Dec 4, 2012 by thanvigandhi_1

1 answer

Using the fact that $\sin (A + B) = \sin A \cos B + \cos A \sin B$ and the differentiation, obtain the sum formula for cosines.

asked Dec 4, 2012 by thanvigandhi_1

1 answer

Using mathematical induction prove that $ \large\frac{d}{dx}\normalsize (x^{\large n}) = nx^{\large{n-1}} $ for all positive integers $n$.

asked Dec 4, 2012 by thanvigandhi_1

1 answer

If $f (x) = |\; x\; |^3$, show that $f ″(x)$ exists for all real $x$ and find it.

asked Dec 4, 2012 by thanvigandhi_1

1 answer

If $x = a (\cos t + t\sin t)$ and $y = a (\sin t - t \cos\: t)$, find $ \large\frac{d^2y}{dx^2}$

asked Dec 4, 2012 by thanvigandhi_1

1 answer