Chapter 2 Limit, Continuity and Derivative-II

2.1 Exercise 1(iii)

2.1.1 Question 1

Examine the continuity of the following functions at the specified point:

• $f\left(x\right)=\left|x\right|$ at $x=0$

• $f\left(x\right)=1,0$ or $-1$ according as $x>=$ or $<0$ at $x=0$

• $f\left(x\right)=\{\begin{array}{cc}-x& \text{when}x\le 0\\ x& \text{when}0<x<1\text{at}x=0\\ 2-x& \text{when}x\ge 1\end{array}$

• $f\left(x\right)=\frac{x-1}{1+e^{1/(x-1)}},f\left(1\right)=0$ at $x=1$

• $f\left(x\right)=\frac{x^4+x^3+2x^2}{{\sin }^{2}x},f\left(0\right)=0$ at $x=0$

2.1.2 Question 2

Are the following functions continuous at the origin?

• $f\left(x\right)=\{\begin{array}{cc}x\sin \frac{1}{x},& x\ne 0\\ 0,& x=0\end{array}$

• $f\left(x\right)=\{\begin{array}{cc}\sin \frac{1}{x},& x\ne 0\\ 0,& x=0\end{array}$

• $f\left(x\right)=\{\begin{array}{cc}(1+x{)}^{1/x}& \text{when}x\ne 0\\ 1& \text{when}x=0\end{array}$

• $f\left(x\right)=(1+3x{)}^{1/x},x\ne 0,f(0)=e^3$

2.1.3 Question 3

• For what value of $k$ is the function

$--$ $f\left(x\right)=\{\begin{array}{cc}{x}^2-1& \text{for}x<3\\ 2kx& \text{for}x\ge 3\end{array}$

$--$continuous at every value of $x$.

• Determine the values of $a$ and $b$ so that the function $f$ defined below is continuous everywhere:

$--$ $f\left(x\right)=\{\begin{array}{cc}1& \text{when}x\le 3\\ ax+b& \text{when}3<x<5\\ 7& \text{when}x\ge 5\end{array}$

2.2 Exercise 1(iv)

2.2.1 Question 1

• Determine whether $f$ is continuous and has a derivative at origin where

$-$ $f\left(x\right)=\{\begin{array}{cc}2+x& \text{if}x\ge 0\\ 2-x& \text{if}x<0\end{array}$

• Examine for Continuity at $x=a$ the function $f$ where

$-$ $f\left(x\right)=\{\begin{array}{cc}\frac{x^2}{a}-a& \text{,}0<x<a\\ 0& \text{,}x=a\\ a-\frac{a^3}{x^2}& \text{,}x>a\end{array}$

$--$Also examine if the function is derivable at $x=a$.

• Show that $f\left(x\right)=x\left|x\right|$ is differentiable at $x=0$.

• Show that the function $f$ defined as follows is continuous at $x=1$ and $x=2$ and that it is derivable at $x=2$ but not at $x=1$

$-$ $f\left(x\right)=\{\begin{array}{cc}x& \text{for}x<1\\ 2-x& \text{for}1\le x<2\\ -2+3x-x^2& \text{for}x>2\end{array}$

• Examine the continuity and derivability of the function $\varphi$ defined as follows:

$-$ $\varphi \left(x\right)=\{\begin{array}{cc}1& \text{when}x\in (-\infty ,0)\\ 1+\sin x& \text{when}x\in [0,\pi /2)\\ 2+(x-\pi /2{)}^2& \text{when}x\in [\pi /2,\infty )\end{array}$

2.2.2 Question 2

Find $f^{\prime }\left(0\right)$ for the following functions:

• $f\left(x\right)=x^2\sin \left(\frac{1}{x}\right)$ for $x\ne 0$

• $f\left(x\right)=e^{\cos x}$

• $f\left(x\right)=a\sin \left(\frac{x}{a}\right)$

2.2.3 Question 3

Find from first principles the derivatives of the following functions:

• $e^{\sin x}$

• $e^{\sqrt{x}}$

• $x^x$

• $\log \sin x$

• ${\tan }^{-1}x$

• $\sin x^2$

2.2.4 Question 4

Find the derivatives of:

• $\sin \varphi \left(x\right)$

• ${\tan }^3(a{x}^2+b)$

• $\log (x+\sqrt{x^2+a^2})$

• $\sec \left({\tan }^{-1}x\right)$

• ${\tan }^{-1}\frac{\sqrt{1+x^2}-1}{x}$

• $e^{x^2}$

• $e^{\sqrt{\cot x}}$

• $\log \sec (ax+b{)}^3$

• $\tan \log \cos e^{x^2}$

• $(\sin x{)}^{\log x}$

• $e^{x^x}$

• $(\text{sinh}x{)}^{\text{tanh}x}$

2.2.5 Question 5

Find $\frac{dy}{dx}$ of:

• $x=a(\cos t+t\sin t),y=a(\sin t-t\cos t)$

• $x=a\frac{a-t^2}{1+t^2},y=b\frac{2t}{1+t^2}$

2.2.6 Question 6

Find $\frac{dy}{dx}$ of:

• $x^3+y^3=3axy$

• $\tan xy=\sec (x^2+y^2)$

• $x^y=y^x$

• $x^m{y}^m=(x+y{)}^{m+n}$

2.2.7 Question 7

If $\sqrt{1-x^2}+\sqrt{1-y^2}=k(x-y)$ prove that $\frac{dy}{dx}=\frac{\sqrt{1-y^2}}{\sqrt{1-x^2}}$.