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:
\(\boldsymbol{f(x)=\left|x\right|}\) at \(x=0\)
\(\boldsymbol{f(x)=1, 0}\) or \(\boldsymbol{-1}\) according as \(\boldsymbol{x> =}\) or \(\boldsymbol{< 0}\) at \(\boldsymbol{x=0}\)
\(\boldsymbol{f(x) = \begin{cases} -x & \text{when } x\leq 0\\ x & \text{when } 0<x<1 \text{ at } x =0 \\ 2-x & \text{when } x \geq 1 \end{cases}}\)
\(\boldsymbol{f(x) = \frac{x-1}{1+e^{1/(x-1)}}, f(1)=0}\) at \(x=1\)
\(\boldsymbol{f(x) = \frac{x^4 + x^3 +2x^2}{\sin^2 x}, f(0)=0}\) at \(x=0\)
2.1.2 Question 2
Are the following functions continuous at the origin?
\(\boldsymbol{f(x) = \begin{cases} x\sin \frac{1}{x}, & x\neq 0\\ 0, & x =0 \end{cases}}\)
\(\boldsymbol{f(x) = \begin{cases} \sin \frac{1}{x}, & x\neq 0\\ 0, & x =0 \end{cases}}\)
\(\boldsymbol{f(x) = \begin{cases} (1+x)^{1/x} & \text{when } x\neq 0\\ 1 & \text{when } x =0 \end{cases}}\)
\(\boldsymbol{f(x) = (1+3x)^{1/x}, x \neq 0, f(0) = e^3}\)
2.1.3 Question 3
- For what value of \(k\) is the function
\(\phantom{--}\) \(\boldsymbol{f(x) = \begin{cases} x^2-1 & \text{for } x <3 \\ 2kx & \text{for } x \geq 3 \end{cases}}\)
\(\phantom{--}\)continuous at every value of \(x\).
- Determine the values of \(a\) and \(b\) so that the function \(f\) defined below is continuous everywhere:
\(\phantom{--}\) \(f(x) = \begin{cases} 1 & \text{when } x \leq 3 \\ ax+b & \text{when } 3<x<5\\ 7 & \text{when } x \geq 5 \end{cases}\)
2.2 Exercise 1(iv)
2.2.1 Question 1
- Determine whether \(f\) is continuous and has a derivative at origin where
\(\phantom{-}\) \(f(x) = \begin{cases} 2+x & \text{if } x \geq 0 \\ 2-x & \text{if } x<0 \end{cases}\)
- Examine for Continuity at \(x=a\) the function \(f\) where
\(\phantom{-}\) \(f(x) = \begin{cases} \frac{x^2}{a}-a & \text{, } 0<x<a \\ 0 & \text{, } x=a\\ a-\frac{a^3}{x^2} & \text{, } x>a \end{cases}\)
\(\phantom{--}\)Also examine if the function is derivable at \(x=a\).
Show that \(f(x)=x \left|x\right|\) 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 \(x=1\)
\(\phantom{-}\) \(f(x) = \begin{cases} x & \text{for } x<1 \\ 2-x & \text{for } 1\leq x<2\\ -2+3x-x^2 & \text{for } x>2 \end{cases}\)
- Examine the continuity and derivability of the function \(\phi\) defined as follows:
\(\phantom{-}\) \(\phi(x) = \begin{cases} 1 & \text{when } x \in (-\infty, 0) \\ 1+\sin x & \text{when } x \in [0, \pi/2)\\ 2 + (x-\pi/2)^2 & \text{when } x \in [\pi/2,\infty) \end{cases}\)
2.2.2 Question 2
Find \(f'(0)\) for the following functions:
\(\boldsymbol{f(x) = x^2\sin\left(\frac{1}{x}\right)}\) for \(x \ne 0\)
\(\boldsymbol{f(x) = e^{\cos x}}\)
\(\boldsymbol{f(x) = a \sin \left(\frac{x}{a}\right)}\)
2.2.3 Question 3
Find from first principles the derivatives of the following functions:
\(\boldsymbol{e^{\sin x}}\)
\(\boldsymbol{e^{\sqrt{x}}}\)
\(\boldsymbol{x^x}\)
\(\boldsymbol{\log \sin x}\)
\(\boldsymbol{\tan^{-1}x}\)
\(\boldsymbol{\sin x^2}\)
2.2.4 Question 4
Find the derivatives of:
\(\boldsymbol{\sin \phi(x)}\)
\(\boldsymbol{\tan^3 (ax^2+b)}\)
\(\boldsymbol{\log (x+\sqrt{x^2+a^2})}\)
\(\boldsymbol{\sec (\tan^{-1}x)}\)
\(\boldsymbol{\tan^{-1}\frac{\sqrt{1+x^2}-1}{x}}\)
\(\boldsymbol{e^{x^2}}\)
\(\boldsymbol{e^\sqrt{\cot x}}\)
\(\boldsymbol{\log \sec (ax+b)^3}\)
\(\boldsymbol{\tan \log \cos e^{x^2}}\)
\(\boldsymbol{(\sin x)^{\log x}}\)
\(\boldsymbol{e^{x^x}}\)
\(\boldsymbol{(\textbf{sinh}x)^{\textbf{tanh}x}}\)
2.2.5 Question 5
Find \(\frac{dy}{dx}\) of:
\(\boldsymbol{x = a (\cos t + t \sin t), y = a(\sin t - t\cos t)}\)
\(\boldsymbol{x = a\frac{a-t^2}{1+t^2}, y = b\frac{2t}{1+t^2}}\)