MTH341
The \(n\)th derivative \(f^{(n)} of the function \(f: \mathbb{R}\rightarrow \mathbb{R}\) defined by \(f(x)=|x|\) for all \(x\in\mathbb{R}\) is
\(f'(x)=\left\{\begin{array}{rcl}-x&\mbox{if}&x> 0\\x&\mbox{if}&x<0\end{array}\right}\).
\(f'(x)=\left\{\begin{array}{rcl}-1&\mbox{if}&x> 0\\1&\mbox{if}&x<0\end{array}\right}\).
\(f'(x)=\left\{\begin{array}{rcl}x&\mbox{if}&x> 0\\-1&\mbox{if}&x<0\end{array}\right}\).
\(f ‘(x)=\left\{\begin{array}{rcl}1&\mbox{if}&x> 0\\-1&\mbox{if}&x<0\end{array}\right}\).
A real function \(f\) defined on an interval \([a,b]\) with \(a<c<b\) where \(c\) is a point of the interval, is said to be differentiable at the point \(x=c\) if
\(\(f'(c)=lim_{x}\frac{f(x)-f(b)}{x-c}\) exist and is infinite
\(\(f'(c)=lim_{x\rightarrow c}\frac{f(x)-f(c)}{x-c}\) exist and is finite
\(\(f'(c)=lim_{x\rightarrow c}\frac{f(c)-f(b)}{a-b}\) exist
\(\(f'(c)=lim_{x\rightarrow c}\frac{f(a)-f(c)}{b-c}\) exist and is finite
Evaluate \(lim_{x\rightarrow 0}\frac{\sqrt{2}-2Cos(\frac{\pi}{4}+x)}{x}\)
\(\sqrt{2}\)
\(2\)
\(2Cos x\)
\(2Cos(\frac{\pi}{4})\)
Let \(\phi(x)=\left\|\begin{array}{rcl}f(x)&g(x)&h(x)\\-f(a)&g(a)&h(a)\\f(b)&g(b)&h(b)\end{array}\right|\), \(g(x)=x\) and \(h(x)\) for all \(x\in[a,b]\), to deduce the Lagrange’s mean value theorem from the Generalized mean value theorem we must obtain the determinant
\(\left\|\begin{array}{rcl}1&g'(x)&0\\f(a)&a&1\\f(b)&a&1\end{array}\right|=0\)
\(\left\|\begin{array}{rcl}f'(x)&g'(x)&0\\f(a)&a&1\\f(b)&b&1\end{array}\right|=0\)
\(\left\|\begin{array}{rcl}f'(x)&g'(x)&0\\f(a)&g(a)&1\\f(b)&g(b)&1\end{array}\right|=0\)
\(\left\|\begin{array}{rcl}f'(x)&1&0\\f(a)&a&1\\f(b)&b&1\end{array}\right|=0\)
Applying Cauchy’s mean value theorem to the function \(f\) and \(g\) defined as \(f(x)=x^2\) and \(g(x)=x\) for all \(x\in[a,b]\), gives
\(c=\frac{1}{2}(a+b)\)
\(c=a^2+b\)
\(c=a+b\)
\(c=\frac{a}{2}+b\)
