Univariate Calculus

A univariate function is defined as \(f:\R\to\R\)

Continuity of a univariate function \(f\)

A function \(f:\R\to\R\) is continous if the \(lim_x\to x^* f(x)\) exists and is equal to \(f(x^*)\).

Tip

If there are no sudden breaks in the graph or the graph does not break in the domain specified than the function is continous in the specified domain.

Differentiability of a univariate function \(f\)

A function \(f:\R\to\R\) is differentiable if the \(f'(x^*) = lim_{h\to 0} \frac{f(x^*+h)-f(x^*)}{h}\) exists.

Warning

If \(f\) is not continous at \(x^*\), then the function is not differentiable at \(x^*\). But if a function is not differentiable at \(x^*\),then it is not necessary that the function is not continous at \(x^*\).

Tip

If the graph of the function has no sudden breaks or no sharp corners in the specified domain, then the function is differentiable in the specified domain.

Derivatives and Linear Approximation of a function \(f\)

Let \(f:\R\to\R\) be a differentiable function.

Then the derivate of the function \(f\) at \(x^*\) is defined as

\[ f'(x^*) = lim_{x\to x^*}\cfrac{f(x)-f(x^*)}{x-x^*}\\ \ \\ \text{Let } f'(x^*) \approx \cfrac{f(x)-f(x^*)}{x-x^*}\text{ (Around } x=x*)\\\ \\ f(x) \approx f(x^*)+f'(x*)(x-x^*) \]

Now \(f(x^*)+f'(x*)(x-x^*)\) is known as linear approximation \(L_{x^*}[f](x)\) of \(x\) around \(x=x^*\).

Tip

$f(x)\approx L_{x^}f $, where $ L_{x^}f =f(x^)+f'(x)(x-x^*)$

Linear Approximations And Tangent Lines

Graph of $ L_{x^*}[f]$, i.e. \(G_{ L_{x^*}[f]}\) is a tangent to the graph of \(f\), i.e., \(G_f\) at the point \((x^*,f(x^*))\)

Some rules and Advanced Applications

Higher order Approximations

Product Rule

Let \(f(x) = g(x).h(x)\) and \(x^*=0\). We need to find \(f'(x)\).

Now, using linear approximation and approximating \(g(x)\) and \(f(x)\) seprately.

\[ f(x) \approx (g(0) + xg'(0))(h(0) + xh'(0))\\ \ \\ f(x) \approx g(0)h(0) + x[g'(0)h(0)+h'(0)g(0)]+x^2g'(0)h'(0) \dots (1)\\ \ \\ \ \\ \text{ We know that }, L_{x^{*}}[f](0) = f(0) + xf'(0)\dots(2) \]

Comparing \((1)\) with \((2)\) we get \(f'(0) = g'(0)h(0) + g(0)h'(0)\)

Chain rule

Let \(f(x) = g(h(x))\).

Now approximating \(h(x)\) around \(0\). We get, \(f(x) \approx g(h(0) + xh'(0))\)

Now approximating \(g(x)\) around \(h(0)\). We get, \(f(x) \approx g(h(0)) + g'(h(0))h'(0)x \dots (1)\)

We know that \(L_{x^*}[f](0) = g(h(0))+f'(0)x \dots(2)\)

Comparing \((1)\) and \((2)\), We get, \(f'(0) = g'(h(0)).h'(0)\)

Maxima, Minima And Saddle Point

Since \(L_{x^*}[f] = f(x^*)+f'(x^*)(x-x^*)\) and if \(f'(x^*) = 0\), then the linear approximation of \(f(x)\) becomes a constant and this happens only at either the maxima, minima or a saddle point.

Hence if \(f'(x^*) = 0\), then \(x^*\) is a critical point.