Courses
MDE-MET-01 Calculus and Linear Algebra for Graduate Students
Single Variable Calculus

Calculus of a Single Variable

Definition: Derivative of a function

The derivative of a function $f(x)$ is given by

$$ f^{\prime}\left( x \right) = \lim_{h \rightarrow 0} \dfrac{f\left( x + h \right) - f\left(x\right)}{h}. $$

Note the following

  1. $f\left(x\right) = c \quad \Rightarrow f^{\prime}\left( x \right) = 0$
  2. $f\left(x\right) = x^a \quad \Rightarrow f^{\prime}\left( x \right) = a x^{a-1}$
  3. $f\left(x\right) = a^x \quad \Rightarrow f^{\prime}\left( x \right) = a^x \ln a$
  4. $f\left(x\right) = \log_{b}x \quad \Rightarrow f^{\prime}\left( x \right) = \dfrac{1}{x \log_e b}$
  5. $f\left(x\right) = \sin\left(x\right) \quad \Rightarrow f^{\prime}\left( x \right) = \cos\left(x\right)$
  6. $f\left(x\right) = \cos\left(x\right) \quad \Rightarrow f^{\prime}\left( x \right) = -\sin\left(x\right)$

Thus for $f(x)=a^x$, when $a=e$ then, $f=e^x$ and $f^{\prime}\left( x \right) = f\left( x \right) = e^x$.

Similarly, for $f\left(x\right) = \log_{b}x$ when $b=e$, i.e. $f\left(x\right) = \log_e x = \ln x$, so $f^{\prime}\left( x \right) = \dfrac{1}{x}$.

Definition: Summation Rule

For a function of the form $f = g+h$,

$$ \dfrac{\mathrm{d}\left(h+g\right)}{\mathrm{d} x} = h^{\prime}\left( x \right) + g^{\prime}\left( x \right). $$
Definition: Product Rule

For a function which is of the form $f = g h$

$$ \dfrac{\mathrm{d}\left(hg\right)}{\mathrm{d} x} = h^{\prime}\left( x \right)g\left(x\right) + h\left(x\right)g^{\prime}\left( x \right). $$
Definition: Quotient Rule

For a function which is of the form $f = g /h$

$$ \dfrac{\mathrm{d}f\left(x\right)}{\mathrm{d} x} = \dfrac{h\left(x\right)g^{\prime}\left( x \right) - h^{\prime}\left( x \right)g\left(x\right)}{ \left( h^\prime \left(x\right) \right)^2 }. $$
Definition: Chain Rule

The chain rule enables the derivative of a function which can be expressed as a composition of two differentiable functions. Thus, for a function which is of the from $f = g \left( h\left(x\right) \right)$

$$ \dfrac{\mathrm{d}f\left(x\right)}{\mathrm{d} x} = g^{\prime}\left( h \right) h^{\prime}\left(x\right). $$

Another expression is

$$ \dfrac{\mathrm{d}f\left(x\right)}{\mathrm{d} x} = \dfrac{\mathrm{d}g}{\mathrm{d} h} \dfrac{\mathrm{d}h}{\mathrm{d}x}. $$

This form can be understood as stating that if a function $f$ is written in terms of $g$, which itself depends on the variable $x$ (that is both $f$ and $g$ are dependent variables), then $f$ depends on $x$ as well, via the intermediate variable $g$.

For example, the function could be $\sin\left( x^2 \right)$, then write the function as $h(x) = f(g(x))$, where $f(x)=\sin(y(x))$ and $y(x)=x^2$, then the derivative is $f^{\prime} = 2x \cos\left( x^2\right)$.

Definition: Critical Point of a Function
If $f^{\prime} \left( x_0 \right) = 0$ for some $x_0$, then this point is called a critical point of $f$.

Critical points are candidates for being local maxima or minima for the function.

Global Maximum

If $f(x)$ is a continuous function of a closed, bounded interval, then it always attains a global maximum and global minimum on that interval.

Linear Approximations

The tangent line at a point $x_0$ is the linear approximation of $f(x)$ at the point $x_0$, that is

$$ y = f\left(x_0 \right) + f^{\prime} \left(x_0 \right) \left( x - x_0 \right). $$

This is of the form $y = mx + b$ where the gradient is $m=\left(x_0 \right)$ and the intercept is given by $b= f\left(x_0 \right) - x_0 f^{\prime} \left(x_0 \right)$.

Newton Methods

Second Derivatives

Assuming $x_0$ is a critical point of $f(x)$ then $f^{\prime}\left( x_0 \right)=0$, so the Taylor expansion about $x_0$ is

$$ f\left(x_0 + x \right) \approx f\left(x_0\right) + \dfrac{1}{2!} f^{\pprime}\left(x_0 \right) x^2 + \ldots $$

Then

  • If $f^{\prime\prime}\left(x_0 \right) < 0$, then $x_0$ is a local maxima
  • If $f^{\prime\prime}\left(x_0 \right) > 0$, then $x_0$ is a local minima
  • If $f^{\prime\prime}\left(x_0 \right) = 0$, then the test fails.