The function $f(x) = e^{2x} + 2x^2 - 4$ is approximated with least-squares regression, based on 21 samples of $f$ at $x=\{0,1,2,\dots,20 \}$.  The regressor is a linear combination of the following 6 basis functions:

$\varphi_0(x) = 1, \quad   \varphi_3(x) = e^x$
$\varphi_1(x) = x, \quad   \varphi_4(x) = e^{2x}$
$\varphi_2(x) = x^2, \quad \varphi_5(x) = \sin x$
What is the error in the regressor at $x=1/2$?

Consider i) interpolation and ii) regression of a two-dimensional function $f(\textbf{x})$ where $f:\mathbb{R}^2\rightarrow\mathbb{R}$.  Assume that the approximating function $p(\textbf{x})$ in both cases is a linear combination of $N$ distinct basis functions $\phi_i(\mathbf{x})$, and that $f$ is sampled at $M$ distinct locations.  The interpolation conditions for i) lead to a system of linear equations, with system matrix $A$.

Consider the existence and uniqueness of solutions to these two problems.  What condition is required on i) and ii) respectively, to guarantee unique solutions for each problem?

A $\it cubic$ spline with natural boundary-conditions is used to intepolate only $\it two$ data-points, $(x_0, f_0)$ and $(x_1, f_1)$, resulting in an interpolant $s(x)$.  Which one of the following statements is true?

We interpolate the function $f(x,y) = x^2 + y^2$ using square elements (patches) from a uniform grid with spacing of $1$ along both axes, with a node at the origin -- i.e. $(x_i,y_j) = (i,j)$, $i,j\in \{0,\dots,N\}$.  We use the polynomial basis $(1, x, y, xy)$ on each of these square elements.  Let the resulting approximation to $f(x,y)$ be $s(x,y)$.  What is $s(x,0)$ on the interval $[0,N]$ equal to?  [Let $X = (i, f(i,0))$ for $i\in\{0,\dots,N\}$ be the data (nodes and values) along the x-axis.]

We attempt to approximate the function $f(x) = x^2$ using samples at the nodes $x_i = [-10,0,10]$ with an interpolant of the form

where $r_i := |x-x_i|$ and $\epsilon = 1000$. Which of the following relations is/are true?

$I$: $a_3 = a_1$

$II$: $a_2 \approx 0$
$III$: $s(100) \approx 0$

For a sufficiently smooth function $f(x)$, consider the two finite-difference formulae approximating $f''(0)$:

$f''(0) = \frac{f(h) - 2f(0) + f(-h)}{h^2} + \mathcal{O}(h^p)$

$f''(0) = \frac{f'(h) - f'(-h)}{2h} + \mathcal{O}(h^q)$

What are the constants $p$ and $q$?

Apply the forward-difference scheme:

$\mathcal{F}(x) = \frac{f(x+h) - f(x)}{h} \approx f'(x)$

to evaluate the derivative of a 2nd-degree polynomial

$f(x) = ax^2 + bx + c.$

For any choice of step-size $h$, which $\it one$ of the following is true of the error $\epsilon := f'(x) - \mathcal{F}(x)$?

Consider quadrature rules for the unit triangle (denoted $\triangle$), with nodes at its corners, namely: $\textbf{x}_0 = (0,0)$, $\textbf{x}_1 = (1,0)$, $\textbf{x}_2= (0, 1)$.  The rules are of the form:

for some weights $w_i$.  Shape functions for this triangular patch are $p_0(x,y) = 1-x-y$, $p_1(x,y) = x$ and $p_2(x,y) = y$, which take the value $1$ at the corresponding node, and $0$ at all other nodes.  Given this information, what values of the weights $(w_0,w_1,w_2)$ integrate all functions of the form $f(\textbf{x}) = a + bx + cy$ on $\triangle$ exactly?

In statistics, $\it weighted$ integrals of functions $f(x)$ are of interest, for example

which is approximated by a quadrature rule in the standard form in the above equation.  Suppose that $N=2$ and the nodes are $\textbf{x} = (x_0,x_1,x_2)= (-1,0,1)$.  What weights $(w_0,w_1,w_2)$ lead to the quadrature rule being exact for all polynomials of degree 2 and less?  [Note: $\int_{-\infty}^{\infty} e^{-x^2}\,\mathrm{d}x = \sqrt{\pi}$ and $\int_{-\infty}^{\infty} x^2 e^{-x^2}\,\mathrm{d}x = \sqrt{\pi}/2$.]

Until now we have considered quadrature rules that integrate polynomials of degree $d$ exactly.  Using the same principles derive a quadrature rule that integrates the functions $\phi = [1, \sin(x), \cos(x)]$ exactly on the interval $x\in [0,\frac{\pi}{2}]$.  For nodes use $x = (0, \frac{\pi}{4},\frac{\pi}{2})$.  What are the corresponding weights?