Quiz Chapter 05.05: Spline Method of Interpolation

1. The following $n$ data points, $\left( x_{1}, y_{1} \right), \, \left( x_{2}, y_{2} \right), \, ... \, , \, \left( x_{n}, y_{n} \right)$ are given. For conducting quadratic spline interpolation the $x$ -data needs to be

equally spaced

in ascending or descending order

integers

positive

2. In cubic spline interpolation,

the first derivatives of the splines are continuous at the interior data points

the second derivatives of the splines are continuous at the interior data points

the first and the second derivatives of the splines are continuous at the interior data points

the third derivatives of the splines are continuous at the interior data points

3. The following incomplete $y$ vs. $x$ data is given

$x$	$1$	$2$	$4$	$6$	$7$
$y$	$5$	$11$	????	????	$32$

The data is fit by quadratic spline interpolants given by

- - $f (x) = ax - 1, \, 1 \leq x \leq 2$
  - $f (x) = -2x^{2} + 14x - 9, \, 2 \leq x \leq 4$
  - $f (x) = bx^{2} + cx + d, \, 4 \leq x \leq 6$
  - $f (x) = 25x^{2} - 303x + 928, \, 6 \leq x \leq 7$

where $a, \, b, \, c,$ and $d$ , are constants. The value of $c$ is most nearly

-303.00

-144.50

-0.0000

14.000

4. The following incomplete $y$ vs. $x$ data is given

$x$	$1$	$2$	$4$	$6$	$7$
$y$	$5$	$11$	????	????	$32$

The data is fit by quadratic spline interpolants given by

- - $f (x) = ax - 1, \, 1 \leq x \leq 2$
  - $f (x) = -2x^{2} + 14x - 9, \, 2 \leq x \leq 4$
  - $f (x) = bx^{2} + cx + d, \, 4 \leq x \leq 6$
  - $f (x) = ex^{2} - fx + g, \, 6 \leq x \leq 7$

where $a, \, b, \, c, \, d, \, e, \, f,$ and $g$ , are constants. The value of $\dfrac{df}{dx}$ at $x = 2.6$ is most nearly

-144.50

-4.0000

-3.6000

12.200

5. The following incomplete $y$ vs. $x$ data is given

$x$	$1$	$2$	$4$	$6$	$7$
$y$	$5$	$11$	????	????	$32$

The data is fit by quadratic spline interpolants given by

- - $f (x) = ax - 1, \, 1 \leq x \leq 2$
  - $f (x) = -2x^{2} + 14x - 9, \, 2 \leq x \leq 4$
  - $f (x) = bx^{2} + cx + d, \, 4 \leq x \leq 6$
  - $f (x) = 25x^{2} - 303x + 928, \, 6 \leq x \leq 7$

where $a, \, b, \, c,$ and $d$ , are constants. What is the value of $\, \displaystyle\int_{1.5}^{3.5} f (x) \, dx$ ?

23.50

25.67

25.75

28.00

6. A robot needs to follow a path that passes through six data points as shown in the figure.

MC05.05 Question 6 Chart

To find the shortest path that is also smooth, you would recommend which of the following?

Pass a fifth order polynomial through the data.

Pass linear splines through the data

Pass quadratic splines through the data

Regress the data to a second order polynomial