Quiz Chapter 02.03: Differentiation of Discrete Functions

1. The definition of the first derivative of a function $f \left( x \right)$ is

{f}' \left( x \right) = \dfrac{f \left( x + \Delta x \right) + f \left( x \right)}{\Delta x}

{f}' \left( x \right) = \dfrac{f \left( x + \Delta x \right) - f \left( x \right)}{\Delta x}

{f}' \left( x \right) = \lim_{x\rightarrow 0} \dfrac{f \left( x + \Delta x \right) + f \left( x \right)}{\Delta x}

{f}' \left( x \right) = \lim_{x\rightarrow 0} \dfrac{f \left( x + \Delta x \right) - f \left( x \right)}{\Delta x}

2. Using forward divided difference with a step size of $0.2$ , the derivative of the function at $x=2$ is given as

$x$	$1.8$	$2.0$	$2.2$	$2.4$	$2.6$
$f \left( x \right)$	$6.0496$	$7.3890$	$9.0250$	$11.023$	$13.464$

6.697

7.389

7.438

8.179

3. A student finds the numerical value of ${f}' \left( x \right) = 20.219$ at $x=3$ using a step size of $0.2$ . Which of the following methods did the student use to conduct the differentiation if it is given in the table below?

$x$	$2.6$	$2.8$	$3.0$	$3.2$	$3.4$	$3.6$
$f \left( x \right)$	$e^{2.6}$	$e^{2.8}$	$e^{3}$	$e^{3.2}$	$e^{3.4}$	$e^{3.6}$

Backward Divided Difference

Calculus, that is, exact

Central Divided Difference

Forward Divided Difference

4. The upward velocity is given as a function in the table below.

$t \left( s \right)$	$10$	$15$	$20$	$22$
$v \left( m/s \right)$	$22$	$36$	$57$	$10$

To find the acceleration at $t=17 \, s$ , a scientist finds a second order polynomial approximation for velocity, and then differentiates it to find the acceleration. The estimate of the acceleration at $t=17$ s in m/s² most nearly is

4.060

4.200

8.157

8.498

5. The velocity of the rocket is given as a function of time in the table below.

$t \left( s \right)$	$0$	$0.5$	$1.2$	$1.5$	$1.8$
$v \left( m/s \right)$	$0$	$213$	$223$	$275$	$300$

Allowed to use forward, backward, or central divided difference approximation of the first derivative, your best estimate for the acceleration $a=\dfrac{dv}{dt}$ of the rocket in m/s² at $t=1.5$ seconds is

83.33

128.33

173.33

183.33

6. In a circuit with an inductor of inductance $L$ , a resistor with resistance $R$ , and a variable voltage source $E \left( t \right)$ , where $E \left( t \right) = L \dfrac{di}{dt} + Ri$ , the current, $i$ , is measured at several values of time as

Time $t \left( s \right)$	$1.00$	$1.01$	$1.03$	$1.1$
Current, $i$ (amperes)	$3.10$	$3.12$	$3.18$	$3.24$

If $L=0.98$ Henries and $R=0.142$ ohms, your choice for most accurate answer for $E \left( 1.00 \right)$ would be

0.98 \times \dfrac{3.24-3.10}{0.1} + \left( 0.142 \right) \left( 3.10 \right)

0.142 \times 3.10

0.98 \times \dfrac{3.12-3.10}{0.01} + \left( 0.142 \right) \left( 3.10 \right)

0.98 \times \dfrac{3.12-3.10}{0.01}