Chapter 3, Problem 3.3E

Interpretation Introduction

(a)

Interpretation:

The pulse width of the given function is to be determined.

Concept introduction:

The pulse width of a function is calculated by equating the function to 0 and then obtaining the value of the x - coordinate.

Interpretation Introduction

(b)

Interpretation:

The given function of $u\left(t\right)$ is to be expressed as the sum of the simpler functions.

Concept introduction:

For a function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$

Here, $F\left(s\right)$ represents the Laplace transform, $s$ is a variable which is complex and independent, $f\left(t\right)$ is any function of time which is being transformed, and $\mathcal{L}$ is the operator which is defined by an integral.

Interpretation Introduction

(c)

Interpretation:

The Laplace transform of the simpler function determined in part (b) is to be calculated.

Concept introduction:

For a function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$

Here, $F\left(s\right)$ represents the Laplace transform, $s$ is a variable which is complex and independent, $f\left(t\right)$ is any function of time which is being transformed, and $\mathcal{L}$ is the operator which is defined by an integral.

Interpretation Introduction

(d)

Interpretation:

The area under the pulse for the given function is to be calculated.

Concept introduction:

Area of a right-angled triangle is calculated as:

$A_{\text{triangle}}=\frac{1}{2}\times \left(\text{height}\right)\times \left(\text{width}\right)$