1b. Products Involving Unit Step Functions

When combined with other functions defined for `t > 0`, the unit step function "turns off" a portion of their graph.

Later, on this page...

Time shifting

The concept is related to having a switch in an electronic circuit open for a period of time (so there is no current flow), then the switch is closed (so the current begins to flow).

Example 1 - Products with Unit Functions

(a) If `f(t) = sin\ t`, then the graph of `g(t) = sin\ t · u(t − 2π)` is

Product sine curve and unit function

The `sin\ t` portion starts at `t = 2π`, because we have multiplied `sin\ t` by `u(t − 2π)`.

We use the dot (`·`) for multiplication so that it is easier to read.

(b) If `f(t) = 10e^(-2t)`, then the graph of `g(t) = 10e^(-2t)· u(t − 5)` is

Product exponential curve and unit function

The portion `10e^(-2t)` starts at `t = 5`.

Product of u(t) vs. Shifting the Function Along the t-axis

Note the differences between the following:

`f(t) · u(t)`, where the `f(t)` part begins at `t = 0`.

`f(t) · u(t − a)`, where the `f(t)` part begins at `t = a`.

`f(t − a) · u(t)`, where the `f(t)` part has been shifted to the right by `a` units and begins at `t = 0`.

`f(t − a) · u(t − a)`, where the `f(t)` part has been shifted to the right by `a` units and begins at `t = a`.

Let's see some examples.

Example 2

Let `f(t) = 4t + 2` and `a = 1`. We see different combinations of shifting with different starting points.

(a) `g_1(t) = f(t) · u(t) = (4t + 2) · u(t)`

Example product unit function

In this example, the `4t + 2` part starts at `t = 0`.

(b) `g_2(t) = f(t) · u(t − a) = (4t + 2) · u(t − 1)`

Example product unit function

In this example, the `4t + 2` part starts at `t = 1`.

(c) `g_3(t) = f(t − a) · u(t) = (4(t − 1) + 2) · u(t) = (4t − 2) · u(t)`

Example product unit function

In this example, the `4t + 2` part has been shifted 1 unit to the right and starts at `t = 0`.

(d) `g_4(t) = f(t − a) · u(t − a) = (4t − 2) · u(t − 1)`

Example product unit function

In this example, the `4t + 2` part has been shifted 1 unit to the right (like example (c)) and starts at `t = 1`.

Example 3

Let `f(t) = sin\ t` and `a = 0.7` and we combine them to shift our graph and start at different times, similar to what we did in Example 1.

(a) `g_1(t) = sin\ t · u(t)`

Example product sine and unit function

In this example, the `sin\ t` part starts at `t= 0`.

(b) `g_2(t) = sin\ t · u(t − 0.7)`

Example product sine and unit function

In this example, the `sin\ t` part starts at `t = 0.7`.

(c) `g_3(t) = sin (t − 0.7) · u(t)`

Example product sine and unit function

In this example, the `sin\ t` part has been shifted `0.7` units to the right, and it starts at `t=0`.

(d) `g_4(t) = sin (t − 0.7) · u(t − 0.7)`

Example product sine and unit function

In this example, the `sin\ t` part has been shifted `0.7` units to the right, and it starts at `t = 0.7`.

Exercises

Need Graph Paper?

rectangular grid
Download graph paper

Rewrite the following functions in a suitable way and then sketch the functions:

1. `f(t) = u(t) + (1 − t) · u(t − 1)` `+ (t − 2) · u(t − 2)`

2. `f(t) = t^2 · u(t) − (t^2− 4) · u(t − 2)`

3. `f(t) = u(t) + (sin\ t − 1) · u(t − π/2) − (sin\ t + 1) · u(t − (3π)/2) + u(t − 2π)`

4. `f(t) = 3t^2· u(t)` `+ (12 − 3t^2) · u(t − 2)` ` + (4t − 40) · u(t − 4)` ` − 4(t − 7) · u(t − 7)`

Scientific Notebook Aside...

NOTE: To graph unit step functions using Scientific Notebook, we must realise that SNB recognises "Heaviside(t)", but not u(t).

So we need to define `u(t)` as `"Heaviside"(t)`, so SNB will graph it properly. Simply type `u(t) = "Heaviside"(t)`, and click on the "New definition" button. Nothing seems to happen, but if you click on the "Show Definitions" button you will see that it is defined. Now you can graph unit step functions in terms of `u(t)`.

Graph of `y = 3·u(t − 4)`

SNB graph of unit step function

Didn't find what you are looking for on this page? Try search:

Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Ready for a break?

 

Play a math game.

(Well, not really a math game, but each game was made using math...)

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.

Share IntMath!

Calculus Lessons on DVD

 

Easy to understand calculus lessons on DVD. See samples before you commit.

More info: Calculus videos

Loading...
Loading...