Questions: Consider the piecewise function f(t)= 4 if t<-8 3t+7 if -8 ≤ t<4 3cos(7t) if 4 ≤ t<8 74e^(3t) if t ≥ 8 This can be written using step functions as : f(t)=□+u-8(t) · □+ u4(t) · □+u8(t) · □

Transcript text: Consider the piecewise function \[ f(t)=\left\{\begin{array}{cc} 4 & t<-8 \\ 3 t+7 & -8 \leq t<4 \\ 3 \cos (7 t) & 4 \leq t<8 \\ 74 e^{3 t} & t \geq 8 \end{array}\right. \] This can be written using step functions as : \[ \begin{array}{l} f(t)=\square+u_{-8}(t) \cdot \square+ \\ u_{4}(t) \cdot \square+u_{8}(t) \cdot \square \end{array} \]

Solution

Solution Steps

To express the given piecewise function using step functions, we need to break down each segment of the function and represent it in terms of the Heaviside step function \( u_c(t) \), which is 0 for \( t < c \) and 1 for \( t \geq c \).

For \( t < -8 \), the function is \( f(t) = 4 \).
For \( -8 \leq t < 4 \), the function is \( f(t) = 3t + 7 \). This segment starts at \( t = -8 \), so we need to subtract the value of the previous segment at \( t = -8 \) to ensure continuity.
For \( 4 \leq t < 8 \), the function is \( f(t) = 3 \cos(7t) \). This segment starts at \( t = 4 \), so we need to subtract the value of the previous segment at \( t = 4 \) to ensure continuity.
For \( t \geq 8 \), the function is \( f(t) = 74 e^{3t} \). This segment starts at \( t = 8 \), so we need to subtract the value of the previous segment at \( t = 8 \) to ensure continuity.

Solution Approach

Start with the first segment \( f(t) = 4 \).
Add the second segment \( (3t + 7 - 4) \cdot u_{-8}(t) \).
Add the third segment \( (3 \cos(7t) - (3 \cdot 4 + 7)) \cdot u_{4}(t) \).
Add the fourth segment \( (74 e^{3t} - 3 \cos(7 \cdot 8)) \cdot u_{8}(t) \).

Step 1: Define the Piecewise Function

The piecewise function \( f(t) \) is defined as follows:

\[ f(t) = \begin{cases} 4 & \text{if } t < -8 \\ 3t + 7 & \text{if } -8 \leq t < 4 \\ 3 \cos(7t) & \text{if } 4 \leq t < 8 \\ 74 e^{3t} & \text{if } t \geq 8 \end{cases} \]

Step 2: Express Using Step Functions

We can express \( f(t) \) using Heaviside step functions \( u_c(t) \):

\[ f(t) = 4 + (3t + 7 - 4) u_{-8}(t) + (3 \cos(7t) - (3 \cdot 4 + 7)) u_{4}(t) + (74 e^{3t} - 3 \cos(7 \cdot 8)) u_{8}(t) \]

Step 3: Evaluate the Function at Specific Points

To understand the behavior of \( f(t) \), we can evaluate it at key points:

For \( t = -10 \): \[ f(-10) = 4 \]
For \( t = -8 \): \[ f(-8) = 3(-8) + 7 = -24 + 7 = -17 \]
For \( t = 0 \): \[ f(0) = 3 \cdot 0 + 7 = 7 \]
For \( t = 4 \): \[ f(4) = 3 \cos(7 \cdot 4) = 3 \cos(28) \]
For \( t = 8 \): \[ f(8) = 74 e^{3 \cdot 8} = 74 e^{24} \]

Step 4: Calculate Specific Values

Calculating the values for \( f(4) \) and \( f(8) \):

For \( f(4) \): \[ f(4) \approx 3 \cdot 0.8480 = 2.544 \]
For \( f(8) \): \[ f(8) \approx 74 \cdot 2.648 \times 10^{10} \approx 1.961 \times 10^{12} \]