Question

The formula \(I(t)=\frac{\sin t}{e^{t}}\) is the formula for a decaying alternating current. a. Complete the following table with the appropriate values. $$ \begin{array}{|l|l|} \hline t & \frac{\sin t}{e^{t}} \\ \hline 0 & \text { (i) } \\ \hline \frac{\pi}{2} & \text { (ii) } \\ \hline \pi & \text { (iii) } \\ \hline \frac{3 \pi}{2} & \text { (iv) } \\ \hline 2 \pi & \text { (v) } \\ \hline 2 \pi & \text { (vi) } \\ \hline 3 \pi & \text { (vii) } \\ \hline \frac{7 \pi}{2} & \text { (viii) } \\ \hline 4 \pi & \text { (ix) } \\ \hline \end{array} $$ b. Using only the values in the table, determine where the tangent line to the graph of \(I(t)\) is horizontal.

Short answer

Horizontal tangents occur at \( t = 0, \pi, 2\pi, 3\pi, 4\pi \).

Step-by-step solution

Calculate the Value for t = 0

For \( t = 0 \), compute \( I(0) \):\[I(0) = \frac{\sin 0}{e^{0}} = \frac{0}{1} = 0\]Thus, the value for \( t = 0 \) is 0.

Calculate the Value for t = \(\frac{\pi}{2}\)

For \( t = \frac{\pi}{2} \), compute \( I\left(\frac{\pi}{2}\right) \):\[I\left(\frac{\pi}{2}\right) = \frac{\sin \frac{\pi}{2}}{e^{\frac{\pi}{2}}} = \frac{1}{e^{\frac{\pi}{2}}}\]Thus, the value for \( t = \frac{\pi}{2} \) is \( \frac{1}{e^{\frac{\pi}{2}}} \).

Calculate the Value for t = \(\pi\)

For \( t = \pi \), compute \( I(\pi) \):\[I(\pi) = \frac{\sin \pi}{e^{\pi}} = \frac{0}{e^{\pi}} = 0\]Thus, the value for \( t = \pi \) is 0.

Calculate the Value for t = \(\frac{3\pi}{2}\)

For \( t = \frac{3\pi}{2} \), compute \( I\left(\frac{3\pi}{2}\right) \):\[I\left(\frac{3\pi}{2}\right) = \frac{\sin \frac{3\pi}{2}}{e^{\frac{3\pi}{2}}} = \frac{-1}{e^{\frac{3\pi}{2}}}\]Thus, the value for \( t = \frac{3\pi}{2} \) is \( \frac{-1}{e^{\frac{3\pi}{2}}} \).

Calculate the Value for t = \(2\pi\)

For \( t = 2\pi \), compute \( I(2\pi) \):\[I(2\pi) = \frac{\sin 2\pi}{e^{2\pi}} = \frac{0}{e^{2\pi}} = 0\]Thus, the value for \( t = 2\pi \) is 0.

Calculate the Value for t = \(3\pi\)

For \( t = 3\pi \), compute \( I(3\pi) \):\[I(3\pi) = \frac{\sin 3\pi}{e^{3\pi}} = \frac{0}{e^{3\pi}} = 0\]Thus, the value for \( t = 3\pi \) is 0.

Calculate the Value for t = \(\frac{7\pi}{2}\)

For \( t = \frac{7\pi}{2} \), compute \( I\left(\frac{7\pi}{2}\right) \):\[I\left(\frac{7\pi}{2}\right) = \frac{\sin \frac{7\pi}{2}}{e^{\frac{7\pi}{2}}} = \frac{1}{e^{\frac{7\pi}{2}}}\]Thus, the value for \( t = \frac{7\pi}{2} \) is \( \frac{1}{e^{\frac{7\pi}{2}}} \).

Calculate the Value for t = \(4\pi\)

For \( t = 4\pi \), compute \( I(4\pi) \):\[I(4\pi) = \frac{\sin 4\pi}{e^{4\pi}} = \frac{0}{e^{4\pi}} = 0\]Thus, the value for \( t = 4\pi \) is 0.

Determine Horizontal Tangent Lines

The tangent line is horizontal where the derivative of \( I(t) \), denoted \( I'(t) \), equals 0. Evaluate the known values:For \( t = 0, \pi, 2\pi, 3\pi, 4\pi \), \( I(t) = 0 \).Therefore, the tangent line to the graph of \( I(t) \) is horizontal at \( t = 0, \pi, 2\pi, 3\pi, 4\pi \).

Key concepts

Alternating Current

Alternating current (AC) is a type of electrical flow where the direction of the current reverses periodically. This flow can be harnessed to power our electrical systems efficiently. Unlike direct current (DC) which flows only in one direction, AC is more compatible with transformers, allowing electricity to be transmitted over long distances.
The behavior of alternating current can be described mathematically using functions, one of which is the decaying alternating current modeled in the exercise with the formula:

• \( I(t) = \frac{\sin t}{e^{t}} \) combines both trigonometric (sine) and exponential functions.

With each cycle, the alternating current's amplitude decreases, a behavior captured by the exponential decay term \( e^t \). As time \( t \) increases, \( e^t \) grows, leading to a decrease in the overall value of \( I(t) \).

• This decay reflects a continual reduction in current magnitude despite regular oscillations depicted by the sine function.

AC systems often use sine waves, because they naturally describe oscillating currents. Understanding this allows students to relate mathematical functions to real-world technical phenomena.

Trigonometric Functions

Trigonometric functions are fundamental in describing periodic phenomena like waves. The function \( \sin(t) \) is particularly significant in the study of alternating current due to its wave-like properties.

• It oscillates between -1 and 1, creating a smooth, repetitive wave pattern over time.

In the context of the given function \( I(t) = \frac{\sin t}{e^{t}} \), \( \sin(t) \) represents the wave's oscillation aspect, indicative of the alternating component of the current.
Trigonometric identities, such as the fact that \( \sin(t) = 0 \) at multiples of \( \pi \) (0, \( \pi, 2\pi \), etc.), are key to solving the exercise:

• This identity explains why \( I(t) = 0 \) at these points, leading to horizontal tangent lines where the rate of change of the function's graph is zero.

By leveraging the periodic nature of \( \sin(t) \), students can accurately predict and model the changes in alternating current systems.

Exponential Functions

Exponential functions feature prominently in calculus and describe processes that grow or decay at rates proportional to their current size. In this exercise, \( e^{t} \) plays a crucial role as it models the decay of the current:

• The function \( e^t \) grows rapidly as \( t \) increases, resulting in a reduction of \( \frac{\sin t}{e^{t}} \) over time.

The decay seen in \( I(t) = \frac{\sin t}{e^{t}} \) models real-world phenomena where an initial strength decreases exponentially, like radioactive decay or cooling.
As time progresses:

• The exponential term’s growth outpaces the oscillating sine term, reducing the overall influence of \( \sin(t) \) and diminishing \( I(t) \).

Understanding exponential decay is vital for comprehending how systems lose energy over time, providing students with insights into natural processes and engineering applications.