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Chapter 7: Problem 7

$$e^{2 t} \cos 3 t$$

Short Answer

Expert verified
The expression \(e^{2 t} \cos 3 t\) represents a function that oscillates due to the \(\cos 3t\) factor, but its peaks and troughs grow exponentially because of the \(e^{2t}\) factor. It does not represent an equation to be solved, but rather a function to be understood in terms of its composite parts.

Step by step solution

01

Identify the functions

The given expression, \(e^{2 t} \cos 3 t\), is a product of two different functions - an exponential function \(e^{2t}\) and a cosine function \(\cos 3t\). Exponential functions are known for their rapid growth rate while cosine functions oscillate between -1 and 1, and display a wave-like pattern.
02

Understand the functions separately

The exponential function \(e^{2t}\) will grow exponentially as \(t\) increases, doubling its rate with each unit increment in \(t\). On the other hand, \(\cos 3t\) will oscillate between -1 and 1 with a period of \(\frac{2\pi}{3}\) due to the factor of 3 in front of \(t\). This means that it completes a cycle three times faster than a regular cosine function.
03

Combine the functions

When these two functions are multiplied together, \(e^{2 t} \cos 3 t\), the resulting function will mimic the oscillation of the cosine function due to its wave-like properties, but each peak and trough of the oscillations will grow according to the exponential function \(e^{2t}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Exponential Functions
Exponential functions are mathematical expressions where a constant base is elevated to the power of a variable, typically denoted as \( e^{x} \). Here, \( e \) stands for Euler's number, which is approximately 2.71828. It is a vital constant in mathematics due to its natural properties.
  • The exponential function \( e^{2t} \) in our expression grows at a constant rate of 2, meaning its rate of growth doubles every time \( t \) increases by 1. This results in an extremely rapid increase as \( t \) gets larger.
  • Exponential functions are widely used in modeling real-world phenomena such as population growth, radioactive decay, and financial interest calculations because of their unique growing nature.
The key aspect of an exponential function is its capability to model processes that undergo consistent growth or decay. In conjunction with other functions, such as trigonometric ones, they can yield complex behaviors.
Unraveling Trigonometric Functions
Trigonometric functions, like cosine, are derived from the geometry of triangles, particularly right-angled triangles. In our equation, the trigonometric element is represented by \( \cos 3t \).
  • Trigonometric functions like the cosine oscillate between fixed values--in the case of cosine, these are -1 and 1.
  • The expression \( \cos 3t \) implies a cosine function with a period decreased to \( \frac{2\pi}{3} \). This happens due to the '3' modifying factor, making it cycle three times in a full period compared to the standard cosine, which has a period of \( 2\pi \).
These oscillating properties make trigonometric functions extremely useful in modeling cyclical behaviors, such as temperature changes, sound waves, and electromagnetic waves. Their periodic nature allows them to fit the "ebb and flow" pattern seen in various natural and human-made systems.
Exploring Function Composition
Function composition involves combining two or more functions to produce a new, complex function. It allows us to understand how the behavior of these functions influences each other. In our case, we have the function \( e^{2t} \cos 3t \).
  • When multiplying \( e^{2t} \) (an exponential function) with the trigonometric function \( \cos 3t \), the composite function maintains the cyclical traits of cosine, as it oscillates, but with an exponentially growing amplitude.
  • The exponential part \( e^{2t} \) causes the amplitude of the waves generated by \( \cos 3t \) to increase rapidly, which means the overall peaks and troughs elevate dramatically as \( t \) grows.
Function composition is a powerful mathematical tool that allows the integration of different types of behaviors within a singular expression. This ability to merge discrete patterns makes composed functions invaluable in fields like physics for describing complex, dynamic systems.

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Most popular questions from this chapter

$$\begin{array}{l}{\text { A mass attached to a spring is released from rest } 1 \mathrm{m}} \\ {\text { below the equilibrium position for the mass-spring sys- }} \\ {\text { tem and begins to vibrate. After } \pi / 2 \text { sec, the mass is }} \\ {\text { struck by a hammer exerting an impulse on the mass. The }} \\ {\text { system is governed by the symbolic initial value problem }}\\\ {\frac{d^{2} x}{d t^{2}}+9 x=-3 \delta\left(t-\frac{\pi}{2}\right)} \\ {\quad x(0)=1, \quad \frac{d x}{d t}(0)=0}\end{array} $$where \(x(t)\) denotes the displacement from equilibrium at time \(t .\) What happens to the mass after it is struck?

36\. Using Theorem 5 in Section 7.3 and the convolution theorem. show that $$\begin{aligned} \int_{0}^{t} \int_{0}^{v} f(z) d z d v &=\mathscr{L}^{-1}\left\\{\frac{F(s)}{s^{2}}\right\\}(t) \\ &=t \int_{0}^{t} f(\boldsymbol{v}) d \boldsymbol{v}-\int_{0}^{t} \boldsymbol{v} f(\boldsymbol{v}) d \boldsymbol{v} \end{aligned}$$, $$F(s)=\mathscr{L}\\{f\\}(s)$$.

\(\int_{0}^{\infty} e^{-2 t} \delta(t-1) d t\)

26\. $$y^{\prime \prime}+2 y^{\prime}-15 y=g(t) ; \quad y(0)=0, \quad y^{\prime}(0)=8$$

\(\begin{array}{ll}{x^{\prime}-x-y=1 ;} & {x(0)=0} \\ {-x+y^{\prime}-y=0 ;} & {y(0)=-5 / 2}\end{array}\)
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