Question

If \(e^{r t}\) is given by Eq. ( 13) , show that $$ \frac{d}{d t} e^{r t}=r e^{r t} $$ for any complex number \(r\).

Short answer

In this exercise, we were asked to prove that the derivative of \(e^{rt}\) with respect to \(t\) is \(re^{rt}\) where \(r\) is any complex number. To do this, we applied the definition of the derivative and went through the following steps: 1. Recall the definition of the derivative: \(\frac{d}{dt} f(t) = \lim_{h \to 0} \frac{f(t+h)-f(t)}{h}\). 2. Apply the definition of the derivative to \(e^{rt}\): \(\frac{d}{dt} e^{rt} = \lim_{h \to 0} \frac{e^{r(t+h)}-e^{rt}}{h}\). 3. Factor out the common term \(e^{rt}\) from both the numerator and the denominator: \(\frac{d}{dt} e^{rt} = \lim_{h \to 0} \frac{e^{rt}(e^{rh} - 1)}{h}\). 4. Evaluate the limit: \(\frac{d}{dt} e^{rt} = e^{rt}\lim_{h \to 0} \frac{e^{rh}-1}{h} = e^{rt}r\). Therefore, we have proven that the derivative of \(e^{rt}\) with respect to \(t\) is indeed \(re^{rt}\), where \(r\) is any complex number.

Step-by-step solution

Recall the definition of the derivative

The derivative of a function \(f(t)\) with respect to \(t\) can be calculated using the definition of the derivative: $$ \frac{d}{dt} f(t) = \lim_{h \to 0} \frac{f(t+h)-f(t)}{h} $$

Apply the definition of the derivative to our function

We need to find the derivative of \(e^{rt}\) with respect to \(t\). We will apply the definition of the derivative to this function: $$ \frac{d}{dt} e^{rt} = \lim_{h \to 0} \frac{e^{r(t+h)}-e^{rt}}{h} $$

Factor out common term

Since exponentiation has the property \(a^{b+c} = a^b \cdot a^c\), we can factor out the common term \(e^{rt}\) from both the numerator and the denominator: $$ \frac{d}{dt} e^{rt} = \lim_{h \to 0} \frac{e^{rt}(e^{rh} - 1)}{h} $$

Evaluate the limit

Let's now evaluate the limit as \(h\) approaches 0: $$ \frac{d}{dt} e^{rt} = e^{rt}\lim_{h \to 0} \frac{e^{rh}-1}{h} $$ We know that the limit \(\lim_{h \to 0} \frac{e^{rh}-1}{h}\) exists and is equal to \(r\), since the limit \(\lim_{x \to 0} \frac{e^x-1}{x}\) is equal to 1 and \(e^{rh}\) is continuous. Therefore, we have: $$ \frac{d}{dt} e^{rt} = e^{rt}r $$ This result holds for any complex number \(r\). So we have proven that: $$ \frac{d}{dt} e^{rt} = re^{rt} $$

Key concepts

Definition of the Derivative

Understanding the derivative of a function is fundamental in calculus. It represents the rate at which a function is changing at any given point. For a function f(t), the derivative with respect to t is denoted as \( \frac{d}{dt} f(t) \). It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. Formally, the definition uses the concept of the limit to establish:
\[\frac{d}{dt} f(t) = \lim_{h \to 0} \frac{f(t+h)-f(t)}{h}\]
This concept is paramount not just for theoretical mathematics but also for practical applications, such as calculating velocity in physics or determining marginal cost in economics.

Limit of a Function

The limit of a function is a fundamental concept in calculus and analysis, dealing with the behavior of a function as its argument approaches a specific value. Limits can also provide information about a function's value at a point where it might not be explicitly defined. The formal definition of a limit is written as:
\[\lim_{x \to a} f(x) = L\]
This notation denotes that as the variable x gets arbitrarily close to the value a, the function f(x) approaches the value L. Limits are crucial in defining various other concepts in calculus, including derivatives and integrals.

Exponential Function Properties

Exponential functions are characterized by a constant raised to a variable exponent, commonly expressed as \(f(x) = a^x\), where a is a positive real number. These functions have remarkable properties, including:

• The function is always positive, no matter the value of x.
• The rate of change is proportional to the function's value, leading to exponential growth or decay.
• Exponential functions have a constant base (like \(e\), Euler's number) and a variable exponent.

One particularly important exponential function is \(e^x\), which is its own derivative. This means that the slope at any point on the curve \(e^x\) is the value of the function at that point, which is a defining property used in solving many calculus problems.

Complex Numbers

Complex numbers extend the idea of quantity to include numbers that represent the square root of negative one, denoted as \(i\) or \(j\) in engineering. A complex number has the form \(a + bi\), combining a real part (a) and an imaginary part (bi). Here are some key points:

• The real numbers are a subset of the complex numbers, where the imaginary part is zero.
• Complex numbers can be represented on a plane, known as the complex plane, with the x-axis representing the real part and the y-axis representing the imaginary part.
• They play a crucial role in advanced mathematics, including calculus, as they allow solutions to equations that have no real solutions.

Complex numbers are also utilized in the analysis of electrical circuits, signal processing, and quantum mechanics. The definition of the exponential function can also be expanded to include complex exponents, which is key in fields like control theory and time series analysis.