Problem 6 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Interpret \(e^{t^{2}}\) as \(e^{\left(t^{2}\right)}\). Argue that $$\lim _{b \rightarrow a} \frac{e^{\left(b^{2}\right)}-e^{\left(a^{2}\right)}}{b^{2}-a^{2}}=e^{\left(a^{2}\right)}$$ What is the ambiguity in the notation \(e^{a^{2}}\). (Consider \(4^{3^{2}}\).) Use parenthesis, they are cheap. However, common practice is to interpret \(e^{t^{2}}\) as \(e^{\left(t^{2}\right)}\).

Step-by-Step Solution

Answer

The limit evaluates to \(e^{(a^2)}\). Parentheses clarify ambiguity in exponent notation.

Step 1: Understand the Notation

The expression given is \(e^{t^{2}}\). It is interpreted as \(e^{(t^2)}\), indicating that the exponent is \(t^2\) itself. Similar ambiguity can arise in expressions like \(4^{3^2}\), which should be interpreted as \(4^{(3^2)} = 4^9\). Parentheses clarify this interpretation problem.

Step 2: Examine Limit Expression

We are given \( \lim _{b \rightarrow a} \frac{e^{(b^{2})}-e^{(a^{2})}}{b^{2}-a^{2}} \). Notice that this resembles the definition of a derivative of the function \(e^{x}\) at \(x = a^2\), where the change in \(x\) is \(b^2 - a^2\).

Step 3: Apply Derivative of Exponential Function

The derivative of \(e^{x}\) with respect to \(x\) is \(e^{x}\). Therefore, when considering \( \lim_{x \to c} \frac{f(x) - f(c)}{x - c} \) for \(f(x) = e^{x}\), the limit is \(e^{c}\). Substituting \(c = a^2\), we find that the limit evaluates to \(e^{(a^2)}\).

Step 4: Conclude with Interpretation

Based on the calculation, \(\lim _{b \rightarrow a} \frac{e^{(b^{2})}-e^{(a^{2})}}{b^{2}-a^{2}} = e^{(a^{2})}\). The notation should include parentheses to avoid any ambiguities.

Key Concepts

Exponential Functions

Exponential functions are a special type of mathematical function where a constant base is raised to a variable exponent. The most common base used in exponential functions is the mathematical constant \(e\), approximately equal to 2.71828. This base is significant because it appears naturally in many areas of mathematics, such as calculus, complex analysis, and number theory.

• In an exponential function, the variable appears in the exponent, not as a coefficient or a multiplier. For example, in the expression \(e^{t^2}\), the exponent is \(t^2\), which signifies that \(t^2\) dictates how many times the base \(e\) is multiplied by itself.
• Exponential functions grow very quickly because their rate of increase (the slope) is proportional to their current value. This property leads to their use in modeling processes involving growth or decay, like populations, radioactive substances, and investments.
• The notation of exponential functions can sometimes lead to ambiguity. For example, expressions like \(e^{t^2}\) can be misinterpreted as \((e^t)^2\). Using parentheses, as in \(e^{(t^2)}\), clarifies that the exponent is the entire expression \(t^2\).

Understanding the basic properties and notation of exponential functions is crucial in many mathematical problems and applications.

Limit of a Function

The limit of a function at a particular point describes the value that the function approaches as the input gets closer to that point. Limits are foundational in calculus and help us understand the behavior of functions as they approach specific values.

• The notation \( \lim_{x \to c} f(x) = L \) means that as \(x\) approaches \(c\), the function \(f(x)\) approaches \(L\). Limits help define derivatives and integrals, which are the cornerstones of calculus.
• In our problem, we looked at the limit \( \lim_{b \to a} \frac{e^{(b^2)} - e^{(a^2)}}{b^2 - a^2} \). This expression resembles the definition of a derivative. Here, the numerator \(e^{(b^2)} - e^{(a^2)}\) represents the change in the function values, and the denominator \(b^2 - a^2\) is the change in its input variables.
• Limits allow us to handle points where expressions seemingly behave unpredictably, such as division by zero or indeterminate forms. By analyzing the behavior of a function near these points, we can often find a meaningful value.

Grasping the concept of limits is essential for understanding how calculus operates at both basic and advanced levels.

Derivative of Exponential Functions

The derivative of a function tells us the rate at which the function's value is changing. With exponential functions, particularly those involving the base \(e\), the derivative has some noteworthy properties.

• For the exponential function \(f(x) = e^x\), its derivative is simply itself: \(f'(x) = e^x\). This is unique to the base \(e\), as most other functions do not have the property where the derivative is the same as the original function.
• In the problem we're discussing, the expression \( \lim_{b \to a} \frac{e^{(b^2)} - e^{(a^2)}}{b^2 - a^2} \) is equivalent to finding the derivative of \(e^{x}\) evaluated at \(x = a^2\), which results in \(e^{(a^2)}\). The calculation illustrates the concept of a derivative by using the limit definition.
• Derivatives are vital in calculus for problems involving rates of change, optimization, and analyzing functions' behavior. Being comfortable with finding derivatives of exponential functions helps solve many real-world problems, such as those involving exponential growth and decay.

Understanding how to compute and interpret derivatives is critical for anyone studying calculus, as they are used across numerous scientific disciplines.