Question

Derivatives from limits The following limits represent \(f^{\prime}(a)\) for some function \(f\) and some real number \(a\) a. Find a possible function \(f\) and number \(a\). b. Evaluate the limit by computing \(f^{\prime}(a)\). $$\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}$$

Short answer

Function: \(f(x) = e^x\) Value of \(a\): \(a = 0\)

Step-by-step solution

Identify Function and Derivative Point#

Compare the given limit to the definition of the derivative: $$\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$$ From the given limit, it seems that \(a = 0\) since \(x \rightarrow 0\). To find \(f(x)\), we need to match the form of the numerator to \(f(x)-f(a)\): \(e^x - 1\) matches \(f(x) - f(0)\). From this, we can infer that \(f(x) = e^x\) and \(f(0) = 1\). So, we can select \(f(x) = e^x\) and \(a = 0\).

Key concepts

Limits and Derivatives

Understanding the connection between limits and derivatives is a foundational concept in calculus. In essence, the derivative of a function at a certain point measures the rate at which the function's value is changing at that point. It is defined as the limit of the average rate of change over an interval as that interval becomes infinitesimally small.

The process to find a derivative using limits involves setting up a ratio of the difference in function values to the difference in input values, \(\frac{f(x)-f(a)}{x-a}\), and then taking the limit as \(x\) approaches \(a\). This ratio is known as the difference quotient, and as the interval becomes smaller, the ratio approaches the derivative \(f'(a)\). Through this process, we can observe the immediate behavior of functions as their input changes, giving us insights into how functions grow or decay, how fast they do so, and the nature of their curvature.

Exponential Functions

Exponential functions are a special type of function where the variable is in the exponent. The function \(e^x\) is the most important exponential function in calculus, thanks to its unique properties. It is its own derivative, which means that \(\frac{d}{dx}e^x = e^x\).

Exponential functions are used to model growth and decay processes such as population growth, radioactive decay, and compound interest. They have a constant percentage rate of change, which makes them distinctive compared to linear functions with a constant rate of change. Understanding the behavior of exponential functions is critical for solving real-world problems and is a stepping stone to discussing more complex topics, like logarithmic functions and modeling.

Limit Definition of Derivative

The limit definition of a derivative is a formal way of stating what the derivative of a function is. According to this definition, the derivative of \(f\) at \(a\), denoted \(f'(a)\), is defined as \(\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}\) provided this limit exists.

When using the limit definition of a derivative, we might encounter different types of functions and limits; one important category includes limits that result in an indeterminate form, like \(0/0\). Techniques like rationalization, L'Hopital's Rule, or reimagining the function can be employed to evaluate these limits. Mastery of the limit definition of the derivative allows for a deeper understanding of the concept of instantaneous rate of change, which is applicable across various disciplines in science and engineering.

Calculus Practice Problems

Solving practice problems is an excellent way to deepen your understanding of calculus concepts. These problems help students to apply theoretical knowledge to concrete examples, reinforcing the connection between the abstract idea of a derivative and its practical applications.

For example, evaluating the limit \(\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}\) might appear on a calculus practice problem set as it involves understanding the limit definition of a derivative. The difficulty of calculus practice problems can range from the application of straightforward rules to solving complex, multi-step problems that require a thorough understanding of several calculus concepts. Consistent practice not only strengthens problem-solving skills but also builds the student's confidence in tackling new and challenging mathematical scenarios.