Problem 2 - Calculus: An Intuitive and Physical Approach

Question

Find the slope of \(y=e^{x}\) at \(x=0\).

Step-by-Step Solution

Answer

The slope of \(y=e^{x}\) at \(x=0\) is 1.

Step 1: Identify the Function

Recognize that the function provided is an exponential function, given by the equation: \(y=e^{x}\).

Step 2: Calculate the Derivative

The slope of the curve at any point is given by the derivative of the function with respect to x. For the function \(y=e^{x}\), differentiate it with respect to x to find its derivative. The derivative of \(e^{x}\) with respect to x is \(e^{x}\) itself.

Step 3: Evaluate the Derivative at x=0

To find the slope at \(x=0\), substitute \(x=0\) into the derivative. The slope at this point is \(e^{0} = 1\).

Key Concepts

Exponential Functions

Exponential functions, like the one in our exercise, are mathematical expressions that describe phenomena that change rapidly and grow or decay at a constant percentage rate. The general form of an exponential function is f(x) = a^x, where a is the base and x represents the exponent, which is the variable.

Unlike linear functions with a steady rate of change, the rate of change in exponential functions increases or decreases relative to the function's current value. They are widely used in diverse fields including biology for population growth, finance for compound interest, and physics for radioactive decay among others. In the given problem, the function y=e^x is a special case where the base a is e, the mathematical constant approximately equal to 2.71828. This specific exponential function has remarkable properties that make it a foundational concept in calculus.

Derivatives of Exponential Functions

The derivative of a function at a point is a fundamental concept in calculus, describing the rate at which the function value is changing at that particular point. For exponential functions, taking a derivative helps us understand how quickly the function is growing or decaying.

The unique aspect of the exponential function with base e is that its derivative is equal to itself. Mathematically, if f(x) = e^x, then the derivative f'(x), is also e^x. This property simplifies calculations and plays an integral role in many areas of mathematics. For example, in compound interest problems, it helps us determine how fast an investment is growing over time.

Finding Slope at a Point

The slope of a curve at a particular point indicates the steepness or incline and is calculated by evaluating the function's derivative at that point. In the context of exponential functions, finding the slope at any point gives us the instantaneous rate of change at that specific value of x.

To find the slope of the function y=e^x at x=0, we substitute x=0 into the derivative e^x. Since e^0 equals 1, the slope at x=0 is 1. This tells us the function is increasing at a rate of 1 unit on the y-axis for every 1 unit it moves along the x-axis at the origin, indicative of a 45-degree angle at that point on the curve.