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The concept of a derivative is a fundamental part of calculus that allows us to understand how a function behaves at any given point. When you take the derivative of a function, you are finding the rate at which the function's values change. This rate of change is also known as the slope of the tangent line. In this exercise, we found the derivative of the function \( y = e^{-x} \) to determine the slope of its tangent line at any point \( x \). The derivative is calculated as \( \frac{dy}{dx} = -e^{-x} \).
Let's break that down:

• The expression \( e^{-x} \) is an exponential function with the base \( e \) and exponent \( -x \).
• The derivative, represented as \( \frac{dy}{dx} \), measures how \( y \) changes with respect to \( x \).
• By differentiating \( e^{-x} \), we find \( -e^{-x} \), indicating that the function decreases as \( x \) increases. This negative sign is crucial because it suggests the curve slopes downward.

Understanding derivatives is essential as it provides the tool to find tangent lines at specific points, enabling us to analyze and predict behavior of functions at a microscopic level.

The exponential function is a type of mathematical function with the form \( a^x \) or \( e^x \), where \( e \) is Euler's number (approximately 2.71828). In our particular exercise, we are dealing with \( y = e^{-x} \), an exponential decay function.Here are some key points about exponential functions:

• An exponential function grows or decays at a rate proportional to its current value. This quality makes it resemble processes like population growth or radioactive decay.
• For \( y = e^{-x} \), as \( x \) increases, the value of \( y \) decreases, which is typical of exponential decay.
• When \( x = 0 \), \( y \) equals 1, as \( e^0 = 1 \), forming a starting point or an intersection with the y-axis.
• This function will never reach zero but will continue to get closer to it, exemplifying asymptotic behavior.

Exponential functions are not only foundational in mathematics but also have applications across various fields like finance, sciences, and engineering. They help in modeling a wide range of real-world scenarios where growth or decay is involved.

Point-slope form is a simple and intuitive way to write the equation of a line when you know a point on the line and its slope. It is expressed as: \[ y - y_1 = m(x - x_1) \] This form is preferred for its ability to directly incorporate data regarding a point \((x_1, y_1)\) and a known slope \(m\).In this exercise, we use point-slope form to find the equation of the tangent line at the point \((-1, e)\) on the curve. Here's a step-by-step breakdown:

• First, we identify the point: \((x_1, y_1) = (-1, e)\). This comes from solving \( y = e^{-x} \) at \( x = -1 \).
• Next, we use the slope \( m = -e \), found by evaluating the derivative at \( x = -1 \).
• Plugging these values into the point-slope equation gives us: \[ y - e = -e(x + 1) \]. Simplifying further, the equation becomes \( y = -ex \).

Point-slope form is incredibly useful as it gives direct and specific insights, allowing easy conversion to other forms, such as slope-intercept form, making plotting and understanding linear equations smoother.