Problem 75 - Calculus and its applications

Question

Find the slope of the line tangent to the graph of \(f(x)=2 e^{-3 x}\) at the point (0,2)

Step-by-Step Solution

Answer

The slope of the tangent line at the point (0,2) is -6.

Step 1: Identify the function and find its derivative

Given the function is \(f(x) = 2e^{-3x}\). To find the slope of the tangent line, we first need the derivative of \(f(x)\), denoted as \(f'(x)\). The derivative of \(2e^{-3x}\) can be found using the chain rule. The outer function is a constant multiple \(2\), and the derivative of \(e^{-3x}\) is \(-3e^{-3x}\). Therefore, \(f'(x) = 2 \times (-3e^{-3x}) = -6e^{-3x}\).

Step 2: Evaluate the derivative at the given point

To find the slope of the tangent line at a specific point, we evaluate the derivative at \(x = 0\). Plugging \(x = 0\) into \(f'(x)\), we get: \(f'(0) = -6e^{-3\times0}\). Since \(e^0 = 1\), it simplifies to \(f'(0) = -6 \times 1 = -6\).

Key Concepts

Understanding Derivatives

In calculus, a derivative represents the rate at which a function is changing at any given point. It is a fundamental concept that underlies many techniques in calculus, especially concerning how one value is changing in relation to another. When computing derivatives, we're essentially finding the slope of the curve; this tells us how steep the curve is at any particular point.

Here’s a succinct breakdown:

• The derivative of a constant is zero because constants don't change.
• The derivative of a power function, like \(x^n\), results in \(n imes x^{n-1}\).
• An exponential function like \(e^{x}\) has a special property where its derivative is itself.

In our example, the function is \(f(x) = 2e^{-3x}\). The derivative of this function shows not just how \(f(x)\) changes, but also how the exponential decay (denoted by negative exponent) is affecting it.

Tangent Line and Its Meaning

A tangent line is a straight line that touches a curve at just one distinct point. Importantly, this line will have the same slope as the curve at that particular point, making the concept of tangent lines particularly useful as they allow us to approximate the behavior of curves at given points.

Consider these insights into tangent lines:

• The tangent line is used to approximate the curve near the point of tangency.
• In essence, the tangent line "hugs" the curve very closely around the point.
• This line can describe the immediate change and give insights into the function's behavior.

In the original exercise, \(f(x) = 2e^{-3x}\) at \(x = 0\) and \(f(0) = 2\). The slope of the line tangent to the curve at this point is determined by the derivative evaluated at \(x = 0\). That slope represents how fast the value of the function is changing at that point, and it is found to be \(-6\).

Chain Rule - A Powerful Tool

The chain rule is a core technique used in calculus for finding derivatives of composite functions. When a function is composed of two or more functions, the chain rule provides a way to compute the derivative of the overall function easily.

Here's how to remember and apply the chain rule:

• To use the chain rule, identify the "outer" and "inner" functions. For \(f(x) = 2e^{-3x}\), "2" and \(e\) represent outer functions while \(-3x\) is the inner function.
• Differentiating the outer function with respect to the inner function gives one part of the derivative.
• Then, multiply this by the derivative of the inner function.

In the given exercise, the chain rule helps to find that \(-6e^{-3x}\) is the derivative of \(2e^{-3x}\). Knowing when and how to apply this rule simplifies the process of dealing with more complex functions, making the chain rule an incredibly useful method in calculus.