Question

In Exercises 35 and \(36,\) find the slope of the tangent line to the graph of the function at the point (0,1) . (a) \(y=e^{2 x}\) (b) \(y=e^{-2 x}\)

Short answer

The slope of the tangent line to the graph of the function \(y=e^{2 x}\) at the point (0,1) is 2 and the slope of the tangent line to the graph of the function \(y=e^{-2 x}\) at the point (0,1) is -2.

Step-by-step solution

Differentiate the Function

First, take the derivative of the given function. For (a) the function is \(y=e^{2 x}\), and for (b) the function is \(y=e^{-2 x}\). Using the chain rule, the derivative of \(y=e^{2 x}\) is \(y'=2e^{2 x}\), and the derivative of \(y=e^{-2 x}\) is \(y'= -2e^{-2 x}\).

Find the Slope of the Tangent Line

Second, plug the x-value of the given point (0,1) into the derivative to find the slope of the tangent line. For (a), the slope of the tangent line at the point (0,1) would be \(m_a = 2e^{2*0} = 2\). For (b), the slope of the tangent line at the point (0,1) would be \(m_b = -2e^{-2*0} = -2\).

Check the Original Function

Finally check that the point (0,1) is actually on the original function. For (a) {y=e^{2 x}}, if we substitute x=0, we get \(y=e^{2*0} = e^0 = 1\), and similarly for (b) {y=e^{-2 x}}, if we substitute x=0 get \(y=e^{-2*0} = e^0 = 1\). In both cases, the y-coordinate is 1 which is the same as the given point (0,1). So the slopes computed in the previous step are indeed for the tangent line to the functions at the given point.

Key concepts

Derivative of a Function

Imagine you're driving and you want to know how quickly you're accelerating at a particular moment. In mathematics, when we're dealing with functions, to find out how fast the function is changing at any given point, we use derivatives. A derivative is like a snapshot of a function's rate of change at a specific instant.

For the problem at hand, we examine the slope of the tangent line to the graph of a function at a particular point. The slope is determined by the derivative of the function at that point. To visualize this, think about the graph of a hill: the steepness at any point on that hill is analogous to the slope of the tangent line to the hill's curve at that point.

For the exercise given, finding the derivative provides us with the information we need to understand how steeply the graph of the exponential function is rising or falling at the point (0,1).

Chain Rule

Now, how do we calculate the derivative when the function is a bit more complex? Enter the chain rule—a crucial tool in calculus. Think of it like unpacking a Russian doll, where each layer is a function nested within another function.

When you want to differentiate a composite function (a function inside another function), say, something like \(e^{2x}\), you can't just proceed as if it's a simple power or exponential function. You need to take the derivative of the outer function and multiply it by the derivative of the inner function. This process is exactly what we refer to as the chain rule.

In the textbook solution, the chain rule enabled us to differentiate \(e^{2x}\) and \(e^{-2x}\) easily by considering \(2x\) and \(-2x\) as the inner functions and differentiating accordingly. This gave us the slopes of the tangent lines at the specific point (0,1) for both functions.

Exponential Function

When we speak of exponential functions, we're often describing situations that involve growth or decay, like populations, investments, or even the spreading of a virus. These functions can be identified by their base 'e', which is the irrational and transcendental number approximately equal to 2.71828. The expression \(e^{x}\) represents an exponential function.

Exponential functions grow or decay at a rate proportionate to their current value. This characteristic makes them incredibly useful in both natural and social sciences. In the context of our problem, the function \(e^{2x}\) is an increasing exponential function, while \(e^{-2x}\) is a decreasing exponential function. Understanding the behavior of these functions provides insight into the nature of the slope at any given point on their graphs—specifically, how sharply they increase or decrease.