Problem 59 - Thomas Calculus

Question

In Exercises \(53-60,\) you will explore some functions and their inverses together with their derivatives and linear approximating functions at specified points. Perform the following steps using your CAS: a. Plot the function \(y=f(x)\) together with its derivative over the given interval. Explain why you know that \(f\) is one-to-one over the interval. b. Solve the equation \(y=f(x)\) for \(x\) as a function of \(y,\) and name the resulting inverse function \(g\) . c. Find the equation for the tangent line to \(f\) at the specified point \(\left(x_{0}, f\left(x_{0}\right)\right) .\) d. Find the equation for the tangent line to \(g\) at the point \(\left(f\left(x_{0}\right), x_{0}\right)\) located symmetrically across the \(45^{\circ}\) line \(y=x\) (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line. e. Plot the functions \(f\) and \(g\) , the identity, the two tangent lines, and the line segment joining the points \(\left(x_{0}, f\left(x_{0}\right)\right)\) and \(\left(f\left(x_{0}\right), x_{0}\right) .\) Discuss the symmetries you see across the main diagonal. $$ y=e^{x}, \quad-3 \leq x \leq 5, \quad x_{0}=1 $$

Step-by-Step Solution

Answer

Function is one-to-one because it is continuously increasing. Inverse is \(g(y) = \ln(y)\). The tangent lines are \(y=ex\) and \(y=\frac{1}{e}x\).

Step 1: Plot the Function and Its Derivative

Plot the function \( y = e^x \) over the interval \(-3 \leq x \leq 5\). The derivative of \( y = e^x \) is also \( y' = e^x \). Since the exponential function is continuously increasing on this interval (as its derivative is always positive), we know that \( f(x) = e^x \) is one-to-one over this interval.

Step 2: Solve for the Inverse Function

To find the inverse of \( y = e^x \), solve for \( x \) in terms of \( y \): \( y = e^x \) implies \( x = \ln(y) \). Thus, the inverse function is \( g(y) = \ln(y) \).

Step 3: Find the Tangent Line to f

The equation of the tangent line to \( f \) at \( (x_0, f(x_0)) = (1, e^1) \) is given by \( y = f'(x_0)(x - x_0) + f(x_0) \). Since \( f'(1) = e^1 = e \), the tangent line at this point is \( y = e(x - 1) + e = ex - e + e = ex \).

Step 4: Find the Tangent Line to g

The point \((f(x_0), x_0)\) is \((e, 1)\) for the inverse function. The slope of the tangent line to \( g \) at this point is \( \frac{1}{f'(x_0)} \), due to Theorem 1. Since \( f'(1) = e \), the slope is \( \frac{1}{e} \). Thus, the tangent line at \((e, 1)\) is \( y = \frac{1}{e}(x - e) + 1 = \frac{1}{e}x - 1 + 1 = \frac{1}{e}x \).

Step 5: Plot Functions and Tangent Lines

Now, plot the functions \( f(x) = e^x \) and its inverse \( g(y) = \ln(y) \), the line \( y = x \) (the identity line), and the tangent lines \( y = ex \) and \( y = \frac{1}{e}x \). Additionally, plot the line segment joining \((1, e)\) and \((e, 1)\). Notice the symmetry of \( f \) and \( g \) about the line \( y = x \), and that the tangent lines at these points also exhibit this symmetry.

Key Concepts

Derivative

A derivative represents the rate at which a function is changing at any given point. It's like looking at how steep a hill is at a particular spot when you're hiking. You notice the change in height as you walk along the path, and similarly, a derivative tells us how a function's value changes with an infinitesimal change in the input.

For the exponential function, such as \(y = e^x\), the derivative is unique. It is exactly the same as the function itself: \(y' = e^x\). This is what makes the exponential function so fascinating and useful in various practical applications, such as compound interest and population growth. Since \(e^x\) is always positive, it is always increasing. This ensures that the function is one-to-one, meaning every output is associated with a unique input over any interval we choose.

If you plot \(y = e^x\) and its derivative \(y' = e^x\) on the same graph, they will overlap, demonstrating this identity characteristic of the derivative of exponential functions.

Tangent Line

The tangent line to a function at a given point is like the best possible straight-line approximation of the function at that point. Imagine if you parked your bike on a hill; the tangent line is the direction your bike would start rolling if you let it go.

To find a tangent line for a given function, you need two things:

• The point of tangency, or where the tangent line touches the curve.
• The slope of that line at the point of tangency.

For the function \(f(x) = e^x\) at the point \((1, e)\), the derivative at that point provides the slope: \(f'(1) = e\). This gives us the equation of the tangent line as \(y = e(x - 1) + e\). Simplifying, this becomes \(y = ex\), a neat linear equation showing how the curve behaves just at that point.

Tangent lines help in understanding the behavior of complex curves by presenting a simple visual and mathematical representation of the function's local behavior.

Exponential Function

The exponential function is characterized by a constant ratio of change. The classic form, \(y = e^x\), grows increasingly fast as \(x\) increases. It features prominently across both biology and finance, as it can model growth phenomena like populations or investments.

What makes \(e^x\) so unique is how everything about it involves itself. Its derivative and integral are both \(e^x\), indicating a continuous and constant rate of growth. That is why \(e^x\) is called an invariant under differentiation. Exponential functions never dip and keep accelerating upwards, showcasing their rapid growth tendencies.

This function has some fascinating properties:

• The rate of growth at any point is proportional to its current value, which models real-world situations effectively.
• It crosses the y-axis at 1, since \(e^0 = 1\).

These properties are why exponential functions serve as essential building blocks in mathematical modeling and real-world computations.

Natural Logarithm

The natural logarithm, denoted as \(\ln(y)\), is the inverse function of the exponential function \(e^x\). It answers the question: "To what power must \(e\) be raised, to produce the number \(y\)?" Essentially, logarithms are the undo buttons for exponentials.

If you have an equation in the form \(y = e^x\), you can rearrange it to solve for \(x\) using the natural logarithm: \(x = \ln(y)\). This maneuver from exponential to logarithmic form is handy in situations where you need to solve for exponents.

The natural logarithm has several key features:

• It only accepts positive input values, as negatives or zero would not make sense in the context of "how many times to multiply a positive number."
• \(\ln(1) = 0\), since \(e^0 = 1\).
• The slope (derivative) of \(\ln(y)\) provides insight into how quickly the logarithmic function grows, starting rapidly for small \(y\) and slowing as \(y\) increases.

In calculus and algebra, the natural logarithm is a powerful tool for managing multiplicative processes and turning them into more easily understandable additive ones.