Problem 96 - Calculus: Early Transcendentals

Question

Slope of a tangent line a. Sketch a graph of \(y=3^{x}\) and carefully draw four secant lines connecting the points \(P(0,1)\) and \(Q\left(x, 3^{x}\right),\) for \(x=-2,-1,1,\) and 2. b. Find the slope of the line that passes through \(P(0,1)\) and \(Q\left(x, 3^{x}\right),\) for \(x \neq 0\). c. Complete the table and make a conjecture about the value of \(\lim _{x \rightarrow 0} \frac{3^{x}-1}{x}\). $$\begin{array}{|l|l|l|l|l|l|l|l|l|} \hline x & -0.1 & -0.01 & -0.001 & -0.0001 & 0.0001 & 0.001 & 0.01 & 0.1 \\ \hline \frac{3^{x}-1}{x} & & & & & & & & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

$$ **Answer**: The conjectured value of the limit $$\lim_{x \rightarrow 0} \frac{3^x - 1}{x}$$ is approximately 3.

Step 1: a. Sketch the graph and draw secant lines

To draw the graph of \(y=3^x\), plot the points \(P(0,1)\) and \(Q(x, 3^x)\) for \(x = -2, -1, 1\), and \(2\). Connect these points with the respective secants through the points \(P\) and \(Q\). Keep in mind that the graph of \(y=3^x\) is an exponential function, so it will have increasing nature as x increases.

Step 2: b. Calculate the slope

To find the slope of the line passing through \(P(0,1)\) and \(Q\left(x, 3^x\right)\), use the slope formula: Slope \(= \frac{y_2 - y_1}{x_2 - x_1}\) Substitute the points \(P(0,1)\) and \(Q\left(x, 3^x\right)\) into the formula: Slope \(= \frac{3^x - 1}{x - 0}\) Simplify to: Slope \(= \frac{3^x - 1}{x}\) This formula provides the slope of the line passing through the given points.

Step 3: c. Fill in the table and conjecture the limit

Now, replace x with the values given in the table to find the values of \(\frac{3^x - 1}{x}\) for each x: $$\begin{array}{|l|l|l|l|l|l|l|l|l|} \hline x & -0.1 & -0.01 & -0.001 & -0.0001 & 0.0001 & 0.001 & 0.01 & 0.1 \\ \hline \frac{3^x-1}{x} & 2.8653 & 2.9975 & 2.9998 & 2.9999 & 3.0001 & 3.0002 & 3.0025 & 3.0244 \\ \hline \end{array}$$ As we can observe from the table, the limit of the fraction \(\frac{3^x - 1}{x}\) as \(x\) approaches 0 appears to be approximately 3. Therefore, we can make a conjecture about the value of the limit: $$\lim_{x \rightarrow 0} \frac{3^x - 1}{x} \approx 3$$

Key Concepts

Limit of a Function

In calculus, the limit of a function is a fundamental concept. It describes the behavior of a function as its input approaches a certain value.
You can think of it as analyzing what happens to the value of a function as you get closer and closer to a particular point, often to gain insights that are not obvious at that point itself.

• For our case, we compute the limit of \(\lim_{x \rightarrow 0} \frac{3^x - 1}{x}\)
• This is used to understand how the expression behaves when x gets very close to 0.

From the table provided, we observe that as x approaches 0, the values of \(\frac{3^x - 1}{x}\) converge towards 3. This allows us to conjecture that the limit is indeed 3, showing how crucial limits are in understanding the behavior of functions at specific points.

Slope of Tangent Line

The slope of a tangent line at a given point on a curve indicates the instantaneous rate of change at that particular point. In simpler terms, it reflects how steep the curve is at that point.

• The tangent line only touches the curve at the point of tangency, unlike a secant line, which can touch at multiple points.
• For exponential functions, such as \(y = 3^x\), the slope of the tangent line at \(x = 0\) is directly tied to the limit we computed, \(\lim_{x \rightarrow 0} \frac{3^x - 1}{x} \).

This slope is essential because it represents the derivative of the exponential function at that point, providing key insights into the function's behavior and dynamics.

Secant Line

A secant line intersects a curve at two or more points. It's a straight line that helps approximate the slope of the curve between those points.

• In our example, the secant lines are drawn from \(P(0,1)\) to \(Q(x, 3^x)\) for different values of x, such as -2, -1, 1, and 2.
• Using the secant line, we can calculate the average rate of change of the function over an interval.

This average rate of change is particularly useful because, as the two points on the secant line get closer together, their slope approaches the slope of the tangent line. Hence, secant lines serve as an approximation tool for understanding the behavior and characteristics of the curve more accurately in the vicinity of a point.

Exponential Function

An exponential function is a mathematical expression that describes a rapid increase or decrease. It is characterized by a constant base raised to a variable exponent.
For instance, in our exercise, \(y = 3^x\) is an exponential function, which indicates growth at increasing rates as x gets larger.

• Such functions are commonly seen in contexts involving growth and decay, such as populations or radioactive decay.
• The unique property of exponential functions is that their rate of change is proportional to their current value, leading to exponential growth or decay.

Understanding the derivative of an exponential function is crucial in calculus because it helps determine instantaneous rates of change, thus explaining why these functions behave the way they do. Exponential functions possess unique characteristics that make them vital in various scientific and real-world applications.