Problem 53 - Calculus Early Transcendentals

Question

a. Sketch a graph of \(y=2^{x}\) and carefully draw three secant lines connecting the points \(P(0,1)\) and \(Q\left(x, 2^{x}\right),\) for \(x=-3,-2\) and -1 b. Find the slope of the line that passes through \(P(0,1)\) and \(Q\left(x, 2^{x}\right),\) for \(x \neq 0\) c. Complete the table and make a conjecture about the value of \(\lim _{x \rightarrow 0^{-}} \frac{2^{x}-1}{x}\).

Step-by-Step Solution

Answer

Question: For a function defined by \(y = 2^x\), and given the points \(P(0,1)\) and \(Q(x, 2^x)\) where \(x\) values are -3, -2, and -1, draw the graph with secant lines connecting the points, calculate the slope formula, complete the table, and make a conjecture about the limit as \(x \rightarrow 0^{-}\) for \(\frac{2^x-1}{x}\). Answer: After sketching the graph of the function \(y = 2^x\) and drawing the secant lines connecting points \(P = (0,1)\) and \(Q = (x, 2^x)\) for \(x=-3, -2, -1\), the slope of the line between these points is given by \(m = \frac{2^x-1}{x}\). Complete the table with the respective slopes for each \(x\) value and observe the trend as \(x \rightarrow 0^{-}\). By analyzing the table, conjecture that the limit \(\lim_{x \rightarrow 0^-} \frac{2^x-1}{x} = \) [conjectured value].

Step 1: Draw the graph of \(y = 2^x\)

Begin by plotting some points on the \(x\)-\(y\) plane, obtaining the corresponding values of \(y\) given \(x\). Then, connect the points with a smooth curve.

Step 2: Draw the secant lines

Using the given values of \(x\), draw the secant lines that connect points \(P = (0,1)\) and \(Q = (x, 2^x)\) for \(x=-3, -2, -1\) on the graph. #Part b: Finding the slope of the secant lines#

Step 3: Determine the slope between two points

To find the slope between points P and Q, apply the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), using the given coordinates \(P(0,1)\) and \(Q(x, 2^x)\). Substitute these coordinates into the formula and simplify. #Part c: Completing the table and conjecturing the limit#

Step 4: Calculate the values for the table

For each value of \(x\) in the table, compute \(\frac{2^x-1}{x}\). This is the slope of the secant line for the respective value of \(x\).

Step 5: Analyze the trend and make a conjecture

Observe the calculated values in the table as \(x \rightarrow 0^-\) and make a conjecture about the limit \(\lim_{x \rightarrow 0^-} \frac{2^x-1}{x}\).

Key Concepts

Secant Lines

A secant line is a straight line that connects two points on a curve. In the context of graphing exponential functions such as the curve of \( y = 2^x \), secant lines help us examine the change in the curve over an interval. For instance, using points like \( P(0,1) \) and \( Q(x, 2^x) \), we can visualize how the function behaves as \( x \) changes.

• This line gives a visual representation of the average rate of change between the two points.
• In the exercise, secant lines were drawn for \( x = -3, -2, -1 \).
• Each secant line offers insight into the function's behavior over smaller intervals closer to zero.

This understanding is a step towards comprehending tangents at the point, which examines the instantaneous rate of change, versus the broader overview provided by secant lines.

Slope Calculation

The slope of a line measures how steep it is. It's calculated between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). In our example, the slope of secant lines connecting points \( P(0,1) \) and \( Q(x, 2^x) \) can be determined explicitly.

• Here, \( x_1 = 0 \) and \( y_1 = 1 \); \( x_2 = x \) and \( y_2 = 2^x \).
• The slope thus becomes \( m = \frac{2^x - 1}{x} \).
• This gives an average rate of change of the function between point \( P \) and any chosen point \( Q \).

Calculating slopes for different \( x \) values leads to a deeper understanding of how the function's rate of change varies as \( x \) approaches zero.

Limits and Conjecture

A limit in mathematics describes what value a function approaches as the input approaches a particular point. When analyzing the limit \( \lim_{x \to 0^-} \frac{2^x - 1}{x} \), we seek to understand the behavior of the function as \( x \) gets very close to zero from the negative side.

• Completing the table for \( x = -3, -2, -1 \) and so forth provides slopes from different secant lines.
• By observing resulting values, a pattern emerges as \( x \) approaches zero.
• Such trends help formulate a conjecture, or an educated guess, about the value of the limit.

This conjecture becomes crucial for understanding more complicated aspects of calculus, especially when dealing with instantaneous rates of change and the idea of derivatives as a limit of slopes.