Problem 99 - Calculus Single Variable

Question

Plot \(y=\exp (x)\) for \(0 \leq x \leq 2\). Let \(P(c)\) denote the point \((c, \exp (c))\) on the graph. The purpose of this exercise is to graphically explore the relationship between \(\exp (c)\) and the slope of the tangent line at \(P(c) .\) For \(c=1 / 2,1\), and \(3 / 2,\) calculate the slope \(m(c)\) of the secant line that passes through the pair of points \(P(c-0.001)\) and \(P(c+0.001) .\) For each \(c,\) calculate \(|\exp (c)-m(c)|\) to see that \(m(c)\) is a good approximation of \(\exp (c) .\) Add the three secant lines to your viewing window. For each of \(c=1 / 2,1,\) and \(3 / 2,\) add to the viewing window the line through \(P(c)\) with slope \(\exp (c)\). As we will see in Chapter \(3,\) these are the tangent lines at \(P(1 / 2), P(1)\) and \(P(3 / 2)\). It is likely that they cannot be distinguished from the secant lines in your plot.

Step-by-Step Solution

Answer

Plot \( y = \exp(x) \), calculate and compare secant slopes \( m(c) \) to \( \exp(c) \), and add secant and tangent lines for \( c=1/2, 1, 3/2 \).

Step 1: Plot the Function

First, plot the function \( y = \exp(x) \) for the range \( 0 \leq x \leq 2 \). This plot will allow us to visualize the exponential curve and know where to add secant and tangent lines.

Step 2: Find Points on the Graph

Identify the points \( P(c) \) for the given values of \( c \). For example, for \( c = \frac{1}{2} \), \( P(c) = \left(\frac{1}{2}, \exp\left(\frac{1}{2}\right)\right) \), for \( c = 1 \), \( P(c) = (1, \exp(1)) \), and for \( c = \frac{3}{2} \), \( P(c) = \left(\frac{3}{2}, \exp\left(\frac{3}{2}\right)\right) \).

Step 3: Calculate Secant Line Slope

Next, calculate the slope of the secant line \( m(c) \) that passes through points \( P(c-0.001) \) and \( P(c+0.001) \) for each \( c \). Use the formula: \[ m(c) = \frac{\exp(c+0.001) - \exp(c-0.001)}{0.002} \] for each \( c \).

Step 4: Compare Slope and Exponential Value

For each \( c \), compute the absolute difference \( |\exp(c) - m(c)| \) to determine how closely the slope of the secant line approximates \( \exp(c) \).

Step 5: Add Secant Lines to Plot

Draw the secant lines on the plot using the points calculated in Step 3.

Step 6: Plot Tangent Lines

For each \( c \), draw the tangent line that passes through \( P(c) \) with a slope of \( \exp(c) \). These tangent lines should coincide closely with the secant lines already plotted.

Key Concepts

Tangent Line

A tangent line is a straight line that just "touches" a curve at one single point. This line has the same slope as the curve it touches at the point of contact. For exponential functions like the one given, this is very insightful. At the point \(c\), where \(x = c\), the slope of the tangent is exactly equal to \(\exp(c)\). This means it describes how steep the function \(y = \exp(x)\) is at that particular point. Visualizing tangent lines can help to understand the local behavior of a curve. When you draw a tangent line through the point \(P(c)\) with the chosen slope, it gives you the best linear approximation of the function at that point.

• Tangent lines linearly approximate the curve at a point.
• They share the same slope as the curve at the contact point.

This straightforward relationship is why tangent lines are significant in calculus and why they're useful for understanding function behavior.

Secant Line

A secant line, unlike a tangent line, doesn't just touch the curve at one point. Instead, it passes through two distinct points on the curve, creating a chord. For the exercise, the points \(P(c-0.001)\) and \(P(c+0.001)\) were used to create the secant line around a point of interest \(P(c)\).
The secant line is helpful because it approximates the tangent line's slope. With very small differences between the points (as in 0.001 here), the secant line can act as a close neighbor to the tangent, offering similar insights into the slope and behavior of the function near that point.

• Secant lines provide a slope that bridges two points.
• They approximate tangent lines when the points are sufficiently close.

By comparing these slopes with that of the actual tangent, one can see how well the secant line serves as an estimate for function behavior.

Slope Calculation

Slope is a measure of steepness or the inclination of a line. For the tangent line, the slope at a specific point on the graph \(y = \exp(x)\) is simply calculated as \(\exp(c)\), representing the derivative or rate of change of the function at that point. For secant lines, the slope calculation involves two points and demonstrates how quickly the function changes between them.
The formula used for secant slope is \[m(c) = \frac{\exp(c+0.001) - \exp(c-0.001)}{0.002}\] where the difference in function values is divided by the horizontal change \(0.002\). This estimate provides a non-instantaneous measure of slope that can be tested against the precise tangent slope for accuracy.

• Slope for tangent lines is the derivative or rate of change at a point.
• Slope for secant lines is an average rate of change between two points.

Through slope calculations, we gain insight into the function's behavior and its rate of change.

Graphing Functions

Graphing functions is an essential way to visually interpret mathematical relationships and behaviors. By graphing \(y = \exp(x)\) from \(x = 0\) to \(x = 2\), it provides a clear picture of how rapidly exponential functions grow. When graphing these exponential functions, it's crucial to place them accurately in the viewing window for proper interpretation.
Adding tangent and secant lines to the graph provides even deeper insights:

• Tangent lines help visualize local behavior at specific points.
• Secant lines show approximate behavior over small intervals.

The overlapping of these lines on the graph, especially close to the points of interest, indicates how closely the secant line approximates the tangent line. Graphing all these together allows for a comprehensive comparison and comprehension of different rates of change across the curve.