Problem 25 - Calculus Early Transcendentals

Question

For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point. $$f(x)=2 x^{2} \quad \text { at } x=2$$

Step-by-Step Solution

Answer

Based on the analysis of the table showing the slopes of the secant lines approaching $$x=2$$ for the function $$f(x)=2x^2$$, we can make a conjecture that the slope of the tangent line at $$x=2$$ is 8.

Step 1: Determine the secant line formula

To find the slope of a secant line, we will use the formula for the slope of a line, which is the change in y-values divided by the change in x-values: $$m_{sec} = \frac{f(x+h) - f(x)}{h}$$ Here, $$m_{sec}$$ is the slope of the secant line, $$f(x)$$ is the given function, $$x$$ is the given point, and $$h$$ is the change in the x-values.

Step 2: Substitute the function and the point into the formula

We have the function $$f(x) = 2x^2$$ and the point $$x = 2$$. Substitute these values into the formula: $$m_{sec} = \frac{f(2+h) - f(2)}{h}$$ $$m_{sec} = \frac{2(2 + h)^2 - 2(2)^2}{h}$$

Step 3: Simplify the formula

Simplify the formula: $$m_{sec} = \frac{2(4 + 4h + h^2) - 8}{h}$$ $$m_{sec} = \frac{8 + 8h + 2h^2 - 8}{h}$$ $$m_{sec} = \frac{2h^2 + 8h}{h}$$

Step 4: Create a table of slopes for secant lines

Now we can create a table of slopes for different secant lines using values of $$h$$ approaching 0 (both from positive and negative sides): $$ \begin{array}{c|c} h & m_{sec} \\ \hline -0.1 & 7.8 \\ -0.01 & 7.98 \\ -0.001 & 7.998 \\ 0.001 & 8.002 \\ 0.01 & 8.02 \\ 0.1 & 8.2 \\ \end{array} $$

Step 5: Make a conjecture about the tangent line slope

Analyze the table and observe how the slopes are getting closer to a specific value as $$h$$ gets closer to 0. From the table, it appears that the slope of the tangent line is approaching 8. Thus, the conjecture is that the slope of the tangent at $$x = 2$$ is 8.

Key Concepts

Secant Line

A secant line is a line that connects two distinct points on a curve. Imagine you have a curvy road and decide to measure the direct distance between two points along it. That straight line measurement is like the secant line.
The slope of a secant line provides us with a way to estimate the average rate of change of the function between these two points. This concept is extremely useful in calculus as a way to approach and understand how functions behave.

• In the function given in the exercise, the equation for the secant line’s slope is formulated as \( m_{sec} = \frac{f(x+h) - f(x)}{h} \)
• By substituting specific values, we can begin to see how the slope changes as the distance between our chosen points, represented by \( h \), gets smaller.

Understanding secant lines provides foundational insight needed to explore more instantaneous rates of change, like those involving the concept of tangent lines.

Slope Calculation

Calculating the slope of a secant line involves determining the ratio of the change in the y-values to the change in the x-values. This ratio is expressed using the slope formula \( m = \frac{\Delta y}{\Delta x} \), where \( \Delta y \) and \( \Delta x \) represent the changes in the y-values and x-values, respectively.
In our exercise:

• We used the function \( f(x) = 2x^2 \) and calculated the difference between the function value at \( x + h \) and \( x \), which gives us \( \Delta y \).
• The change in x-values is represented by \( h \), making our formula \( m_{sec} = \frac{2((2+h)^2) - 8}{h} \).

Simplifying provides clearer insights into how the slope evolves:

• Our formula, after simplification, is \( m_{sec} = \frac{2h^2 + 8h}{h} \).
• As \( h \) approaches zero, the slope of the secant line approaches the actual tangent slope, revealing valuable information about the function’s behavior at a given point.

Limit of Function

The concept of limits is crucial to finding the exact slope of a tangent line. A limit evaluates what happens to a function’s value as the input approaches a certain point. This gives us an honest look into a function’s behavior at very small scales.
In our problem, as \( h \) becomes closer to zero - meaning that the two points on the curve get exceedingly near each other - the secant slope \( m_{sec} \) moves towards the slope of the tangent line.

• This is observed in the table where decreasing values of \( h \) lead to \( m_{sec} \) values approaching 8.
• The limit here shows us that the slope of the tangent line, as \( h \) approaches zero, is indeed 8.

Through limits, we transition from an average rate of change (secant line) to the instantaneous rate of change (tangent line), capturing how a function behaves precisely at a single point.