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Chapter 2: Problem 28

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. $$f(x)=x^{3}-x \quad \text { at } x=1$$

Short Answer

Expert verified
Answer: The conjecture for the slope of the tangent line at the point (x=1) is given by the limit as h approaches 0: $$m_{tan} = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h}$$

Step by step solution

01

Understand the concept of secant lines and tangent lines

A secant line is a line that intersects a curve at two distinct points, while a tangent line is a line that touches a curve at just one point. In this problem, we need to find the slope of the secant lines and then conjecture the slope of the tangent line at the indicated point `(x=1)`.
02

Find the slope of secant lines

To find the slope of the secant lines, we need to choose two points on the curve. The first point is `(x=1, y=f(1))`, and the second point will be `(x+h, y=f(1+h))`. Let's set an interval h for the x-value. The slope of the secant line between these points is given by: $$m_{sec} = \frac{f(1+h) - f(1)}{h}$$
03

Create a table of slopes for secant lines

For different values of h, we can calculate the slope of secant lines following the formula above and create a table as follows: | h | `f(1) = 1^3 - 1` | `f(1 + h)` | `m_{sec}` | |---|--------------|---------|--------| | 1 | 0 | `f(2)` | \(\frac{f(2) - f(1)}{1}\) | | 0.1 | 0 | `f(1.1)` | \(\frac{f(1.1) - f(1)}{0.1}\) | | 0.01 | 0 | `f(1.01)` | \(\frac{f(1.01) - f(1)}{0.01}\) | | ... | ... | ... | ... | We should observe a pattern from these slopes, which suggest the slope of the tangent line when `h` become very small or approach to `0`.
04

Conjecture the slope of the tangent line

Based on our table of slopes in step 3, we can make a conjecture about the slope of the tangent line as h approaches 0. The value of the slope of the tangent line is the limit of the slope of secant lines as `h -> 0`: $$m_{tan} = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h}$$ This is our conjecture for the slope of the tangent line at the point `(x=1)` on the curve `f(x) = x^3 - x`.

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