Let a>1 be some fixed number. Let h>0 be some small number. Write out a formula for the slope of the secant line to the graph of x=p(t) between the point at t=a and the point at t=a+h. You don't need to simplify this formula yet.

To find the slope of the secant line to the graph of \(x = p(t)\) between the points at \(t = a\) and \(t = a + h\), we use the formula for the slope between two points on a curve. The slope of a secant line is given by the difference in the function values divided by the difference in the independent variable.

The two points on the curve are:
- At \( t = a \): \( \left(a, p(a)\right) \)
- At \( t = a + h \): \( \left(a + h, p(a + h)\right) \)

The slope of the secant line, \(m_{\text{secant}}\), is:

\[
m_{\text{secant}} = \frac{p(a + h) - p(a)}{(a + h) - a}
\]

Since \( (a + h) - a = h \), the formula simplifies to:

\[
m_{\text{secant}} = \frac{p(a + h) - p(a)}{h}
\]

This is the unsimplified formula for the slope of the secant line between \(t = a\) and \(t = a + h\).

Would you like further explanation or examples?

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Here are 5 related questions to extend this concept:

1. How does the secant line relate to the derivative of the function at \(t = a\)?
2. What happens to the slope of the secant line as \(h\) approaches zero?
3. How can you interpret the secant line geometrically in terms of the graph of \(p(t)\)?
4. What is the formula for the tangent line's slope at \(t = a\) for the same function \(p(t)\)?
5. How would the secant slope formula change if we consider the points \(t = a\) and \(t = b\) instead of \(t = a\) and \(t = a + h\)?

**Tip:** The secant line approximates the slope of the tangent line as \(h\) becomes smaller. The derivative is the limit of the secant line's slope as \(h\) approaches zero.

Let a > 1 be some fixed number. Let h > 0 be some small number. Write out a formula for the slope of the secant line to the graph of x = p(t) between the point at t = a and the point at t = a + h. You don't need to simplify this formula yet.

Discover how to calculate the slope of the secant line for a function p(t) between t = a and t = a + h. This problem uses key calculus concepts like secant lines, the slope formula, and introduces the connection to derivatives as h approaches zero. Ideal for students in Grades 11-12 or early college calculus.

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