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.