Questions: Use the definition for the slope of a tangent line below to explain how slopes of secant lines approach the slope of the tangent line at a point. mtan = limx -> a (f(x)-f(a))/(x-a) Given the point (a, f(a)) and any point (x, f(x)) near (a, f(a)), the slope of the line joining these points is. The limit of this quotient as x approaches a is the slope of the line at the point (a, f(a)), provided that this limit exists.

Use the definition for the slope of a tangent line below to explain how slopes of secant lines approach the slope of the tangent line at a point.
mtan = limx -> a (f(x)-f(a))/(x-a)

Given the point (a, f(a)) and any point (x, f(x)) near (a, f(a)), the slope of the line joining these points is. The limit of this quotient as x approaches a is the slope of the line at the point (a, f(a)), provided that this limit exists.

Transcript text: Use the definition for the slope of a tangent line below to explain how slopes of secant lines approach the slope of the tangent line at a point. \[ m_{\tan }=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a} \] Given the point ( $a, f(a))$ and any point $(x, f(x))$ near $(a, f(a))$, the slope of the $\square$ line joining these points is $\square$ The limit of this quotient as $x$ approaches $\square$ is the slope of the $\square$ line at the point (a, $\mathrm{f}(\mathrm{a}))$, provided that this limit exists.

Solution

Solution Steps

To explain how the slopes of secant lines approach the slope of the tangent line at a point, we can use the definition of the derivative. The slope of the secant line between the points $(a, f(a))$ and $(x, f(x))$ is given by the difference quotient $\frac{f(x) - f(a)}{x - a}$. As $x$ approaches $a$, this difference quotient approaches the derivative of $f$ at $a$, which is the slope of the tangent line at that point.

Step 1: Define the Function and Points

Let $ f(x) $ be a function defined in the neighborhood of the point $ a $. We are interested in the slope of the secant line joining the points $ (a, f(a)) $ and $ (x, f(x)) $.

Step 2: Difference Quotient

The slope of the secant line is given by the difference quotient: \[ m_{\text{secant}} = \frac{f(x) - f(a)}{x - a} \]

Step 3: Limit as $ x $ Approaches $ a $

To find the slope of the tangent line at the point $ (a, f(a)) $, we take the limit of the difference quotient as $ x $ approaches $ a $: \[ m_{\tan} = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \]

Step 4: Resulting Expression

The limit results in the derivative of $ f $ at the point $ a $: \[ m_{\tan} = f'(a) \]

Final Answer

Thus, the slopes of the secant lines approach the slope of the tangent line at the point $ (a, f(a)) $ as $ x $ approaches $ a $: \[ \boxed{m_{\tan} = f'(a)} \]

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