Question Statement

Find the slope of the tangent line to the function $f(x)=-\frac{1}{2}x+3$ at the point $(a,f(a))$ using the limit definition of the derivative.

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Background and Explanation

This problem applies the limit definition of the derivative to find the slope of a tangent line to a linear function. The limit definition calculates the instantaneous rate of change by evaluating the limit of the difference quotient as the interval $\Delta x$ approaches zero.

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Solution

Given the function: $f(x)=-\frac{1}{2}x+3$

First, we evaluate the function at $x+\Delta x$: $f(x+\Delta x)=-\frac{1}{2}(x+\Delta x)+3$

The slope of the tangent line is defined by the limit: $m_{\tan }={\lim }_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$

Substituting our expressions for $f(x+\Delta x)$ and $f(x)$: $m_{\tan }={\lim }_{\Delta x\to 0}\frac{\left(-\frac{1}{2}(x+\Delta x)+3\right)-\left(-\frac{1}{2}x+3\right)}{\Delta x}$

Expanding the numerator: $={\lim }_{\Delta x\to 0}\frac{-\frac{1}{2}x-\frac{1}{2}\Delta x+3+\frac{1}{2}x-3}{\Delta x}$

Simplifying by combining like terms (the $-\frac{1}{2}x$ and $+\frac{1}{2}x$ cancel, as do the $+3$ and $-3$): $={\lim }_{\Delta x\to 0}\frac{-\frac{1}{2}\Delta x}{\Delta x}$

Canceling $\Delta x$ from the numerator and denominator: $={\lim }_{\Delta x\to 0}\left(-\frac{1}{2}\right)$

Evaluating the limit (since the expression no longer depends on $\Delta x$): $m_{\tan }=-\frac{1}{2}$

Therefore, the slope of the tangent line at any point $(a,f(a))$ is $-\frac{1}{2}$.

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Key Formulas or Methods Used

• Limit definition of the derivative (slope of tangent): $m_{\tan }={\lim }_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$
• Function substitution: Evaluating $f(x+\Delta x)$ by replacing $x$ with $(x+\Delta x)$ in the function expression

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Summary of Steps

1. Identify the given function $f(x)=-\frac{1}{2}x+3$
2. Calculate $f(x+\Delta x)$ by substituting $(x+\Delta x)$ into the function
3. Set up the limit definition of the derivative (the difference quotient)
4. Substitute both $f(x+\Delta x)$ and $f(x)$ into the numerator
5. Expand and simplify the numerator by distributing and combining like terms
6. Cancel $\Delta x$ from the numerator and denominator
7. Evaluate the limit as $\Delta x\to 0$ to obtain the slope $-\frac{1}{2}$