Question Statement

Find the slope of the tangent line to the function $f(x)=2x-1$ at the point $(4,7)$.

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Background and Explanation

The slope of a tangent line at a point on a curve is defined as the limit of the difference quotient as the interval approaches zero. For linear functions, this process confirms that the slope remains constant at every point on the line.

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Solution

We begin with the given function and apply the limit definition of the derivative to find the slope of the tangent line.

Given: $f(x)=2x-1$

First, determine $f(x+\Delta x)$ by substituting $(x+\Delta x)$ into the function: $f(x+\Delta x)=2(x+\Delta x)-1$

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 the expressions for $f(x+\Delta x)$ and $f(x)$: $\begin{array}{rl}{m}_{\tan }& =\underset{\Delta x\to 0}{\lim }\frac{[2(x+\Delta x)-1]-[2x-1]}{\Delta x}\\ & =\underset{\Delta x\to 0}{\lim }\frac{2x+2\Delta x-1-2x+1}{\Delta x}\\ & =\underset{\Delta x\to 0}{\lim }\frac{2\Delta x}{\Delta x}\end{array}$

Canceling $\Delta x$ from the numerator and denominator: $m_{\tan }={\lim }_{\Delta x\to 0}2=2$

Therefore: $\text{Slope of tangent}=2$

Since $f(4)=2(4)-1=7$, the point $(4,7)$ lies on the line, and the slope of the tangent line at this specific point is $2$.

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Key Formulas or Methods Used

• Limit definition of the derivative (slope of tangent): $m_{\tan }={\lim }_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$

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Summary of Steps

1. Write the given function $f(x)=2x-1$
2. Compute $f(x+\Delta x)=2(x+\Delta x)-1$ by substituting $(x+\Delta x)$ into the original function
3. Apply the limit definition of the slope of the tangent line
4. Substitute the function expressions into the difference quotient and simplify the numerator
5. Cancel $\Delta x$ and evaluate the limit as $\Delta x\to 0$ to obtain $m_{\tan }=2$