Question

Use the limit definition to find the slope of the tangent line to the graph of \(f\) at the given point. $$ f(x)=6-2 x ;(2,2) $$

Short answer

The slope of the tangent line is -2.

Step-by-step solution

Write out the Difference Quotient

The difference quotient uses the variable \(h\) to represent the mutable distance along the x-axis from the given point to another on the function's curve. The formula for the difference quotient is \(\frac{f(x+h)-f(x)}{h}\).

Substitute \(x\) for the Given x-coordinate and \(f(x)\) for the Given y-coordinate

We have \(x=2\) and \(f(x)=2\). Therefore, we obtain \(\frac{f(2+h)-2}{h}\)

Substitute \(f(2+h)\) in the Difference Quotient

The function \(f(x)\) is given as \(6-2x\). Therefore, \(f(2+h)\) is \(6-2(2+h)\), or \(6-4-2h\), or \(2-2h\). So, we now have \(\frac{2-2h-2}{h}\) or \(\frac{-2h}{h}\)

Simplify

The expression simplifies to -2.

Take the Limit

Because the constant does not have \(h\) in its expression, the limit as \(h\) approaches 0 is -2.