Approximate the value of $g(6)$ using the tangent line to $g(x)$ at $x=5$

Expert intro: This is a standard linear approximation problem. The tangent line gives the best local estimate of the function near the point of tangency.

Detailed walkthrough

To approximate $g(6)$ using the tangent line at $x=5$, use the linearization formula

$L(x)=g(5)+g'(5)(x-5).$

Then evaluate at $x=6$:

$g(6)\approx L(6)=g(5)+g'(5)(6-5).$

So

$g(6)\approx g(5)+g'(5).$

If the problem provides numerical values for $g(5)$ and $g'(5)$, substitute them directly into this expression. The tangent line estimate works well when $6$ is close to $5$, because the function behaves almost linearly over a small interval.

💡 Pitfall guide

A common mistake is to use the slope $g'(5)$ by itself and forget the base value $g(5)$. Another mistake is to plug in $x=6$ into the original function instead of the tangent-line formula.

🔄 Real-world variant

If you need to approximate another nearby value, such as $g(4.8)$ or $g(5.2)$, use the same tangent-line formula:

$g(x)\approx g(5)+g'(5)(x-5).$

The only thing that changes is the $x$ value you substitute.

🔍 Related terms

tangent line, linear approximation, linearization