Tangent Lines and Linear Approximation

Local Linearization

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A tangent line gives the instantaneous direction of a curve at one point, and linear approximation uses that line to estimate nearby function values. These ideas are central in calculus because they connect geometry, rates of change, and practical estimation. When a function is complicated, its tangent line often provides a much simpler local model. This helps in physics, engineering, economics, and any setting where small changes matter.

At $x = a$ , the tangent line has slope $f'(a)$ and passes through the point $(a, f(a))$ . Its equation is $y = f(a) + f'(a)(x - a)$ , and this same expression is called the linear approximation $L(x)$ . Near $x = a$ , $L(x)$ is usually close to $f(x)$ , but the approximation becomes less accurate farther away. The quality of the estimate depends on how much the curve bends, which is related to the second derivative.

Key Facts

The slope of the tangent line at $x = a$ is $f'(a)$ .
Point-slope form of the tangent line: $y - f(a) = f'(a)(x - a)$ .
Linear approximation formula: $L(x) = f(a) + f'(a)(x - a)$ .
Use $f(a + h)$ approximately equal to $f(a) + f'(a)h$ for small $h$ .
If $f''(x)$ is small near $a$ , the tangent line often gives a better local approximation.
The approximation is exact at $x = a$ because $L(a) = f(a)$ .

Vocabulary

Tangent line: A line that matches the slope of a curve at a specific point and locally follows the curve there.
Derivative: The derivative $f'(a)$ is the instantaneous rate of change of a function at $x = a$ .
Linear approximation: A nearby estimate of a function using the tangent line formula $L(x) = f(a) + f'(a)(x - a)$ .
Point of tangency: The point $(a, f(a))$ where the tangent line touches the curve and shares its slope.
Local behavior: How a function acts close to a chosen point rather than over its entire graph.

Common Mistakes to Avoid

Using the tangent line far from the point of tangency, which is wrong because linear approximation is only reliable near $x = a$ where the curve and line stay close.
Confusing $f(a)$ with $f'(a)$ , which is wrong because $f(a)$ is the function value while $f'(a)$ is the slope at that point.
Writing the tangent line as $y = f'(a)x + f(a)$ , which is wrong unless $a = 0$ because the correct formula must account for the shift from $x = a$ .
Forgetting to plug in the base point a before estimating, which is wrong because the approximation depends on the specific point where the tangent line is built.

Practice Questions

1 Find the tangent line and linear approximation for $f(x) = x^2$ at $a = 3$ . Then use it to estimate $f(3.1)$ .
2 Let $f(x) = \sqrt{x}$ and $a = 9$ . Find $L(x)$ , then use it to estimate $\sqrt{9.2}$ .
3 A function has a large positive second derivative near $x = a$ . Explain whether its tangent line is likely to stay close to the curve over a wide interval around $a$ .