The Big Idea: Curves Look Like Lines Up Close
Most functions you encounter in calculus are curved. But if you zoom in sufficiently close to a point on a differentiable curve, the curve begins to look almost like a straight line.
This idea is called local linearity. Near a point , a differentiable function behaves approximately like its tangent line at that point.
That observation is extremely useful. Instead of calculating a difficult value of a function exactly, you can use a nearby value that you already know and the derivative at that nearby point to make a very good estimate.
The Tangent Line as an Approximation
The tangent line to at has the familiar point-slope form:
When we use this tangent line to approximate the original function near , we usually write the approximation as:
The expression is the linearization of at .
There is an important distinction to keep in mind:
- The tangent line is an actual line.
- The linearization is the formula we use to approximate the function.
- The approximation is the estimated value obtained by plugging a nearby input into that linearization.
Why the Formula Works
Suppose you know the exact value of and the derivative at . If you move a small horizontal distance away from , the tangent line predicts that the corresponding change in the function is approximately:
So the new function value is approximately the old value plus the estimated change:
This is the same idea behind the tangent line formula, just written in a form that is especially useful for approximation problems.
A Step-by-Step Method
- Choose a convenient nearby point. Look for a value of where the function is easy to evaluate exactly.
- Find the derivative. Calculate .
- Evaluate the function and derivative at the nearby point. Find and .
- Write the linearization. Use .
- Use the linearization to estimate the requested value. Substitute the desired input into .
- State that the result is an approximation. The tangent line usually does not give the exact value unless the function happens to be linear.
Worked Examples
Use a tangent line approximation to estimate .
Use local linearity to estimate .
Use a tangent line approximation to estimate .
Suppose , , and . Use the tangent line at to approximate .
When Is the Approximation Good?
The tangent line is generally a better approximation when the requested input is close to the point of tangency.
Why? The tangent line matches the function at the point of tangency in two important ways:
- It has the same function value.
- It has the same instantaneous rate of change.
So, right near the point, the curve and its tangent line are moving together. As you move farther away, however, the curvature of the original function becomes more important and the approximation can become less accurate.
The symbol means "approximately equal to." It is important on an AP Calculus response because the tangent-line result is an estimate, not generally an exact value.
Common Mistakes
The whole point of local linearity is to replace a difficult calculation with an easy one. Choose a nearby point where the function value is known or easy to calculate. For square roots, perfect squares are especially useful; for trigonometric functions, familiar angles are often ideal.
The slope in the tangent line must be evaluated at the base point. Do not substitute the target input into the derivative when constructing the linearization.
The linearization is not just . It must include both the known function value and the derivative: .
Unless the function is actually linear, the tangent line and the function are not identical. Write , not , when reporting a tangent-line estimate.
Practice Problems
Use a tangent line approximation to estimate .
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Use local linearity to estimate .
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Find the linearization of at , and use it to approximate .
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Suppose and . Use the tangent line at to approximate .
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