A tangent line approximation replaces a curved function with the straight line that just touches it at a chosen point. This is useful because lines are much easier to calculate with than curves, especially for quick estimates. The approximation is best near the tangent point, where the function and tangent line share the same value and slope.
As you move farther away, the curve can bend away from the line, creating error.
The size of the error depends mainly on distance from the tangent point and on the curvature of the function. The second derivative measures how quickly the slope is changing, so it controls how strongly the graph bends. Taylor's theorem shows that the error is roughly proportional to the square of the distance from the tangent point.
This is why doubling the distance can make the error about four times larger when curvature stays similar.
Key Facts
- Linear approximation at x = a: L(x) = f(a) + f'(a)(x - a)
- Approximation error: Error = f(x) - L(x)
- Second-order error estimate: Error ≈ (1/2)f''(a)(x - a)^2
- Taylor error form: f(x) - L(x) = (1/2)f''(c)(x - a)^2 for some c between a and x
- If f''(x) > 0, the graph is concave up and the tangent line usually lies below the curve near the point.
- If |x - a| doubles, the tangent line error is often about 4 times larger when f'' is nearly constant.
Vocabulary
- Tangent line
- A line that touches a curve at a point and has the same slope as the curve at that point.
- Linear approximation
- An estimate of a function near a point using the tangent line at that point.
- Approximation error
- The difference between the true function value and the value predicted by the tangent line.
- Second derivative
- The derivative of the derivative, which measures how the slope of a function is changing.
- Concavity
- The bending direction of a graph, determined by whether the second derivative is positive or negative.
Common Mistakes to Avoid
- Using the tangent line far from the tangent point: this is wrong because linear approximation is only reliable near the point where it is made.
- Ignoring the second derivative: this is wrong because two functions can have the same tangent line but very different errors if their curvatures are different.
- Assuming the error grows linearly with distance: this is wrong because the leading error term usually contains (x - a)^2.
- Dropping the sign of the error without thinking: this is wrong because the sign tells whether the tangent line estimate is above or below the true curve.
Practice Questions
- 1 For f(x) = x^2 at a = 3, find the tangent line approximation L(x) and compute the exact error at x = 3.2.
- 2 Use the error estimate Error ≈ (1/2)f''(a)(x - a)^2 for f(x) = sin x at a = 0. Estimate the error at x = 0.1, and compare it to the exact error sin(0.1) - 0 using a calculator.
- 3 A function has a large positive second derivative near a tangent point. Explain whether the tangent line approximation is likely to be above or below the curve nearby, and why the error grows as you move away.