Close to the point of tangency, a curve and its tangent are almost the same. The line is far easier to evaluate.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The formula
L(x) = f(a) + f′(a)(x − a). It is the tangent line at a, used as a stand-in for the function.
Choosing the base point
Pick an a nearby where the function is easy. To estimate √4.1, use a = 4, where the value is exactly 2.
Which way it errs
A concave-down curve sits below its tangent, so the estimate is too high. A concave-up curve gives an estimate that is too low.
Accuracy fades
The further from a, the worse the estimate. Linear approximation is a local tool.
Differentials
The change in the line is dy = f′(x) dx, which estimates the change in the function for a small change in input.
Use the tangent instead of the curve
Near a point, a differentiable function is almost its tangent line. Replacing the curve with the line gives a good approximation for nearby inputs, and the line is much easier to evaluate.
The formula
f(x) ≈ f(a) + f′(a)(x − a). It is the equation of the tangent line at a, used as a stand-in for f. The approximation is exactly the function value plus the slope times the step.
The error grows with distance
The approximation is excellent near a and degrades as you move away. How fast depends on the concavity, which is why the second derivative appears in error estimates and in Taylor series.
Where it is used
Newton's method, error propagation in measurement, and the linearised models that dominate engineering. Almost every applied model is a linear approximation to something that is not linear.
Step 2: Try It Yourself
Tap and try it out.
- Point(4, 2)
- Slope of the tangent0.25
Step 3: Watch an Example
One step at a time.
Watch Yusuf Estimate the Square Root of 4.1
Yusuf wants √4.1 without a calculator.
- Step 1
He takes f(x) = √x and the nearby easy point a = 4, where f(4) = 2.
Step 4: Your Turn
Practice makes it stick.
The Estimate
Problem 1 of 2
L(x) = 2 + 0.25(x − 4). What is L(4.2)?
The Direction
Problem 2 of 2
A curve is concave down. Is the tangent estimate too high or too low? 1 high, 2 low.
Estimate With a Line
1 of 8
L(x) = 3 + 2(x − 1). What is L(1.5)?
2 of 8
f(9) = 3 and f′(9) = 1/6. Estimate f(9.6).
3 of 8
f(x) = √x. What is f(25)?
4 of 8
A curve is concave up. Is the estimate high or low? 1 high, 2 low.
5 of 8
L(x) = 2 + 0.25(x − 4). What is L(4)?
6 of 8
dy = f′(x) dx. With f′(x) = 6 and dx = 0.5, what is dy?
7 of 8
Which choices make a linear approximation more accurate?
8 of 8
L(x) = 5 + 3(x − 2). What is L(2.1)?
Step 5: Quick Check
Show what you know.
Question 1 of 2
L(x) = 4 + 0.5(x − 16). What is L(16.4)?
Question 2 of 2
Why does the estimate worsen far from the base point?
What You Learned
- L(x) = f(a) + f′(a)(x − a) uses the tangent line as a stand-in.
- Concave down overestimates; concave up underestimates.
- Accuracy falls away as you move from the base point.