Near the point of tangency a curve and its tangent are almost identical, and 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 f.
Choosing the base point
Pick an a nearby where the function is exactly known. For √4.1, take 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. Concave up gives an underestimate.
The exam asks for the direction
Free-response questions routinely ask whether an approximation is an over- or underestimate. The answer is the sign of f double prime.
Differentials
dy = f′(x) dx estimates the change in the function for a small change in input.
The tangent line as a stand-in
L(x) = f(a) + f′(a)(x − a) approximates f near a. It is simply the tangent line, used to estimate values of a function that is awkward to evaluate directly.
Over or under, decided by concavity
If f is concave up, the tangent lies below the curve and the approximation underestimates. Concave down means it overestimates. The exam asks for this justification, and it comes from the sign of f″.
Differentials
dy = f′(x)dx estimates the change in output from a small change in input. It is the same idea in different notation, and it is how error propagation is computed in the sciences.
The approximation degrades with distance
It is excellent close to a and worsens as you move away, at a rate governed by the second derivative. Using a linearisation far from its base point is a misuse, not merely an inaccuracy.
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 Kofi Estimate a Root
Kofi approximates √4.1 using a tangent line.
- Step 1
He takes f(x) = √x and the nearby exact 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 an over- or underestimate? 1 over, 2 under.
Approximate 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
A curve is concave up. Over- or underestimate? 1 over, 2 under.
4 of 8
L(x) = 2 + 0.25(x − 4). What is L(4)?
5 of 8
dy = f′(x) dx with f′(x) = 6 and dx = 0.5. What is dy?
6 of 8
Which sign of f double prime gives an overestimate? 1 positive, 2 negative.
7 of 8
Which choices improve a linear approximation?
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
What determines whether an approximation is an over- or underestimate?
What You Learned
- L(x) = f(a) + f′(a)(x − a) uses the tangent as a stand-in.
- Concave down overestimates; concave up underestimates.
- Accuracy falls away as you move from the base point.