M171 Calculus I · Unit 2: Computing derivatives · on Exam 2
§1.8 The tangent line approximation
Study notes for The tangent line approximation in M171 Calculus I at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- The tangent line (linearization) at $a$ is $L(x) = f(a) + f'(a)(x-a)$. Near $a$, $f(x)\approx L(x)$ because differentiable functions are locally linear.
- Concavity tells you the error's sign: if $f$ is concave up near $a$, the tangent line lies below the curve, so $L(x)$ underestimates $f(x)$. Concave down: overestimate.
- Farther from $a$ means a worse approximation.
Key formulas
Linearization
$$L(x) = f(a) + f'(a)(x-a)$$
Estimate
$$f(a+h) \approx f(a) + f'(a)\,h$$
Worked example
Given $g(4)=2$, $g'(4)=0.25$ and $g''(x)<0$ for all $x$, estimate $g(4.4)$ and say whether it is an over- or underestimate.
Show the solution
$L(4.4) = 2 + 0.25(0.4) = 2.1$. Since $g$ is concave down, the tangent line lies above the curve, so $2.1$ is an overestimate.
Common mistakes
- Using $f'(x)$ (a function) in the tangent line instead of the number $f'(a)$.
- Forgetting to state over/underestimate when concavity is given: it is usually part of the question.
Exam tipLab 5 is linearization. Practice writing $L(x)$ and evaluating it at a nearby point with a clean sentence about the error.
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