You want to approximate f(x) = sin(x) near a = 0 using a linear approximation. Which estimate will be least accurate?
Asin(0.01)
Bsin(0.1)
Csin(0.5)
Dsin(1.5)
Linear approximation L(x) = f(a) + f'(a)(x−a) gets less accurate as x moves farther from a. At a = 0, sin(x) ≈ x. The estimate for sin(1.5) is farthest from the center a = 0, where the function has curved significantly away from the tangent line. The error grows with both distance from a and the curvature of f near a.
Question 2 Multiple Choice
A function has f(5) = 3 and f'(5) = 2. What is the linear approximation L(5.001)?
A3.001 — because L(x) ≈ f(a) + (x − a)
B3.002 — because L(x) = f(a) + f'(a)(x − a)
C10.002 — because f'(5) × 5.001 ≈ 10
D5.002 — because the tangent line passes through (a, f'(a))
L(x) = f(a) + f'(a)(x − a) = 3 + 2(5.001 − 5) = 3 + 2(0.001) = 3.002. The formula is the point-slope equation of the tangent line at (a, f(a)), evaluated at x. Option A confuses f'(a) with 1; options C and D misapply the formula entirely.
Question 3 True / False
The linear approximation L(x) always overestimates the true value of a concave-down function (f'' < 0) near the base point a.
TTrue
FFalse
Answer: True
When f''(a) < 0, the function curves downward — it bends below the tangent line on both sides of a. This means the tangent line lies above the curve, so L(x) ≥ f(x) near a, giving an overestimate. The second derivative reveals not just the magnitude of the error but its direction, even though f'' doesn't appear in the formula L(x) = f(a) + f'(a)(x − a) itself.
Question 4 True / False
Linear approximation is mainly useful when the exact value of f(x) is substantially unknown to the user.
TTrue
FFalse
Answer: False
Linear approximation is valuable when exact computation is difficult, not merely when the value is unknown. Approximating √4.1 by linearizing at a = 4 provides a quick, accurate estimate without a calculator. But the technique's importance extends further: it is the foundation for differentials, Newton's root-finding method, and Taylor polynomials. Its value is computational convenience and conceptual generativity, not just filling gaps in knowledge.
Question 5 Short Answer
Why does the accuracy of a linear approximation depend on f''(a), even though the formula L(x) = f(a) + f'(a)(x − a) contains no second derivative term?
Think about your answer, then reveal below.
Model answer: f''(a) measures how sharply the function curves away from the tangent line. A large |f''(a)| means the function bends steeply, so the tangent diverges quickly from the curve as x moves away from a. A small |f''(a)| means the function is nearly linear near a, and the approximation stays accurate over a wider range. The leading error term in the Taylor expansion is (1/2)f''(a)(x−a)², so f'' directly governs the approximation error.
The linear approximation is the first-order Taylor polynomial. The 'missing' second-derivative term is precisely what makes it an approximation rather than the exact value. Understanding this reveals why L(x) works so well for √x at a = 4 (small f'' there) but would be less reliable for functions with large curvature — the formula doesn't include the error, but the error is always there, governed by f''.