A sphere's radius is measured as r = 10 cm with a measurement error of dr = 0.2 cm. Using differentials, what is the approximate error in the computed volume V = (4/3)πr³?
AdV = (4/3)π(0.2)³ ≈ 0.034 cm³
BdV = 4πr² dr = 4π(100)(0.2) ≈ 251.3 cm³
CdV = (4/3)π(10.2)³ − (4/3)π(10)³ ≈ 257 cm³
DdV = 4πr³ dr = 4π(1000)(0.2) ≈ 2513 cm³
The differential of V = (4/3)πr³ is dV = 4πr² dr. Substituting r = 10 and dr = 0.2 gives dV = 4π(100)(0.2) ≈ 251.3 cm³. This is the change along the tangent (the differential), which approximates the true change ΔV. Option C computes ΔV exactly — notice dV and ΔV are close but not identical; the differential is the approximation.
Question 2 Multiple Choice
Which statement best describes the relationship between dy and Δy as dx approaches zero?
Ady and Δy both approach zero, and their ratio dy/Δy approaches 1
Bdy approaches Δy exactly for sufficiently small dx, so they become equal
Cdy and Δy both approach zero, but they remain different quantities representing different geometric objects
Ddy approaches zero while Δy remains constant
As dx → 0, both dy and Δy → 0. The key fact is that their ratio dy/Δy → 1, meaning the differential is an ever-better proportional approximation to the actual change. But dy and Δy never become equal for any nonzero dx (unless the function is linear) — dy always follows the tangent line while Δy follows the curve. Option B is the classic confusion: 'sufficiently small' doesn't make them equal, only close in proportion.
Question 3 True / False
In Leibniz notation, dy/dx should be understood as a limit of a ratio, not as an actual ratio of two quantities.
TTrue
FFalse
Answer: False
Once differentials are properly defined, dy/dx IS a literal ratio of two differentials: dy = f′(x) dx, and dividing both sides by dx gives dy/dx = f′(x). This is precisely why Leibniz notation is so powerful — it lets the chain rule look like fraction cancellation (dy/du · du/dx = dy/dx) and makes substitution in integrals meaningful. The differential formalism vindicates the literal ratio interpretation that the limit definition initially discouraged.
Question 4 True / False
If y = x³, then dy = 3x² dx gives the exact change in y for any nonzero value of dx.
TTrue
FFalse
Answer: False
dy = 3x² dx is an approximation — it gives the change along the tangent line, not the actual change along the curve. The true change is Δy = (x + dx)³ − x³ = 3x² dx + 3x(dx)² + (dx)³. The terms 3x(dx)² + (dx)³ are the error: for small dx they are negligible, but for large dx they matter significantly. The differential approximation is exact only when f is linear.
Question 5 Short Answer
Explain why the substitution step in u-substitution (writing du = f'(x) dx) is mathematically legitimate, not just a notational trick.
Think about your answer, then reveal below.
Model answer: When you write u = g(x) and compute du = g′(x) dx, you are computing a differential — a precise object defined as the rate of change of u times a change in x. This means du and dx are related as actual quantities, not just notation. When you substitute into an integral, replacing the integrand's x-expression with u and the 'dx' with 'du/g′(x)', you are performing a genuine change of variable using the differential relationship. The integral's value is preserved because the differential correctly captures how the infinitesimal element transforms under the substitution.
The legitimacy of u-substitution rests on the differential formalism: differentials obey the chain rule algebraically, so du = g′(x) dx can be rearranged and substituted. This is not a symbol-shuffling trick — it is a coherent change of variables where the differential of the new variable correctly accounts for the stretching or compressing of the integration variable. This is why the topic builds toward u-substitution: differentials are the conceptual foundation that makes the technique rigorous.