Source: Stewart, Calculus 7th Ed., Ch. 3.2 | Math 132, MSU
Tags: Uniqueness Theorem, constant function, same derivative, physical law, linear approximation, error bound
Difficulty: Intermediate to Advanced | Prerequisites: Rolle's Theorem, Mean Value Theorem (both in Ch. 3.2), linear approximation (Ch. 2.9), basic differentiation
The Uniqueness Theorem is described in the source as the most important result of Section 3.2. It answers a question that sits beneath all of applied mathematics: if two functions have the same derivative and agree at one point, must they be the same function? The answer is yes, and the proof uses the Mean Value Theorem. This is what guarantees that differential equations in physics produce a single, unambiguous prediction rather than a family of candidates. The section closes with the Linear Approximation Error Estimate, a practical application of the MVT that bounds how far a tangent-line approximation can stray from the true function.
If f'(x) = 0 everywhere, f must be constant. If two functions share the same derivative and agree at one point, they are identical. These results make physical laws well-defined. The linear approximation error is controlled by the second derivative and grows with the square of the distance from the centre point.
Constant function
A function f(x) = C for all x, where C is a fixed number. Its derivative is zero everywhere.
Think of it as a flat horizontal line on the graph.
Uniqueness (of a solution)
The property that a differential equation with a given initial condition has exactly one solution, not multiple competing candidates.
In simple terms, this means the maths gives one answer, not several contradictory ones.
Physical law
A mathematical relationship that predicts exactly one outcome for a given set of initial conditions. The Uniqueness Theorem is what makes differential equations qualify as physical laws.
Linear approximation (tangent-line approximation)
The approximation f(x) is roughly equal to L_a(x) = f(a) + f'(a)(x - a). It replaces the curve with its tangent line near x = a.
Think of it as zooming in so close on the curve that it looks like a straight line.
Error estimate (for linear approximation)
A bound on how far L_a(x) can deviate from f(x). Controlled by the second derivative: if |f''(x)| <= B near a, then the error is at most (B/2)|x - a|^2.
Initial condition
A known value of the function at a specific point, such as f(c) = g(c). Combined with a shared derivative, an initial condition pins down a unique function.
The theorem has three parts, each building on the one before.
Part (a): Zero derivative implies constant
If f(x) is differentiable and f'(x) = 0 for all x in (a, b), then f(x) = C for some constant C.
Proof sketch: Suppose f were not constant. Then f(a_1) differs from f(b_1) for some points in [a, b]. Applying the MVT to [a_1, b_1] gives f'(c) = (f(b_1) - f(a_1))/(b_1 - a_1), which is nonzero. This contradicts f'(x) = 0 everywhere. So f must be constant.
Part (b): Same derivative implies functions differ by a constant
If f'(x) = g'(x) for all x in (a, b), then f(x) = g(x) + C for some constant C.
Proof: Define h(x) = f(x) - g(x). Then h'(x) = f'(x) - g'(x) = 0, so by part (a), h(x) is constant. Therefore f(x) = g(x) + C.
Part (c): Same derivative plus shared value implies identical functions
If f'(x) = g'(x) for all x in (a, b), and f(c) = g(c) for some c in (a, b), then f(x) = g(x) for all x.
Proof: By part (b), f(x) = g(x) + C. Plugging in x = c gives C = f(c) - g(c) = 0. So f(x) = g(x).
The ballistic equation s(t) = s_0 + v_0 t - (1/2)gt^2 gives the height of an object thrown straight up. Its derivative s'(t) = v_0 - gt gives the velocity.
The Uniqueness Theorem (part c) shows that this is the only function with that derivative and initial height s(0) = s_0. If some other function s_tilde(t) had the same derivative and the same initial value, the theorem forces s_tilde(t) = s(t).
This is what makes the equation a physical law: the mathematics produces exactly one prediction, matching the experimental fact that identical launches produce identical trajectories. (The source notes that quantum mechanics operates on different principles, applying calculus to probability distributions rather than definite positions.)
Recall from Section 2.9 that the linear approximation of f(x) centred at x = a is:
f(x) \approx L_a(x) = f(a) + f'(a)(x - a)The error bound states that if |f''(x)| <= B over an interval (a - delta, a + delta), then:
|f(x) - L_a(x)| \leq \frac{B}{2}|x - a|^2The proof defines the error function epsilon(t) = f(x) - L_t(x) = f(x) - f(t) - f'(t)(x - t), treating x as fixed and varying the centre point t. Taking the derivative with respect to t gives epsilon'(t) = -f''(t)(x - t). Applying the MVT to epsilon on [a, x] yields the bound.
The key takeaway: the error is proportional to |x - a|^2, so the approximation improves quadratically as you move closer to the centre. The second derivative controls how quickly the tangent line drifts away from the curve.
Uniqueness Theorem, Part (c):
f'(x) = g'(x) \text{ for all } x \in (a,b) \text{ and } f(c) = g(c) \implies f(x) = g(x)Ballistic equation:
s(t) = s_0 + v_0 t - \tfrac{1}{2}g t^2, \quad s'(t) = v_0 - gtLinear approximation error:
|f(x) - L_a(x)| \leq \frac{B}{2}|x-a|^2 \quad \text{where } |f''(x)| \leq BThe Uniqueness Theorem is the reason physics works as a predictive science. When engineers compute the trajectory of a satellite using Newton's laws, they rely on the mathematical guarantee that the equations produce exactly one answer. Without uniqueness, the same differential equation could yield multiple trajectories, and no amount of calculation would tell you which one the satellite actually follows.
The linear approximation error estimate is used whenever you need to know how trustworthy a tangent-line approximation is, for instance when estimating square roots or cube roots by hand, or bounding rounding errors in numerical computation.
Students sometimes think that "same derivative" alone forces two functions to be equal. It does not: they can differ by a constant. You need part (c), the shared value at one point, to pin down equality.
A common error is applying the error estimate formula with B/2 but forgetting that B must be an upper bound on |f''(x)| over the entire interval, not just at the centre point a.
Some students confuse the Uniqueness Theorem with the existence theorems (Rolle's, MVT). The earlier theorems guarantee that something exists; the Uniqueness Theorem guarantees that something is unique.
The proof of part (a) uses proof by contradiction. Students sometimes struggle to articulate why f being non-constant leads to a contradiction with f'(x) = 0.
The three-part structure of the Uniqueness Theorem is a favourite for "state and prove" exam questions. Know how each part depends on the one before it.
Expect problems asking you to show that two antiderivatives of the same function differ by a constant (a direct application of part b).
The linear approximation error estimate may appear in problems that ask you to bound the error when using a tangent-line approximation for a specific function.
The ballistic equation application connects the Uniqueness Theorem to earlier material from Section 2.7. Be prepared to explain why uniqueness matters physically.
Fill in the blank: If f'(x) = 0 for all x in (a, b), then f(x) = ___.
True or False: If f'(x) = g'(x) for all x in (a, b), then f(x) = g(x). ___
True or False: The Uniqueness Theorem part (c) requires both a shared derivative and a shared value at one point. ___
Fill in the blank: The linear approximation error is bounded by (B/2)|x - a|^___, where B bounds |f''(x)|.
True or False: The ballistic equation is the only function with derivative v_0 - gt and initial height s_0. ___
Answers: 1. A constant C. 2. False (they may differ by a constant). 3. True. 4. 2 (squared). 5. True (by the Uniqueness Theorem, part c).
Q: Suppose f'(x) = 3x^2 for all x, and f(0) = 5. What is f(x)?
A: Any antiderivative of 3x^2 has the form x^3 + C. The condition f(0) = 5 gives C = 5. By the Uniqueness Theorem (part c), f(x) = x^3 + 5 is the only solution.
Q: Explain in your own words why part (b) of the Uniqueness Theorem follows from part (a).
A: If f'(x) = g'(x), then the function h(x) = f(x) - g(x) has h'(x) = 0 everywhere. Part (a) says h must be constant, so f(x) - g(x) = C, meaning f(x) = g(x) + C.
Q: Use the error estimate to bound |f(x) - L_0(x)| for f(x) = cos(x) on the interval (-0.5, 0.5).
A: f''(x) = -cos(x), so |f''(x)| <= 1 on (-0.5, 0.5). Taking B = 1, the error is at most (1/2)|x - 0|^2 = x^2/2. At x = 0.5, the bound is 0.125.
Q: Why does the source describe the Uniqueness Theorem as "the most important result" of Section 3.2?
A: Because it gives the mathematical foundation for physical laws. Without it, a differential equation with an initial condition could have multiple solutions, and the theory would fail to make definite predictions. The Uniqueness Theorem guarantees that one initial condition plus one derivative rule produces exactly one function.
The Uniqueness Theorem completes the chain that began with Rolle's Theorem and ran through the MVT. In later chapters, when the course introduces antiderivatives and the Fundamental Theorem of Calculus, the "+C" in indefinite integrals is a direct consequence of part (b): any two antiderivatives differ by a constant. The error estimate connects to Taylor polynomials (Chapter 11 in Stewart), where higher-order approximations yield tighter bounds using higher derivatives.
Uniqueness Theorem, zero derivative, constant function, antiderivative plus C, same derivative means differ by constant, initial condition, initial value problem, ballistic equation, projectile motion, physical law, determinism, linear approximation, tangent line approximation, error bound, second derivative bound, L_a(x), Stewart 3.2, Math 132, Calculus I, MSU, quantum mechanics footnote