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📝 Constant difference theorem calculus (22 MCQs)

📖 From Calculus • 5. The derivative in Graphing and Applications • 22 questions available

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What is Constant difference theorem calculus?

Definition:
The Constant Difference Theorem states if $f'(x) = g'(x)$ for all $x$ in an interval, then $f(x) = g(x) + C$ for some constant $C$ . This implies functions with identical derivatives differ only by a vertical shift, fundamental to integration theory.

Example:
If $f'(x) = 2x$ and $g'(x) = 2x$ , then $f(x) = x^2 + 3$ and $g(x) = x^2 - 5$ differ by $C = 8$ , confirming $f(x) = g(x) + 8$ .

Reason:
It justifies the $+C$ in indefinite integrals, showing that antiderivatives form a family of parallel curves shifted vertically by constants.

Easy

Medium

Hard

📝 All Constant difference theorem calculus MCQs

Q1. What does the Constant Difference Theorem state about two differentiable functions f and g on an interval when their derivatives are equal?

A.f and g are identical functions

B.f(x) - g(x) is a constant ✅

C.Both functions are constant

D.Their second derivatives are equal

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The theorem asserts that if f′(x)=g′(x) for every x in the interval, then the difference f(x)-g(x) does not change with x; it is a fixed number k. This directly yields the statement that the difference is constant, which is option B.

Q2. According to the geometric interpretation of the Constant Difference Theorem, how are the graphs of f and g related?

A.They intersect at a single point

B.They are reflections across the x‑axis

C.One is a vertical translation of the other ✅

D.They have the same slope at every point

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: When two functions have identical derivatives on an interval, their graphs differ only by a vertical shift. That means one graph can be obtained by moving the other up or down by a constant amount, which matches option C.

Q3. If f′(x)=g′(x) for all x in [a,b] and f(a)=g(a)+5, what must be true about f(b)−g(b)?

A.It equals 5

B.It equals 0 ✅

C.It equals f(a)−g(a)

D.Cannot be determined

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Because the difference f−g is constant on the whole interval, its value at any point equals its value at any other point. Since at x=a the difference is 5, the same value holds at x=b, giving f(b)−g(b)=5.

Q4. Suppose h(x)=f(x)−g(x) is constant on (c,d). Which of the following statements is a direct consequence?

A.h′(x)=0 for all x in (c,d)

B.f′(x)=g′(x) only at the endpoints

C.f and g have the same maximum value

D.h is constant on (c,d) ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: If h is constant, its derivative is zero everywhere in the interval, which is the fundamental property of constant functions. This is captured by option D, stating that h itself is constant.

Q5. Given that f′(x)=g′(x) on (p,q) and that f(p)=g(p)+k, which reasoning correctly explains why f(q)=g(q)+k?

A.By the Mean Value Theorem applied to f−g

B.Because the integral of the derivative over [p,q] is zero ✅

C.Since constant functions have zero derivative

D.Because f and g are linear

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Since f′−g′=0, integrating this zero derivative from p to q gives ∫ₚ^q0 dt=0, which implies f(q)−g(q)=f(p)−g(p)=k. Thus the constant difference is preserved, matching option B.

Q6. Assume f and g satisfy the hypotheses of the Constant Difference Theorem on an interval I. If a point x₀∈I satisfies f(x₀)=g(x₀), what can be inferred about the constant k in f(x)=g(x)+k?

A.k must be zero

B.k equals f′(x₀)−g′(x₀)

C.k equals 0 because f(x₀)=g(x₀) implies constant zero ✅

D.k equals the average of f and g on I

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Because the theorem tells us f(x)=g(x)+k for every x, plugging in x₀ gives k=f(x₀)−g(x₀). If those values are equal, the difference is zero, so k=0, which is option C.

Q7. Let f and g be differentiable on [m,n] with f′(x)=g′(x) for all x. If the area between the graphs of f and g over [m,n] is zero, what does this imply about the constant k?

A.k=0 ✅

B.k equals the length of the interval

C.k is positive

D.No implication

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Zero area means the vertical distance between the curves is zero everywhere, which can only happen when the constant difference is zero. Hence k must be 0, corresponding to option A.

Q8. If f′(x)=g′(x) on an interval J and f−g attains its maximum at an interior point of J, what must be true about the constant difference?

A.The constant is the maximum value

B.The constant must be zero

C.The maximum cannot occur interiorly unless the constant is zero ✅

D.The derivative of f−g is non‑zero there

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: A constant function cannot have a genuine interior maximum unless the constant itself is zero, because any non‑zero constant would be the same everywhere, contradicting the notion of a distinct maximum point. Thus option C is correct.

Q9. Consider two functions where f′(x)=g′(x) for all x∈[α,β]. If a student incorrectly concludes that f(x)=g(x) for all x, which logical flaw does this represent?

A.Confusing equality of derivatives with equality of functions ✅

B.Assuming continuity without proof

C.Misapplying Rolle’s Theorem

D.Ignoring the constant of integration

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The mistake lies in assuming that identical derivatives force the functions themselves to be identical, overlooking the possibility of a non‑zero constant difference. This confusion between derivative equality and function equality is captured by option A.

Q10. Which of the following pairs of functions satisfy the Constant Difference Theorem on ℝ?

A.f(x)=x^2, g(x)=x^2+3

B.f(x)=e^x, g(x)=e^x+5 ✅

C.f(x)=\sin x, g(x)=\cos x

D.f(x)=\ln x, g(x)=\ln x+1

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Both functions in the pair have identical derivatives (e^x for each), and their difference is the constant 5, satisfying the theorem. The pair in option B therefore fulfills the hypothesis and conclusion.

Q11. Given f(x)=\ln(x)+7 and g(x)=\ln(x), which statement follows from the Constant Difference Theorem on (0,∞)?

A.f′(x)=g′(x) and f−g=7 ✅

B.f′(x)≠g′(x)

C.f−g is not constant

D.g is the derivative of f

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Both functions share the derivative 1/x, and their difference is the constant 7 for every x>0, exactly what the theorem predicts. Hence option A correctly reflects the conclusion.

Q12. Two functions satisfy f′(x)=g′(x) on (a,b). Which of the following statements correctly distinguishes the Constant Difference Theorem from the Mean Value Theorem?

A.The Constant Difference Theorem guarantees a constant difference; the MVT guarantees a point where the derivative equals the average slope.

B.Both theorems give the same conclusion about f−g.

C.The MVT requires f and g to be equal at the endpoints.

D.The Constant Difference Theorem is a special case of Rolle’s Theorem applied to f−g. ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: While both theorems involve derivatives, the Constant Difference Theorem is indeed a direct consequence of Rolle’s Theorem applied to the difference f−g, making option D the accurate distinction.

Q13. If h(x)=f(x)−g(x) is known to be constant on [s,t] and h′(x)=0 for all x, which analytical method confirms that f and g have identical antiderivatives up to a constant?

A.Integration by parts

B.Fundamental Theorem of Calculus ✅

C.Differentiation of a constant function

D.Partial fractions

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The Fundamental Theorem of Calculus links the derivative of an antiderivative to the original function, confirming that if the derivative of the difference is zero, the original functions differ only by a constant. This reasoning aligns with option B.

Q14. Suppose f and g are twice differentiable on an interval and satisfy f′(x)=g′(x). Which statement about their second derivatives is necessarily true?

A.f″(x)=g″(x) for all x

B.f″(x)−g″(x) is constant

C.No conclusion can be drawn about second derivatives ✅

D.Both second derivatives must be zero

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Equality of first derivatives does not impose any condition on second derivatives; they may differ arbitrarily while still having the same first derivative. Therefore no definite statement about f″ and g″ can be made, making option C correct.

Q15. Let f(x)=x^3+2x and g(x)=x^3+2x+5. Using the Constant Difference Theorem, determine the set of points where the tangent lines to f and g are parallel.

A.All real numbers ✅

B.No points

C.Only at x=0

D.Only where x=−1

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Since f′(x)=3x^2+2 and g′(x)=3x^2+2, the slopes of their tangents are identical for every x. Hence the tangent lines are parallel at all points on the real line, which corresponds to option A.

Q16. Given functions p and q with p′(x)=q′(x) on (−∞,∞) and p(0)=q(0)+k. If r(x)=p(x)−q(x), which of the following best describes the graph of r?

A.A line with slope k

B.A horizontal line y=k ✅

C.A parabola opening upward

D.An exponential curve

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Because p−q is constant (equal to k) everywhere, its graph is a horizontal line at height k. This matches option B.

Q17. How can the Constant Difference Theorem be used to prove the identity \ $\\sin^{-1}x + \\cos^{-1}x = \\frac{\\pi}{2}\$ for \ $-1\\le x\\le 1\$ ?

A.By showing the derivative of the left side is zero

B.By integrating both sides ✅

C.By applying the Pythagorean identity

D.By differentiating the right side

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Define f(x)=\\sin^{-1}x+\\cos^{-1}x. Differentiating yields f′(x)=1/\\sqrt{1-x^2}-1/\\sqrt{1-x^2}=0, so f is constant. Evaluating at x=0 gives f(0)=\\pi/2, establishing the identity. This reasoning is captured by option B.

Q18. In a physics context, if two position functions s₁(t) and s₂(t) have identical velocity functions on an interval, what does the Constant Difference Theorem tell us about their relative motion?

A.Their displacement differs by a constant, meaning they move together as a rigid body. ✅

B.Their accelerations are equal.

C.They must intersect at the interval’s midpoint.

D.Their velocities are zero.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Equal velocities imply equal derivatives of the position functions, so the difference s₁−s₂ has zero derivative and is therefore a constant. This constant separation means the bodies maintain a fixed offset, as described in option A.

Q19. Consider the family of functions \ $F_k(x)=\\ln(x)+k\$ . Which principle of the Constant Difference Theorem explains why all members of this family have the same derivative?

A.If derivatives are equal, the functions differ by a constant. ✅

B.Derivatives of logarithmic functions are always constant.

C.The theorem does not apply to logarithms.

D.All functions with the same derivative are identical.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Each member differs from another by the constant k, and their derivatives are identical (1/x). The theorem states that when two functions share a derivative, they must differ by a constant, which is exactly the situation here—option A.

Q20. When solving differential equations, why is the Constant Difference Theorem important for determining the general solution of a first‑order linear ODE?

A.It justifies adding an arbitrary constant after integration. ✅

B.It allows us to ignore the integrating factor.

C.It shows that all solutions are identical.

D.It provides the particular solution directly.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: After integrating the ODE, one obtains an antiderivative plus a constant. The Constant Difference Theorem guarantees that any two antiderivatives of the same derivative differ only by a constant, legitimizing the addition of the arbitrary constant in the general solution.

Q21. If two curves are vertical translations of each other on an interval, which of the following must be true about their curvature functions on that interval?

A.The curvature values are identical at corresponding points. ✅

B.The curvature of one is the negative of the other.

C.Curvature cannot be compared without additional information.

D.One curvature is constant while the other varies.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Vertical translation does not affect the shape of a curve, only its position. Curvature depends solely on the shape, so corresponding points on the two curves have the same curvature. This is precisely option A.

Q22. Let f and g be differentiable on (0,π) with f′(x)=g′(x) and f(π/4)=g(π/4)+2. Using the Constant Difference Theorem, which of the following integrals correctly expresses the constant difference?

A.\

\\int_{\\pi/4}^{x} [f′(t)-g′(t)] dt = 2\

B.\

\\int_{\\pi/4}^{x} f′(t) dt = \\int_{\\pi/4}^{x} g′(t) dt + 2\

✅

C.\

\\int_{\\pi/4}^{x} (f′(t)-g′(t)) dt = 0\

D.\

\\int_{0}^{\\pi} (f′(t)-g′(t)) dt = 2\

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Since f′−g′=0, integrating from π/4 to x gives \ $\\int_{\\pi/4}^{x}f′(t)dt = \\int_{\\pi/4}^{x}g′(t)dt\$ . Adding the known constant 2 to the right side yields the equality in option B, which correctly reflects the constant difference.

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