![Q.No. 25 A variable Y is a function of t. Given that Y(t = 0) = 1 and Y(t = 1) = 2, dY/dt in the interval t = [0, 1] can be approximated as __________.](https://www.letstalkacademy.com/wp-content/uploads/2026/01/slide_25-11.png)
Q.No. 25 A variable Y is a function of t. Given that Y(t = 0) = 1 and Y(t = 1) = 2, dY/dt in the interval t = [0, 1] can be approximated as __________.
Y(t=0)=1 and Y(t=1)=2 indicate a net change of 1 over the interval Δt=1. The forward difference approximation for dY/dt is ΔY/Δt = (2-1)/(1-0) = 1.
Solution Explanation
The derivative dY/dt represents the instantaneous rate of change, but with only endpoint values given, finite difference methods provide the standard approximation. The simplest forward difference formula uses the secant slope: dY/dt ≈ [Y(1) – Y(0)] / (1 – 0) = 1. This assumes constant rate over, valid for linear functions and a first-order estimate for others via mean value theorem.
Common Approximations
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Forward difference: [Y(t+Δt) – Y(t)]/Δt at t=0 gives (2-1)/1 = 1.
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Backward difference: Same values yield [Y(1) – Y(0)]/(1-0) = 1 at t=1.
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Central difference: Requires midpoint Y(0.5), unavailable here, so not applicable.
The approximating dY/dt in interval problem arises frequently in competitive exams like IIT JAM, where Y(t=0)=1 and Y(t=1)=2. This tests finite difference approximation for derivatives without the full function.
Step-by-Step Derivation
Start with the definition: dY/dt ≈ ΔY/Δt for small Δt, but here Δt=1. Compute change: ΔY = 2 – 1 = 1, so dY/dt ≈ 1/1 = 1. By mean value theorem, a true derivative exists equaling this average over.
Why Fill-in-the-Blank is 1
Exams expect the basic forward/backward difference: (Y(1)-Y(0))/(1-0)=1. Higher-order methods need more points, unavailable here.
Exam Tips
Practice Taylor expansions for error analysis: actual dY/dt = average + O(Δt) term. For IIT JAM, recognize this as 1 directly.
