Applications of Differential Calculus — Flashcards
Tamil Nadu Board · Class 12 · Mathematics
25 flashcards for Applications of Differential Calculus (Tamil Nadu Board Class 12 Mathematics) to test yourself on key terms and facts.
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Find the instantaneous rate of change of f(x) = 3x² + 2x at x = 1 using the derivative.
Answer
Step 1: Find the derivative: f'(x) = 6x + 2 Step 2: Evaluate at x = 1: f'(1) = 6(1) + 2 = 8 Step 3: The instantaneous rate of change at x = 1 is 8 units. Explanation: The derivative gives the slope of…
A particle's position is given by s(t) = t³ - 6t² + 9t + 1. Find when the particle is at rest.
Answer
Step 1: Velocity v(t) = ds/dt = 3t² - 12t + 9 Step 2: Set v(t) = 0: 3t² - 12t + 9 = 0 Step 3: Factor: 3(t² - 4t + 3) = 0 → 3(t - 1)(t - 3) = 0 Step 4: t = 1 or t = 3 Answer: The particle is at rest at…
When do you use the related rates method? Provide a context example.
Answer
Use related rates when: - Two or more quantities are changing with respect to time - You know the rate of change of one quantity and need to find the rate of change of another - The quantities are rel…
Find the equation of the tangent line to y = x² - 3x + 2 at the point (1, 0).
Answer
Step 1: Find the derivative: dy/dx = 2x - 3 Step 2: Find the slope at x = 1: m = 2(1) - 3 = -1 Step 3: Use point-slope form: y - y₁ = m(x - x₁) Step 4: Substitute (1, 0) and m = -1: y - 0 = -1(x - 1) …
Find the equation of the normal line to the curve y = x³ - 2x at x = 1.
Answer
Step 1: Find dy/dx = 3x² - 2 Step 2: Find the slope of tangent at x = 1: m_tangent = 3(1)² - 2 = 1 Step 3: Slope of normal is perpendicular: m_normal = -1/m_tangent = -1/1 = -1 Step 4: Find y-coordina…
Find the angle between the curves y = x² and y = (x - 3)² at their point of intersection.
Answer
Step 1: Find intersection points: x² = (x - 3)² → x² = x² - 6x + 9 → x = 3/2 Step 2: At x = 3/2: y = (3/2)² = 9/4, so point is (3/2, 9/4) Step 3: Find slope of first curve: dy/dx = 2x → m₁ = 2(3/2) = …
Use Rolle's Theorem to verify that f(x) = x² - 4x has a critical point in (0, 4).
Answer
Rolle's Theorem conditions: Step 1: Check if f is continuous on [0, 4] → Yes, polynomial Step 2: Check if f is differentiable on (0, 4) → Yes, polynomial Step 3: Check if f(0) = f(4): - f(0) = 0² - 4(…
Use the Mean Value Theorem to find c for f(x) = x² on [1, 3].
Answer
Mean Value Theorem: ∃c ∈ (a, b): f'(c) = [f(b) - f(a)]/(b - a) Step 1: Calculate f(3) - f(1) = 9 - 1 = 8 Step 2: Calculate (b - a) = 3 - 1 = 2 Step 3: Average rate of change = 8/2 = 4 Step 4: Find f'(…
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