Master Instantaneous Rate of Change for CBSE Class 11 Applied Mathematics. Learn to calculate derivatives as slopes and rates with solved examples.
The instantaneous rate of change of a function f(x) at a specific point x = a represents the slope of the tangent line to the curve at that point. It is defined as the limit of the average rate of change as the interval approaches zero. This concept is fundamental in physics and economics to describe velocity or marginal cost.
Step 1: Identify f(x) = x² and a = 3.
Step 2: Apply the limit definition: f'(3) = lim(h→0) [(3+h)² - 3²] / h.
Step 3: Expand: lim(h→0) [9 + 6h + h² - 9] / h = lim(h→0) [6h + h²] / h.
Step 4: Simplify: lim(h→0) (6 + h) = 6.
Answer: 6
Instead of evaluating at a single point, we can define the derivative f'(x) as a new function that gives the instantaneous rate of change for any x in the domain. This allows us to determine the slope of the tangent at any point on the curve. For power functions, we use the power rule to simplify calculations.
Step 1: Identify n = 3 and the constant 4.
Step 2: Apply the power rule: f'(x) = 4 · (3x³⁻¹).
Step 3: Multiply constants: 4 · 3 = 12.
Answer: 12x²
In economics, the instantaneous rate of change of the total cost function C(x) with respect to the quantity x is known as the marginal cost. It represents the approximate cost of producing one additional unit. Similarly, marginal revenue is the derivative of the total revenue function.
Step 1: Given C(x) = 5x² + 10x + 500.
Step 2: Differentiate with respect to x: d/dx(5x²) + d/dx(10x) + d/dx(500).
Step 3: Apply rules: 10x + 10 + 0.
Answer: 10x + 10