Advanced Applications of Derivatives for JEE
1. Rate of Change
Rate of change measures how one quantity changes with respect to another. If \( y = f(x) \), then the rate of change is \( f'(x) \).
Example: \( f(x) = 3x^2 \) gives \( f'(x) = 6x \).
2. Monotonicity
If \( f'(x) > 0 \), the function is increasing; if \( f'(x) < 0 \), it is decreasing.
Advanced Applications of Derivatives for JEE
1. Newton-Leibniz Formula
The Newton-Leibniz formula relates the integral of a function to its antiderivative. It states that if \( F(x) \) is the antiderivative of \( f(x) \), then:
\[ \int_a^b f(x)\,dx = F(b) - F(a) \]
Example:
Evaluate \( \int_1^3 2x \, dx \):
\[ \int_1^3 2x \, dx = x^2 \Big|_1^3 = 9 - 1 = 8 \]
2. Jansen's Inequality
Jensen's inequality applies to convex functions, stating that for any convex function \( f \) and real numbers \( x_1, x_2, \dots, x_n \), we have:
\[ f\left(\frac{x_1 + x_2 + \cdots + x_n}{n}\right) \leq \frac{f(x_1) + f(x_2) + \cdots + f(x_n)}{n} \]
Example:
For \( f(x) = x^2 \), prove that \( \left(\frac{x_1 + x_2}{2}\right)^2 \leq \frac{x_1^2 + x_2^2}{2} \).
3. Rolle's Theorem
Rolle's theorem states that if a function \( f(x) \) is continuous on \( [a, b] \), differentiable on \( (a, b) \), and \( f(a) = f(b) \), then there exists at least one \( c \in (a, b) \) such that:
\[ f'(c) = 0 \]
Example:
Apply Rolle’s theorem to \( f(x) = x^2 - 3x + 2 \) in \( [1, 2] \).
Solution: Since \( f(1) = f(2) = 0 \), by Rolle’s theorem, there exists a \( c \in (1, 2) \) such that \( f'(c) = 0 \). Find \( f'(x) = 2x - 3 \). Solving \( 2c - 3 = 0 \), we get \( c = 1.5 \).
4. Lagrange Mean Value Theorem (LMVT)
The Lagrange Mean Value Theorem generalizes Rolle’s theorem. If \( f(x) \) is continuous on \( [a, b] \) and differentiable on \( (a, b) \), then there exists \( c \in (a, b) \) such that:
\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]
Example:
Apply LMVT to \( f(x) = x^2 \) on \( [1, 3] \).
Solution: \( f'(c) = 2c \), and from LMVT, \( 2c = \frac{9 - 1}{3 - 1} = 4 \), giving \( c = 2 \).
5. Cauchy Mean Value Theorem (CMVT)
The Cauchy Mean Value Theorem is a generalization of LMVT. If \( f(x) \) and \( g(x) \) are continuous on \( [a, b] \) and differentiable on \( (a, b) \), then there exists a point \( c \in (a, b) \) such that:
\[ \frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)} \]
Example:
For \( f(x) = x^2 \) and \( g(x) = x^3 \), find \( c \) in the interval \( [1, 2] \) using CMVT.
Solution: From CMVT, we get \( \frac{2c}{3c^2} = \frac{4 - 1}{8 - 1} \Rightarrow \frac{2}{3c} = \frac{3}{7} \), giving \( c = 14/9 \).
6. Analysis of Cubic Functions
A cubic function has the general form \( f(x) = ax^3 + bx^2 + cx + d \). To analyze the cubic, find its critical points and points of inflection by solving \( f'(x) = 0 \) and \( f''(x) = 0 \).
Example:
Analyze the cubic function \( f(x) = x^3 - 3x^2 + 2x \).
Solution: First, find the derivative \( f'(x) = 3x^2 - 6x + 2 \), and solve \( 3x^2 - 6x + 2 = 0 \) to find the critical points. Then, check for concavity using \( f''(x) = 6x - 6 \).
7. Curve Sketching
Curve sketching involves finding the key features of a function: intercepts, critical points, asymptotes, and intervals of increase or decrease.
Example:
Sketch the curve of \( f(x) = \frac{1}{x} \).
Solution: Find the intercepts (none), asymptotes (\( x = 0 \), \( y = 0 \)), and check for increasing/decreasing behavior using the derivative \( f'(x) = -\frac{1}{x^2} \).
Practice Questions
- Prove Rolle’s theorem for \( f(x) = x^2 - 5x + 6 \) in \( [2, 3] \).
- Use LMVT for \( f(x) = \ln x \) in \( [1, e] \).
- Find the critical points and sketch the curve for \( f(x) = x^3 - 6x^2 + 9x \).
- Apply CMVT for \( f(x) = x^2 \) and \( g(x) = x^4 \) on \( [1, 2] \).
- Analyze the cubic function \( f(x) = 2x^3 - 3x^2 + x \).
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