Let ƒ :R→R be a function defined by
\(f(x) = \frac{2e^{2x}}{e^{2x} + e^x}\)
Then \(f\left(\frac{1}{100}\right) + f\left(\frac{2}{100}\right) + f\left(\frac{3}{100}\right) + \ldots + f\left(\frac{99}{100}\right)\)
is equal to ________.
Let ƒ :R→R be a function defined by
\(f(x) = \frac{2e^{2x}}{e^{2x} + e^x}\)
Then \(f\left(\frac{1}{100}\right) + f\left(\frac{2}{100}\right) + f\left(\frac{3}{100}\right) + \ldots + f\left(\frac{99}{100}\right)\)
is equal to ________.
Correct Answer: 99
Solution and Explanation
The correct answer is 99
\(f(x) = \frac{2e^{2x}}{e^{2x} + e^x}\) and \(f(1-x) = \frac{2e^{2-2x}}{e^{2-2x} + e^{1-x}}\)
\(∴\)\( \frac{f(x)+f(1-x)}{2} = 1\)
i.e. f(x)+f(1-x) = 2
Therefore,
\(f(\frac{1}{100}) + f(\frac{2}{100})+...+f(\frac{99}{100})\)
= \(\sum_{x=1}^{49} f\left(\frac{x}{100}\right) + f\left(\frac{1 - \frac{x}{100}}{100}\right) + f\left(\frac{1}{2}\right)\)
\(= 49 \times 2+1 = 99\)
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Concepts Used:
Differential Equations
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
Orders of a Differential Equation
First Order Differential Equation
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
Second-Order Differential Equation
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations