Problem 2 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

For what value of \(a\), the mean rate of change of the function \(f(x)=x^{2}\) in the interval \([a, a+1]\) is 2 ?

Step-by-Step Solution

Answer

The value of 'a' for which the mean rate of change of the function \(f(x) = x^2\) in the interval \([a, a+1]\) is 2 is \(a = \frac{1}{2}\).

Step 1: Write down the formula for the mean rate of change

The mean rate of change (MROC) of a function over an interval is given by the formula: \(MROC = \frac{f(b) - f(a)}{b - a}\) In our case, the function is \(f(x) = x^2\), and the interval is \([a, a+1]\), so the formula becomes: \(MROC = \frac{f(a+1) - f(a)}{(a+1) - a}\)

Step 2: Substitute the given MROC and function into the formula

We are given that the MROC for the function in the interval is 2. So, we can substitute this value in our formula: \(2 = \frac{f(a+1) - f(a)}{(a+1) - a}\) Next, we substitute the function \(f(x) = x^2\): \(2 = \frac{(a+1)^2 - a^2}{(a+1) - a}\)

Step 3: Simplify and solve for 'a'

Now, let's simplify the equation: \(2 = \frac{(a^2 + 2a + 1) - a^2}{1}\) \(2 = 2a + 1\) Subtracting 1 from both sides: \(1 = 2a\) Dividing by 2: \(a = \frac{1}{2}\) So, the value of 'a' for which the mean rate of change of the function \(f(x) = x^2\) in the interval \([a, a+1]\) is 2 is \(a = \frac{1}{2}\).

Key Concepts

Differential Calculus

Differential calculus focuses on understanding how functions change and how they behave at specific moments. One of its main tools is the concept of the derivative, which provides a way to measure the rate at which a function is changing at any point. The mean rate of change is a basic application of these ideas. It gives an average rate of how the function changes between two points. Here is how it connects with the derivative:

• The mean rate of change over an interval \([a, b]\) is given by the formula: \(\frac{f(b) - f(a)}{b - a}\), which is essentially the slope of the secant line connecting \(f(a)\) and \(f(b)\).
• The instantaneous rate of change at a point, however, is given by the derivative, often noted as \(f'(x)\).

Understanding this distinction helps in tackling problems related to how functions behave over an interval versus at a single point.
The exercise leverages these principles by asking for the average change, making it an approachable entry point to the rich field of differential calculus.

Quadratic Functions

Quadratic functions are a fundamental type of polynomial function in which the highest degree of the variable is 2. This form is typically expressed as \(f(x) = ax^2 + bx + c\). In the exercise above, the function given is \(f(x) = x^2\), which is a simple quadratic with:

• No linear term (\(b = 0\)).
• A constant term of 0 (\(c = 0\)).
• The coefficient \(a = 1\) for the quadratic term.

Quadratic functions like \(x^2\) open upwards, forming a U-shaped graph known as a parabola. Analyzing these functions involves examining their symmetry around the y-axis and the vertex, which is the highest or lowest point of the function.
In the exercise, the parabola's shape helps determine how quickly the function values change over specific intervals. This is why understanding the graph of quadratic functions is crucial when exploring their mean rate of change.

Interval Analysis

Interval analysis in calculus involves examining function behavior over specific segments on the x-axis, referred to as intervals. This helps identify how the function's output (or y-value) changes throughout those sections. In this exercise, we focus on the interval \([a, a+1]\). Key points to consider include:

• Functions can increase, decrease, or remain constant over intervals.
• The length of the interval affects the average rate of change.
• Choosing different intervals can alter our insights into the function's behavior.

By exploring interval analysis, students learn to pinpoint where changes in the function occur more rapidly or slowly.
In the given problem, we explore the mean rate of change specifically over the interval \([a, a+1]\). This teaches how intervals contribute to understanding function dynamics comprehensively across specified domains.