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Magazine Chapter 5: Problem 1
If \(v(x)=d f / d x,\) what constants \(C\) make \(1-10\) true? $$ \int_{2}^{b} v(x) d x=f(b)+C $$
Short Answer
Expert verified
The constant \(C\) is \(-f(2)\).
Step by step solution
01
Interpret the Problem
We are given that \(v(x) = \frac{df}{dx}\), which indicates that \(v(x)\) is the derivative of the function \(f(x)\). The problem asks us to find the constants \(C\) such that the integral \(\int_{2}^{b} v(x) \, dx = f(b) + C\) holds.
02
Apply the Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus tells us that if \(v(x) = \frac{df}{dx}\), then \(\int_{a}^{b} v(x) \, dx = f(b) - f(a)\) for any function \(f\) differentiable on \([a, b]\). In this case, \(a = 2\). Hence, \(\int_{2}^{b} v(x) \, dx = f(b) - f(2)\).
03
Relating the Integral to the Expression
We can now directly relate the integral with the given expression: \[ f(b) - f(2) = f(b) + C. \]By matching terms, we find that \[ -f(2) = C. \]Thus, \(C = -f(2)\).
04
Conclusion
The constant \(C\) that makes the statement true is \(C = -f(2)\). This ensures that the left side of the equation, which represents the definite integral from 2 to \(b\), equals the right side, \(f(b) + C\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integral
The concept of a definite integral is crucial in calculus, as it quantifies the accumulation of quantities over a certain interval. Imagine you have a graph of a function, and you want to find the area under this curve between two points on the x-axis, noted as \(a\) and \(b\). The definite integral \(\int_{a}^{b} v(x) \, dx\) computes this area.
Key aspects of definite integrals include:
Key aspects of definite integrals include:
- The limits of integration, \(a\) and \(b\), denote the starting and ending points on the x-axis.
- The result of a definite integral is a number, offering the net area under the curve, accounting for any areas where the function might dip below the x-axis and be negative.
- The process of integration "undoes" differentiation; thus, integrating a derivative returns the original function within the interval \([a, b]\).
Derivative
A derivative is a core component of calculus that reflects the rate at which a quantity changes. It examines how a function's value changes as its input changes. Think of it like speed 鈥� how fast is something changing at any given moment?
For a function \(f(x)\), the derivative is noted as \(\frac{df}{dx}\) or \(f'(x)\). This derivative provides several insights:
For a function \(f(x)\), the derivative is noted as \(\frac{df}{dx}\) or \(f'(x)\). This derivative provides several insights:
- It tells you the slope of the tangent line to the graph of the function at any point \(x\).
- Positive values indicate that the function is increasing at \(x\), while negative values show it's decreasing.
- If the derivative is zero, you may have found a local maximum or minimum (where the function changes direction).
Constant of Integration
When integrating a function, especially its derivative, you will often encounter the constant of integration. This constant, usually denoted as \(C\), arises because the process of differentiation loses any constant terms.
Understanding \(C\) involves:
Understanding \(C\) involves:
- When you integrate a function's derivative, the result could be many different functions, all differing by a constant. This is because the derivative of any constant is zero.
- The constant \(C\) represents this undetermined fixed value that could change without affecting the derivative.
- To solve or evaluate these integrals fully, additional information or boundary conditions are typically needed (such as a known point through which the function passes).
