An even function is a mathematical function where the value of the function at a point is equal to the value of the function at the negative of that point. In other words, the graph of an even function is symmetric about the y-axis.
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For an even function $f(x)$, $f(-x) = f(x)$, meaning the function values are the same for positive and negative inputs.
The graph of an even function is symmetric about the y-axis, meaning the left and right halves of the graph are mirror images of each other.
When integrating an even function over a symmetric interval $[a, b]$, where $a = -b$, the definite integral simplifies to $2 \int_0^b f(x) dx$.
Many common functions, such as $x^2$, $\cos(x)$, and $\cosh(x)$, are examples of even functions.
Identifying whether a function is even, odd, or neither is an important step in applying integration formulas and theorems.
Review Questions
Explain how the property of evenness affects the definite integral of an even function over a symmetric interval.
For an even function $f(x)$ integrated over a symmetric interval $[a, b]$ where $a = -b$, the definite integral simplifies to $2 \int_0^b f(x) dx$. This is because the values of the function are the same for positive and negative inputs, allowing us to combine the integrals over the positive and negative halves of the interval. This property can be leveraged to simplify the calculation of definite integrals involving even functions.
Describe how the concept of evenness is related to the Net Change Theorem in the context of integration.
The Net Change Theorem states that the net change in a function over an interval is equal to the definite integral of the function over that interval. For an even function, the net change over a symmetric interval $[a, b]$ where $a = -b$ is zero, as the positive and negative changes cancel each other out due to the symmetry of the function. This relationship between evenness and the Net Change Theorem is an important consideration when applying integration techniques and theorems.
Analyze how the property of evenness can be used to simplify the integration of functions in calculus problems.
The property of evenness can be leveraged to simplify the integration of functions in several ways. First, for an even function integrated over a symmetric interval, the definite integral can be reduced to $2 \int_0^b f(x) dx$, as discussed earlier. Additionally, recognizing that a function is even can allow for the use of specialized integration formulas and techniques, such as the use of trigonometric identities or the substitution method. Identifying the evenness of a function is a crucial step in efficiently solving integration problems in calculus.
A function is considered odd if the value of the function at a point is the negative of the value of the function at the negative of that point. The graph of an odd function is symmetric about the origin.