Problem 65 - Calculus Early Transcendentals

Question

Fill in the following table with either even or odd, and prove each result. Assume \(n\) is a nonnegative integer and \(f^{n}\) means the \(n\) th power of \(f\). $$\begin{array}{cccc} & f \text { is even } & f \text { is odd } \\\\\hline n \text { is even } & f^{n}\text { is } \underline{\quad}\underline{\quad} & f^{n} \text { is } \underline{\quad} \underline{\quad} \\\n \text { is odd } & f^{n} \text { is } \underline{\quad} \underline{\quad} & f^{n} \text { is } \underline{\quad} \underline{\quad} \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

Based on the analysis and solution provided, we can summarize the results as follows: 1. If \(f\) is even and \(n\) is even, then \(f^n\) is even. 2. If \(f\) is even and \(n\) is odd, then \(f^n\) is even. 3. If \(f\) is odd and \(n\) is even, then \(f^n\) is odd. 4. If \(f\) is odd and \(n\) is odd, then \(f^n\) is odd.

Step 1: Case 1: \(f\) is even and \(n\) is even

If \(f\) is even, that means \(f = 2a\) for some integer \(a\). If \(n\) is also even, then \(n = 2b\) for some integer \(b\). Now, let's compute \(f^n = (2a)^{2b}\). Since \(2b\) is even, we know that any number raised to an even power will be a square. Thus, \((2a)^{2b} = (2^2a^2)^{b} = 2^{2b}a^{2b} = 2c\) where \(c = 2^bah^b\) (\(c\) is an integer). We can see that \(f^n\) is an even number.

Step 2: Case 2: \(f\) is even and \(n\) is odd

If \(f\) is even, that means \(f = 2a\) for some integer \(a\). If \(n\) is odd, then \(n = 2b+1\) for some integer \(b\). Now, let's compute \(f^n = (2a)^{2b+1}\). Since \(2b+1\) is odd, we can rewrite the expression as \((2a)^{2b} (2a) = 2^{2b} a^{2b} (2a) = 2c (2a)\) where \(c = 2^ba^b\) (\(c\) is an integer). We can see that \(f^n\) is an even number.

Step 3: Case 3: \(f\) is odd and \(n\) is even

If \(f\) is odd, that means \(f = 2a+1\) for some integer \(a\). If \(n\) is even, then \(n = 2b\) for some integer \(b\). Now, let's compute \(f^n = (2a+1)^{2b}\). Since \(n\) is even, we know that any number raised to an even power will be a square. Using the binomial theorem, the expanded form of \((2a+1)^{2b}\) is a sum of terms. Every term in the sum, except for the last one, has an even power of \(2a\), which makes each term even. The last term \((1)^{2b}\) will be odd, but the sum of all terms will still be odd. Thus, we can conclude that \(f^n\) is odd.

Step 4: Case 4: \(f\) is odd and \(n\) is odd

If \(f\) is odd, that means \(f = 2a+1\) for some integer \(a\). If \(n\) is odd, then \(n = 2b+1\) for some integer \(b\). Now, let's compute \(f^n = (2a+1)^{2b+1}\). Since \(n\) is odd, we know that any odd number raised to an odd power will be odd. Using the binomial theorem, the expanded form of \((2a+1)^{2b+1}\) is a sum of terms. Every term in the sum, except for the last one, has an even power of \(2a\), which makes each term even. The last term \((1)^{2b+1}\) will be odd, and since the sum of even and odd numbers is odd, we can conclude that \(f^n\) is odd. The final table looks like this: $$\begin{array}{cccc} & f \text { is even } & f \text { is odd } \\\\\hline n \text { is even } & f^{n}\text { is } \underline{\text {even}} & f^{n} \text { is } \underline{\text {odd}} \\\n \text { is odd } & f^{n} \text { is } \underline{\text {even}} & f^{n} \text { is } \underline{\text {odd}} \\\\\hline\end{array}$$

Key Concepts

Even functions

Even functions are those peculiar mathematical entities that exhibit symmetry about the y-axis on a standard Cartesian plane. In simpler terms, if you fold the graph along the y-axis, both halves should align perfectly. Mathematically speaking, a function \( f(x) \) is considered "even" if it satisfies the condition:\[ f(-x) = f(x) \]This means that for every value of \( x \), the output value at \( -x \) is the same.

• An illustrative example of an even function is \( f(x) = x^2 \), where squaring any real number yields the same result whether the input is positive or negative.
• Another common example is the cosine function, \( f(x) = \cos(x) \), which repeats its values symmetrically around the y-axis.

Understanding whether a function is even can be incredibly useful, especially when predicting behavior in equations and graphs. It also simplifies integration and differentiation problems by allowing certain symmetries to be leveraged. To check if a function is even, examine its algebraic form or its graph for y-axis symmetry.

Odd functions

Odd functions are defined by a different kind of symmetry compared to even functions. Instead of symmetry around an axis, odd functions have a reflective symmetry about the origin. This means that if you rotate the graph 180 degrees around the origin, it maps onto itself. Mathematically, a function \( f(x) \) is called "odd" if it satisfies:\[ f(-x) = -f(x) \]In practice, this property ensures that for every input value, the function's value at the opposite input (negative) is the negative of the corresponding output value.

• For example, the function \( f(x) = x^3 \) is odd because \((-x)^3 = -x^3\), satisfying the condition across the domain.
• Another example is the sine function, \( f(x) = \sin(x) \), known for its reflective symmetry about the origin.

Recognizing odd functions is significant in mathematics because they exhibit specific mirrored behaviors. Similar to even functions, odd functions are helpful in calculus, particularly with definite integrals over symmetric intervals, often allowing conclusions about their sum.

Binomial theorem

The binomial theorem is a powerful algebraic principle that expands expressions of powers of binomials (expressions like \((a+b)^n\)) into a sum of terms. Each term in this expansion includes binomial coefficients that can be determined using Pascal’s Triangle or calculated directly using:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]Here, \(n!\) (n factorial) is the product of all positive integers up to \(n\), and \(k\) is the term’s position within the expansion. This theorem is expressed as:\[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k \]Each term in the sum is composed of powers of \(a\) and \(b\), evolving as you move through each term. It's an essential tool for quick computation and is widely used in calculus, algebra, and beyond.

• For instance, if you want to expand \((x+1)^3\): \[ (x+1)^3 = \binom{3}{0}x^3 + \binom{3}{1}x^2\cdot1 + \binom{3}{2}x\cdot1^2 + \binom{3}{3}1^3 = x^3 + 3x^2 + 3x + 1 \]
• The binomial theorem helps simplify calculations, such as multiplying large numbers, and is instrumental in probabilities and statistics.

Learning the binomial theorem allows you to understand the relationships between coefficients and terms in an expansion, offering a systematic way to handle complex algebraic expressions.