Rewrite The Following Expression In Terms Of The Given Function

Let's embark on a journey to master the art of rewriting mathematical expressions in terms of specified functions. This is a fundamental skill in calculus, differential equations, linear algebra, and numerous other branches of mathematics and physics. The core idea revolves around manipulating a given expression to make it look like, or at least involve, the desired function, often revealing underlying relationships or simplifying complex problems.

Understanding the Foundation

Before we dive into examples, let's establish a clear understanding of what it means to "rewrite an expression in terms of a given function." Essentially, we aim to express the original mathematical entity—be it an equation, a formula, or even a set of data points—using the provided function as a building block. This typically involves algebraic manipulations, trigonometric identities, calculus techniques (like differentiation and integration), and sometimes even clever substitutions.

The "given function" acts as our lens. We want to reshape the expression so that the properties and characteristics of the given function become apparent and useful in further analysis or problem-solving.

Tools of the Trade: Techniques and Strategies

Rewriting expressions often requires a combination of techniques. Here's a rundown of some crucial methods:

• Algebraic Manipulation: This is the bedrock. Factoring, expanding, simplifying fractions, and rationalizing denominators are indispensable. Look for opportunities to rearrange terms to match the form of the target function.

• Trigonometric Identities: If the expression involves trigonometric functions, knowing your identities is paramount. Pythagorean identities (sin²θ + cos²θ = 1), double-angle formulas, sum-to-product formulas – these are your allies.

• Calculus Techniques: Differentiation and integration can be powerful tools. Sometimes, differentiating a portion of the expression reveals a derivative related to the target function. Conversely, integrating might introduce the function in a usable form.

• Substitution: Introducing new variables can often simplify a complex expression. Choose substitutions that align with the structure of the target function. This can involve trigonometric substitutions, exponential substitutions, or even more abstract variable changes.

• Partial Fraction Decomposition: When dealing with rational functions (ratios of polynomials), partial fraction decomposition allows you to break down the complex fraction into simpler components, which might be easier to express in terms of the desired function.

• Completing the Square: This technique is particularly useful when dealing with quadratic expressions. Completing the square can reveal hidden structures or allow you to express the quadratic in a form that involves a specific function (e.g., a squared term).

• Series Expansions: For certain functions (like exponential, trigonometric, and logarithmic functions), series expansions (Taylor or Maclaurin series) can provide alternative representations that might be more suitable for rewriting.

Examples: Putting Theory into Practice

Let's solidify our understanding with a series of examples. These examples progressively increase in complexity, demonstrating the various techniques we've discussed.

Example 1: Rewriting in Terms of Sine

Problem: Rewrite the expression cos²(x) in terms of sin(x).

Solution: This is a straightforward application of the Pythagorean identity. We know that:

sin²(x) + cos²(x) = 1

Therefore, we can directly solve for cos²(x):

cos²(x) = 1 - sin²(x)

Answer: cos²(x) = 1 - sin²(x)

Example 2: Rewriting in Terms of Exponential Function

Problem: Rewrite the expression cosh(x) in terms of e^x. Recall that cosh(x) = (e^x + e^-x) / 2.

Solution: The expression is already defined in terms of exponential functions! The goal here might be to simplify a larger expression where cosh(x) appears by substituting its exponential form.

Answer: cosh(x) = (e^x + e^-x) / 2

Example 3: Rewriting a Polynomial in Terms of Another Polynomial

Problem: Rewrite the polynomial p(x) = x^2 + 4x + 5 in terms of q(x) = x + 2.

Solution: The trick here is to realize that q(x)^2 = (x+2)^2 = x^2 + 4x + 4. We can use this to rewrite p(x):

p(x) = x^2 + 4x + 5 = (x^2 + 4x + 4) + 1 = (x + 2)^2 + 1 = q(x)^2 + 1

Answer: p(x) = q(x)^2 + 1

Example 4: Rewriting a Rational Function in Terms of Simpler Fractions

Problem: Rewrite the rational function (x + 1) / (x^2 - 1) in terms of simpler fractions using partial fraction decomposition.

Solution: First, factor the denominator: x^2 - 1 = (x - 1)(x + 1)

Now, we can decompose the rational function into:

(x + 1) / ((x - 1)(x + 1)) = A / (x - 1) + B / (x + 1)

However, we can simplify the left-hand side first:

(x + 1) / ((x - 1)(x + 1)) = 1 / (x - 1)

Therefore, the entire expression simplifies to one term.

Answer: 1 / (x - 1)

Example 5: Rewriting an Integral in Terms of a Specific Function

Problem: Rewrite the integral ∫x*e^(x^2) dx in terms of the function f(x) = e^(x^2).

Solution: This requires a u-substitution. Let u = x^2. Then du = 2x dx, or x dx = (1/2) du.

The integral becomes:

∫x*e^(x^2) dx = ∫e^u * (1/2) du = (1/2) ∫e^u du = (1/2)e^u + C = (1/2)e^(x^2) + C

Since f(x) = e^(x^2), we can write:

(1/2)e^(x^2) + C = (1/2)f(x) + C

Answer: ∫x*e^(x^2) dx = (1/2)f(x) + C where f(x) = e^(x^2) and C is the constant of integration.

Example 6: Rewriting a Differential Equation in Terms of a Transformed Variable

Problem: Consider the differential equation dy/dx + y = x. Rewrite this equation in terms of the new variable z = e^x * y.

Solution: First, we need to find dz/dx. Using the product rule:

dz/dx = d(e^x * y) / dx = e^x * (dy/dx) + y * e^x = e^x * (dy/dx + y)

Now, notice that the term dy/dx + y appears in the original differential equation and is equal to x. Therefore:

dz/dx = e^x * x

This is the rewritten differential equation in terms of z. We have eliminated y and dy/dx.

Answer: dz/dx = x*e^x, where z = e^x * y.

Example 7: Rewriting in Terms of a Bessel Function (Advanced)

Problem: Consider the differential equation x^2 * y'' + x * y' + (x^2 - ν^2) * y = 0 (Bessel's Equation). Rewrite this equation in terms of Bessel functions of the first kind, J_ν(x), which are solutions to this equation.

Solution: This is less about manipulating the equation and more about recognizing its form. Bessel's equation is defined in terms of Bessel functions. Therefore, the equation itself is the rewritten form in terms of those functions. The key here is understanding that J_ν(x) and J_{-ν}(x) (or Y_ν(x) for non-integer ν) are linearly independent solutions to the equation. The general solution can be written as:

y(x) = A * J_ν(x) + B * J_{-ν}(x) (if ν is not an integer) or y(x) = A * J_ν(x) + B * Y_ν(x) (generally)

Where A and B are arbitrary constants, and Y_ν(x) is the Bessel function of the second kind.

Answer: The equation itself is already expressed in terms of Bessel functions because J_ν(x) is defined as a solution to that differential equation. The general solution is a linear combination of Bessel functions of the first and second kind (or first kind with different orders).

Example 8: Rewriting Using Trigonometric Substitution

Problem: Evaluate the integral ∫√(1 - x^2) dx by rewriting it using a trigonometric substitution.

Solution: Let x = sin(θ). Then dx = cos(θ) dθ. The integral becomes:

∫√(1 - sin^2(θ)) * cos(θ) dθ = ∫√(cos^2(θ)) * cos(θ) dθ = ∫cos^2(θ) dθ

Now we need to use the identity cos^2(θ) = (1 + cos(2θ))/2:

∫cos^2(θ) dθ = ∫(1 + cos(2θ))/2 dθ = (1/2)∫(1 + cos(2θ)) dθ = (1/2)(θ + (1/2)sin(2θ)) + C

Finally, we need to rewrite this back in terms of x. Since x = sin(θ), θ = arcsin(x). Also, sin(2θ) = 2sin(θ)cos(θ) = 2x√(1 - x^2). Therefore:

(1/2)(θ + (1/2)sin(2θ)) + C = (1/2)(arcsin(x) + x√(1 - x^2)) + C

Answer: ∫√(1 - x^2) dx = (1/2)(arcsin(x) + x√(1 - x^2)) + C. The integral is rewritten (and solved) in terms of arcsin(x).

Common Pitfalls and How to Avoid Them

• Forgetting Constants of Integration: When integrating, always remember to add the constant of integration, C.
• Incorrectly Applying Identities: Double-check your trigonometric identities and algebraic manipulations. A small error can lead to a completely wrong answer.
• Ignoring Domain Restrictions: Be mindful of the domains of the functions involved. For example, arcsin(x) is only defined for -1 ≤ x ≤ 1.
• Not Simplifying: Always simplify your final answer as much as possible.
• Choosing the Wrong Substitution: Selecting an inappropriate substitution can make the problem more complicated instead of simpler. Practice and experience will help you develop an intuition for choosing the right substitution.

The Power of Rewriting: Why It Matters

Rewriting expressions isn't just an academic exercise. It's a crucial skill with far-reaching applications:

• Simplifying Complex Problems: By expressing an expression in terms of a more familiar function, you can often simplify the problem and make it easier to solve.
• Revealing Hidden Relationships: Rewriting can expose underlying connections between different mathematical concepts.
• Solving Differential Equations: As we saw in the examples, rewriting is essential for finding solutions to differential equations.
• Evaluating Integrals: Trigonometric substitutions and other rewriting techniques are indispensable for evaluating complex integrals.
• Numerical Analysis: Rewriting expressions can improve the accuracy and efficiency of numerical computations.
• Data Analysis: In statistics and data analysis, rewriting expressions is often used to transform data into a more meaningful form.

Conclusion

Rewriting expressions in terms of given functions is a fundamental and powerful technique in mathematics. By mastering the various methods, understanding common pitfalls, and appreciating the wide range of applications, you can unlock a deeper understanding of mathematical relationships and tackle complex problems with greater confidence. Practice is key to developing proficiency. So, grab a pencil, work through examples, and embrace the art of rewriting! The more you practice, the more intuitive these techniques will become, and the more effectively you will be able to apply them to solve a wide variety of mathematical problems. Remember that the goal is not just to find the answer, but also to understand the underlying principles and connections between different mathematical concepts.