Problem 5 - Applied Calculus

Question

For \(g(x)=x^{2}+2 x+3,\) find and simplify: (a) \(g(2+h)\) (b) \(g(2)\) (c) \(g(2+h)-g(2)\)

Step-by-Step Solution

Answer

(a) \(h^2 + 6h + 11\), (b) 11, (c) \(h^2 + 6h\)

Step 1: Substitute into g(x)

To find \(g(2 + h)\), substitute \(2 + h\) for \(x\) in the expression for \(g(x)\): \(g(x) = x^2 + 2x + 3\). This gives: \(g(2 + h) = (2 + h)^2 + 2(2 + h) + 3\).

Step 2: Expand g(2 + h)

Expand \((2 + h)^2\) and \(2(2 + h)\): \((2 + h)^2 = 4 + 4h + h^2\) and \(2(2 + h) = 4 + 2h\). Substitute back into the equation: \(g(2+h) = 4 + 4h + h^2 + 4 + 2h + 3\).

Step 3: Simplify g(2 + h)

Combine like terms in \(g(2 + h) = 4 + 4h + h^2 + 4 + 2h + 3\). This results in \(g(2 + h) = h^2 + 6h + 11\).

Step 4: Calculate g(2)

Substitute \(x = 2\) into \(g(x)\): \(g(2) = 2^2 + 2(2) + 3\). Simplify this to get \(g(2) = 4 + 4 + 3 = 11\).

Step 5: Find g(2+h) - g(2)

Using the expressions found, calculate \(g(2 + h) - g(2)\): \(g(2 + h) = h^2 + 6h + 11\) and \(g(2) = 11\). Thus, \(g(2+h) - g(2) = h^2 + 6h + 11 - 11\).

Step 6: Simplify the Expression

Simplify \(g(2+h) - g(2)\): \(h^2 + 6h + 11 - 11 = h^2 + 6h\).

Key Concepts

Polynomial Functions

Polynomial functions are an essential concept in algebra and provide a foundation for understanding more complex mathematical ideas. A polynomial in one variable like the function \(g(x) = x^2 + 2x + 3\) comprises terms built from powers of \(x\) multiplied by coefficients.

• The term \(x^2\) is called the quadratic term and represents the degree of the polynomial since it's the highest power of \(x\).
• The 2 in front of \(x\) and the 3 are the linear and constant coefficients, respectively.

Polynomials can describe a wide range of phenomena from simple calculations to modeling real-world situations. Understanding polynomials involves evaluating them at specific points or doing calculations like finding \(g(2)\) or \(g(2 + h)\), which include substituting values for \(x\). This substitution lets us examine specific behaviors or outcomes of the polynomial under varied conditions.
Each polynomial is unique in its configuration, creating different curves and intersections when plotted on a graph. Recognizing the elements of a polynomial function helps simplify algebraic manipulation and aids in deeper comprehension of mathematical relationships.

Algebraic Manipulation

Algebraic manipulation involves rearranging and transforming algebraic expressions to achieve a desired outcome. When addressing polynomial functions, we often perform operations like expansion, factorization, and simplification.
To solve for \(g(2 + h)\), substitute \(2 + h\) for \(x\) in the original function; it involves expanding the expression \((2 + h)^2 + 2(2 + h) + 3\).

• When expanding \((2 + h)^2\), you multiply it out to \(4 + 4h + h^2\).
• For the \(2(2 + h)\), distribute the 2, resulting in \(4 + 2h\).

These actions are crucial to transforming the original format into a computed answer.
Algebraic manipulation is not only about calculating the right numbers; it involves understanding which properties of numbers and operations can simplify a problem. Mastering these manipulations empowers solving complex equations systematically and efficiently.

Simplification Process

The simplification process aims to make expressions more manageable by reducing them to their simplest forms. This is often vital in solving any algebraic problem as it enhances understanding and utility.
After substituting \(2 + h\) into \(g(x)\), the resulting expression was \(4 + 4h + h^2 + 4 + 2h + 3\). To simplify:

• Collect like terms, combining \(4\), \(4\), and \(3\) to get 11.
• Add \(4h\) and \(2h\) giving \(6h\).
• The \(h^2\) remains as it doesn’t combine with other terms.

The simplified form becomes \(h^2 + 6h + 11\).
In finding \(g(2+h) - g(2)\), subtract the previously computed \(g(2) = 11\) from \(h^2 + 6h + 11\), resulting in just \(h^2 + 6h\).
Simplifying expressions allow focusing on critical components by removing redundant information, providing a clearer perspective of what’s essential for further calculations or understanding.