Problem 1 - 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) \(g(2+h) = h^2 + 6h + 11\); (b) \(g(2) = 11\); (c) \(g(2+h) - g(2) = h^2 + 6h\).

Step 1: Define and Substitute in the Function

Define the given function as \( g(x) = x^2 + 2x + 3 \). For part (a), substitute \( x = 2 + h \) into the function to find \( g(2+h) \):\[g(2+h) = (2+h)^2 + 2(2+h) + 3\]

Step 2: Expand Polynomial for g(2+h)

Expand \((2+h)^2\) and simplify the expression for \( g(2+h) \):\[(2+h)^2 = 4 + 4h + h^2\]Substitute back into the expression:\[g(2+h) = 4 + 4h + h^2 + 4 + 2h + 3\]

Step 3: Simplify Expression for g(2+h)

Combine like terms in the equation for \( g(2+h) \):\[g(2+h) = h^2 + 6h + 11\]

Step 4: Substitute and Evaluate g(2)

Find \( g(2) \) by substituting \( x = 2 \) into the function:\[g(2) = 2^2 + 2(2) + 3 = 4 + 4 + 3 = 11\]

Step 5: Compute and Simplify g(2+h) - g(2)

Subtract \( g(2) \) from \( g(2+h) \):\[g(2+h) - g(2) = (h^2 + 6h + 11) - 11\]Simplifying gives:\[g(2+h) - g(2) = h^2 + 6h\]

Key Concepts

Function Evaluation

Evaluating a function involves substituting a specific value for the variable in the equation. This process helps us find the output, or result, of the function for that particular input. For example, to evaluate the function \( g(x) = x^2 + 2x + 3 \), you simply replace \( x \) with the given value.Let's consider part (a) of the exercise where we need to find \( g(2 + h) \). Here, you substitute \( 2 + h \) in place of \( x \), resulting in the expression: - \[ g(2+h) = (2+h)^2 + 2(2+h) + 3 \]This substitution step is crucial as it sets up the expression that needs further expansion and simplification. The goal of function evaluation is to find these outputs accurately by substituting and computing the terms.

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial that has been raised to a power in a more extended form. This is particularly relevant when dealing with expressions like \((x + y)^2\). Here, the binomial formula is often used.In the original exercise, part of the task was to expand \((2+h)^2\): - \[ (2+h)^2 = 4 + 4h + h^2 \]This expansion involves using simple arithmetic: - The \(2\) squared is \(4\). - Multiply \(2\) and \(h\), and double it to get \(4h\). - Finally, \(h\) squared gives you \(h^2\).After expanding, the expanded terms are substituted back into the expression for further operations, highlighting the importance of this basic algebraic skill in simplifying complex expressions.

Simplification of Expressions

Simplification of expressions often involves reducing an expression to its simplest form by combining like terms. Like terms are terms that contain the same variable raised to the same power.In our exercise, once the substitution and expansion have been done, the goal is to simplify the expression for \( g(2+h) \): - Start from \[ g(2+h) = 4 + 4h + h^2 + 4 + 2h + 3 \]. - Combine like terms, which gives you: \[ h^2 + 6h + 11 \].For simplifying \( g(2+h) - g(2) \), we subtract \( g(2) \) from \( g(2+h) \): - Begin with \[ g(2+h) - g(2) = (h^2 + 6h + 11) - 11 \]. - Simplify to get \[ h^2 + 6h \].The result is a simplified expression, making it more manageable, especially in calculus-related tasks like finding limits or derivatives.