Problem 62 - Brief Applied Calculus

Question

$$ \text { For each function, find and simplify } f(x+h) \text { . } $$ $$ f(x)=3 x^{2}-5 x+2 $$

Step-by-Step Solution

Answer

Substitute \(x + h\) into \(f(x)\) and simplify: \(3x^2 + 6xh + 3h^2 - 5x - 5h + 2\).

Step 1: Substitute \(x + h\) for \(x\) in \(f(x)\)

Start by substituting \(x + h\) into the function \(f(x)\) instead of \(x\). Our function \(f(x)\) is given as \(3x^2 - 5x + 2\). Replace \(x\) with \(x + h\): \[ f(x+h) = 3(x+h)^2 - 5(x+h) + 2 \]

Step 2: Expand \((x+h)^2\)

Next, expand the expression \((x+h)^2\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\). Here \(a = x\) and \(b = h\):\[ (x+h)^2 = x^2 + 2xh + h^2 \]

Step 3: Simplify \(3(x^2 + 2xh + h^2)\)

Now, substitute the expanded expression of \((x+h)^2\) back into the function and distribute the 3:\[ 3(x^2 + 2xh + h^2) = 3x^2 + 6xh + 3h^2 \]

Step 4: Expand \(-5(x+h)\)

Continue by expanding \(-5(x+h)\) using distribution:\[ -5(x+h) = -5x - 5h \]

Step 5: Combine All Parts

Now, combine all the expanded parts together in \(f(x+h)\):\[ f(x+h) = 3x^2 + 6xh + 3h^2 - 5x - 5h + 2 \]

Step 6: Simplify the Expression

Finally, combine like terms in the expression:\[ f(x+h) = 3x^2 - 5x + 6xh + 3h^2 - 5h + 2 \]

Key Concepts

Polynomial Expansion

Polynomial expansion is about stretching out a polynomial expression into its simpler terms. Imagine you have an elastic band that needs stretching to see all its parts clearly. In the case of polynomial expressions like \((x+h)^2\), you use the expansion rule \((a+b)^2 = a^2 + 2ab + b^2\). This process breaks \((x+h)^2\) into three distinct terms: \(x^2\), \(2xh\), and \(h^2\). It is crucial because it transforms a compact expression into a form that is more manageable and easy to work with. This process also gives a clear view of how each component affects the final expression, preparing it for any further mathematical operations.

Substitution in Functions

Substitution in functions is like changing the ingredient in a recipe to see how it affects the final dish. In this exercise, we substitute \(x + h\) for \(x\) in the function \(f(x) = 3x^2 - 5x + 2\). This means wherever you see \(x\) in the function, you replace it with \(x + h\). So, the function becomes \(f(x+h) = 3(x+h)^2 - 5(x+h) + 2\). This method of substitution is fundamental in calculus and algebra because it allows us to explore how a function behaves when its input changes slightly. Such exploration is the backbone of understanding limits, differentials, and the behavior of functions over time. Always be mindful of applying substitution correctly to ensure you obtain the correct results.

Simplifying Expressions

Simplifying expressions is the act of tidying up. Consider it like organizing a cluttered room: you want to simplify the space to make everything easier to find and understand. For mathematical expressions, this involves combining like terms and reducing the expression to its simplest form. In our exercise, after expansion and substitution, we end up with \(f(x+h) = 3x^2 + 6xh + 3h^2 - 5x - 5h + 2\). The goal here is to condense this to a neat, simplified version: \(3x^2 - 5x + 6xh + 3h^2 - 5h + 2\). Simplification reduces complexity and prepares expressions for further operations. It allows for clearer calculations and facilitates an easier understanding of what the expression represents or how it might behave under further mathematical scrutiny.