Problem 11 - Applied Calculus

Question

For the functions in Problems \(10-14,\) find \(f(5)\). $$f(x)=10 x-x^{2}$$

Step-by-Step Solution

Answer

The value of \( f(5) \) is 25.

Step 1: Understand the Problem

We are given a function \( f(x) = 10x - x^2 \) and need to find the value of \( f(5) \). This means we will substitute \( x = 5 \) into the function and calculate the result.

Step 2: Substitute the Value of x

Substitute \( x = 5 \) into the given function. This gives us \( f(5) = 10(5) - 5^2 \).

Step 3: Calculate the Multiplication

Calculate \( 10 \times 5 \) which equals \( 50 \). The expression becomes \( f(5) = 50 - 5^2 \).

Step 4: Calculate the Exponent

Now calculate \( 5^2 \), which equals \( 25 \). The expression is now \( f(5) = 50 - 25 \).

Step 5: Perform the Subtraction

Subtract \( 25 \) from \( 50 \) to find \( f(5) = 25 \).

Key Concepts

Algebra

In mathematics, algebra is the study of mathematical symbols and the rules for manipulating these symbols. It is a unifying thread of almost all mathematics. When we work with algebra, we often use variables like \( x \) or \( y \) to represent numbers. These variables can stand for unknown values or represent any number in a given set.
In the context of our problem, \( f(x) = 10x - x^2 \), we use the variable \( x \) to depict any potential number we might be interested in. The function shows how \( f(x) \), or a result based on \( x \), is calculated by performing operations on \( x \). Here, algebra helps us express the relation between \( x \) and \( f(x) \) concisely.
When you substitute a specific value into \( x \), you replace the variable with that number, as we did with \( x = 5 \). Algebra allows such manipulations very systematically.

Substitution

Substitution is a key concept in algebra and involves replacing a variable in an expression with a given number to evaluate the expression for that specific value. In simple terms, substitution is like plugging numbers into a formula.
To evaluate the function \( f(x) = 10x - x^2 \) at \( x = 5 \), we simply replace \( x \) with 5 in the expression. This means our function \( f(x) \) becomes \( f(5) = 10(5) - 5^2 \).

• First step: replace \( x \) with 5, showing how the general formula adapts for a specific input.
• Second step: perform the arithmetic - multiplying, squaring, and then subtracting.

By following these steps, we solve the function for a particular instance of \( x \), making substitution a vital tool for finding specific values from general expressions.

Quadratic Functions

A quadratic function is a polynomial function of degree 2, generally written in the form \( ax^2 + bx + c \). This means that the highest power of the variable \( x \) is squared. In our example, the function \( f(x) = 10x - x^2 \) is a quadratic function.

• The term \(-x^2\) designates the quadratic part, indicating that the function will exhibit a parabolic curve when graphed.
• The term \(10x\) is a linear component that affects the slope of the function.

Quadratic functions often have a U-shaped graph known as a parabola, which can open upward or downward. In our function \( f(x) = 10x - x^2 \), the graph would open downwards because the coefficient of \( x^2 \) is negative. Understanding these aspects helps in analyzing how changes in the input \( x \) affect the output \( f(x) \). Quadratic functions are key in various real-world scenarios like optimization problems, projectile motion, and more.