Problem 8 - Applied Calculus

Question

For the functions, 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 Function

The function given is \(f(x) = 10x - x^2\). This is a quadratic function, which means it involves terms with the variable \(x\) raised to the second power or less. To find \(f(5)\), we will substitute 5 in place of every \(x\) in the function.

Step 2: Substitute the Value of x

Replace \(x\) in the function \(f(x) = 10x - x^2\) with 5. So, the expression becomes \(f(5) = 10 \times 5 - 5^2\).

Step 3: Perform the Multiplication

Calculate \(10 \times 5\), which equals 50. Now the expression is \(f(5) = 50 - 5^2\).

Step 4: Perform the Exponentiation

Calculate \(5^2\), which equals 25. Now substitute back into the expression to get \(f(5) = 50 - 25\).

Step 5: Perform the Subtraction

Subtract 25 from 50, which results in 25. Therefore, \(f(5) = 25\).

Key Concepts

Substitution

Substitution is a fundamental mathematical concept often used in algebra and calculus. It involves replacing a variable in an equation or expression with another value or expression. In our exercise, the goal was to evaluate the function \( f(x) = 10x - x^2 \) at \( x = 5 \).

To do this, we take every instance of the variable \( x \) in the function and replace it with the number 5. This process changes the expression from \( 10x - x^2 \) to \( 10 \times 5 - 5^2 \). Substitution is particularly useful because it simplifies the problem, allowing us to convert a general expression into specific instances for easier evaluation.

Understanding substitution helps us tackle a wide range of mathematical problems, from basic algebra to complex calculus, by providing clarity and precision.

Function Evaluation

Function evaluation is another key concept that deals with determining the output of a function for a given input. In simple terms, when you "plug in" a number for a variable in a function, you are evaluating that function at a given point.

For the quadratic function \( f(x) = 10x - x^2 \), evaluating the function at \( x = 5 \) means calculating \( f(5) \). After performing substitution, we are left with the expression \( 10 \times 5 - 5^2 \). This calculates how the function transforms the input 5, resulting in the final value of 25.

Function evaluation is essential as it provides the numerical result when inputs are substituted; enabling us to graph functions or solve real-world problems. It turns the abstract into the concrete, guiding decision making in various mathematics applications.

Mathematical Operations

Mathematical operations are the steps taken to manipulate numbers or expressions to achieve a desired result. In the realm of functions, especially with quadratic expressions, operations such as multiplication, exponentiation, and subtraction are commonly used.

In our function \( f(x) = 10x - x^2 \), after substituting \( x = 5 \), we first perform multiplication: \( 10 \times 5 = 50 \). Then, we proceed with exponentiation: \( 5^2 = 25 \). Finally, subtraction is used to find the difference: \( 50 - 25 = 25 \).

Understanding these operations individually and seeing how they are combined within functions is crucial. It allows solving complex equations more systematically and understanding how different operations affect the values involved.