Problem 22 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=0.002 x^{3}-0.05 x^{2}+0.1 x-20\)

Step-by-Step Solution

Answer

The derivative of the given function \(f(x) = 0.002x^3 - 0.05x^2 + 0.1x - 20\) is: \(f'(x) = 0.006x^2 - 0.1x + 0.1\)

Step 1: Identify each term of the polynomial function

The given polynomial function is: \(f(x) = 0.002x^3 - 0.05x^2 + 0.1x - 20\) We can break this function into four terms: Term 1: \(0.002x^3\) Term 2: \(-0.05x^2\) Term 3: \(0.1x\) Term 4: \(-20\)

Step 2: Apply the power rule to each term

Now, we will apply the power rule to each of these terms: Term 1: \(0.002x^3\), n = 3, so its derivative is: \(0.002(3)x^{(3-1)} = 0.006x^2\) Term 2: \(-0.05x^2\), n = 2, so its derivative is: \(-0.05(2)x^{(2-1)} = -0.1x\) Term 3: \(0.1x\), n = 1, so its derivative is: \(0.1(1)x^{(1-1)} = 0.1\) Term 4: \(-20\), n = 0 (constant), so its derivative is: \(0\)

Step 3: Add the derivatives together

Finally, we'll add the derivatives of each term to find the derivative of the entire function: \(f'(x) = 0.006x^2 - 0.1x + 0.1\)

Step 4: Present the final derivative of the function

The derivative of the given function f(x) is: \(f'(x) = 0.006x^2 - 0.1x + 0.1\)

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that focuses on finding the rate at which a function is changing at any given point. It involves calculating the derivative of a function, which essentially tells us how the output of the function changes as the input changes.
Typically, the process of differentiation is used to find the slope of the tangent line to a curve at a particular point. This provides valuable insights in various real-world applications, such as calculating speed, acceleration, or optimizing functions.

• The derivative of a function is denoted by a prime symbol, for example, the derivative of a function \( f(x) \) is represented as \( f'(x) \).
• We commonly use various rules and techniques to differentiate complex functions, such as the power rule, product rule, and chain rule. Each rule applies to particular cases or combinations of functions.

Differentiation is not only applicable to polynomials but also provides solutions to more complex functions involving trigonometric, exponential, and logarithmic expressions.

Power Rule

The power rule is one of the simplest and most frequently used rules in differentiation. It provides a straightforward way to calculate the derivative of functions in the form of \( x^n \), where \( n \) is any real number.
According to the power rule, if a function is \( f(x) = x^n \), its derivative \( f'(x) \) is found using the formula: \( nx^{n-1} \). This rule holds true for any exponent, whether it's positive, negative, or even fractional.

• Apply the power rule by multiplying the coefficient of \( x \) (if any) with the exponent.
• After multiplying, reduce the exponent by 1.

For example, in our original exercise, the term \( 0.002x^3 \) is differentiated using the power rule, resulting in \( 0.006x^2 \). This demonstrates how straightforward it can be to apply the power rule to each term of a polynomial.

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables raised to some power, combined with coefficients. These expressions can have multiple terms, and each term is a product of a constant coefficient and a power of the variable.
Polynomials can have degrees, which refer to the highest power of the variable within the expression. The degree indicates the general shape and behavior of the graph of the polynomial.

• The simplest example of a polynomial is a linear function, which is of degree 1.
• Quadratic functions, of degree 2, form parabolas and are often used in physics and engineering.
• Cubic functions, like the one in our exercise, are of degree 3 and can have more complex shapes including twists and turns.

Differentiating polynomial functions like \( f(x) = 0.002x^3 - 0.05x^2 + 0.1x - 20 \) involves handling each term separately using rules such as the power rule. As seen in the exercise solution, each term is addressed individually before being combined to form the complete derivative, \( f'(x) = 0.006x^2 - 0.1x + 0.1 \). This straightforward handling makes polynomials some of the easiest functions to differentiate.