Problem 15 - Applied Calculus

Question

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=4.2 q^{2}-0.5 q+11.27$$

Step-by-Step Solution

Answer

The derivative of the function is \(8.4q - 0.5\).

Step 1: Identify the function form

The function given is a polynomial of the form \(f(q) = aq^2 + bq + c\), where the exponents are non-negative integers.

Step 2: Apply the Power Rule

The power rule states that the derivative of a term \(x^n\) is \(nx^{n-1}\). Apply this to each term of the polynomial. For \(4.2q^2\), the derivative is \(2 \cdot 4.2q^{2-1} = 8.4q\). For \(-0.5q\), the derivative is \(-0.5\). For the constant \(11.27\), the derivative is zero.

Step 3: Combine the derivatives

Sum the derivatives of each individual term to find the derivative of the entire function. This gives \(\frac{dy}{dq} = 8.4q - 0.5\).

Key Concepts

Polynomial Function

A polynomial function is a type of mathematical expression that consists of variables called terms, each multiplied by a coefficient. These terms are written as powers, where the exponents are whole numbers. The given function, \[ y = 4.2q^2 - 0.5q + 11.27 \]is a polynomial. Here:- **4.2** is the coefficient of the squared term \(q^2\),- **-0.5** is the coefficient of \(q\), the linear term, and- **11.27** is a constant term.Each term in a polynomial can be easily identified by its degree, which is the exponent of the variable. The degree of the entire polynomial is the highest exponent, which in this case is 2 (from \(4.2q^2\)). A polynomial's degree tells us about its general shape and behavior. Understanding these components is essential for finding derivatives effectively.

Power Rule

The power rule is a fundamental technique in calculus used to derive functions with power terms. It simplifies the process of finding derivatives for polynomial expressions. According to the power rule, if you have a term \(x^n\),its derivative is given by\[ nx^{n-1} \]where:- **n** is the exponent,- and **x** is the variable.Let's apply the power rule to each term in the polynomial:- For **4.2q²**, the derivative calculation is \(2 \times 4.2q^{2-1} = 8.4q\).- For **-0.5q**, apply the rule to get a derivative of \(-0.5\), since the power of \(q\) is 1.This rule helps in quickly differentiating polynomial functions term by term, effectively identifying the slope at any point. Practicing the power rule allows for a more nuanced understanding of how functions change and lets you calculate rates quickly.

Constants in Derivatives

In calculus, when differentiating, constants behave uniquely. A constant in the context of a polynomial function is a term without any variable attached. For example, in the function:\[ y = 4.2q^2 - 0.5q + 11.27 \],**11.27** is a constant term. When taking derivatives:- The derivative of any constant is **always zero**. This is because constants do not change, hence they have no rate of change or slope.In this exercise, - The derivative of **11.27** is **0**.Recognizing constants and knowing their derivatives play a crucial role in simplifying the process of finding the overall derivative of a function. This concept emphasizes that while variables and their coefficients determine how a polynomial function behaves and changes, constants do not contribute to the dynamism of change captured by the derivative.