Problem 4 - Thomas Calculus

Question

Find the first and second derivatives. $$ w=3 z^{7}-7 z^{3}+21 z^{2} $$

Step-by-Step Solution

Answer

First derivative: \( w' = 21z^6 - 21z^2 + 42z \), Second derivative: \( w'' = 126z^5 - 42z + 42 \).

Step 1: Understand the Function

The given function is a polynomial: \[ w = 3z^7 - 7z^3 + 21z^2 \] We need to find its first and second derivatives with respect to \( z \).

Step 2: Apply the Power Rule for the First Derivative

To find the first derivative, \( w' \), we apply the power rule: \( d/dz \> a z^n = a n z^{n-1} \).1. Differentiate \( 3z^7 \): \[ d/dz \>(3z^7) = 3 imes 7 z^{6} = 21z^6 \]2. Differentiate \( -7z^3 \): \[ d/dz \>(-7z^3) = -7 imes 3 z^{2} = -21z^2 \]3. Differentiate \( 21z^2 \): \[ d/dz \>(21z^2) = 21 imes 2 z^{1} = 42z \]Combine these results to get:\[ w' = 21z^6 - 21z^2 + 42z \]

Step 3: Apply the Power Rule for the Second Derivative

Now, differentiate the first derivative to find the second derivative, \( w'' \).1. Differentiate \( 21z^6 \): \[ d/dz \>(21z^6) = 21 imes 6 z^{5} = 126z^5 \]2. Differentiate \( -21z^2 \): \[ d/dz \>(-21z^2) = -21 imes 2 z^{1} = -42z \]3. Differentiate \( 42z \): \[ d/dz \>(42z) = 42 \]Combine these results to get:\[ w'' = 126z^5 - 42z + 42 \]

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that are built from variables and constants, using only addition, subtraction, multiplication, and positive integer exponents. A general polynomial function can be written in the form of

• \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \)

where \( a_n, a_{n-1}, ..., a_1, a_0 \) are constants, and \( n \) is a non-negative integer representing the highest power in the polynomial.
In our exercise, the polynomial function given is:

• \( w = 3z^7 - 7z^3 + 21z^2 \)

This expression involves terms where the variable \( z \) is raised to powers of 7, 3, and 2, respectively. Each term in a polynomial is composed of a coefficient (like 3, -7, and 21) and a power of the variable \( z \).
Understanding polynomial functions is crucial because they serve as the foundation for many concepts in calculus, including the differentiation process used in finding derivatives.

Power Rule

The power rule is a basic and powerful tool for finding derivatives of polynomial functions. It simplifies the process of differentiation when dealing with terms where variables are raised to exponents. The rule states:

• \( \frac{d}{dz} (a z^n) = a n z^{n-1} \)

where \( a \) is a constant, \( z \) is a variable, and \( n \) is a non-negative integer.
When applying the power rule, you multiply the original exponent by the coefficient of that term and then subtract one from the exponent.
In our exercise, we applied the power rule to each term:

• To \( 3z^7 \) we get \( 21z^6 \).
• To \( -7z^3 \) we get \( -21z^2 \).
• To \( 21z^2 \) we get \( 42z \).

This method is universally applicable for differentiating polynomial terms, making it an indispensable technique in calculus.

First Derivative

The first derivative of a function provides information about the rate of change of the function’s value with respect to a change in the variable. It is often denoted as \( w' \) or \( \frac{dw}{dz} \) for a function \( w \) of \( z \).
In physical terms, if a function describes position over time, its first derivative represents velocity.
In our exercise, the first derivative of the polynomial \( w = 3z^7 - 7z^3 + 21z^2 \) is calculated as follows:

• \( w' = 21z^6 - 21z^2 + 42z \)

Each differentiated term tells us how much the function \( w \) changes as \( z \) changes.
The first derivative is used to find critical points, determine where a function is increasing or decreasing, and is a fundamental step towards understanding the overall shape of the function’s graph.

Second Derivative

The second derivative of a function is the derivative of its first derivative. It provides insight into the concavity of the function and how the rate of change of the rate of change occurs. For a function \( w \), the second derivative is denoted by \( w'' \) or even \( \frac{d^2w}{dz^2} \).
A positive second derivative indicates that the function is concave up (shaped like a cup), while a negative second derivative indicates concave down (shaped like a cap).
From the first derivative \( w' = 21z^6 - 21z^2 + 42z \) that we calculated, we find the second derivative by further differentiating:

• \( w'' = 126z^5 - 42z + 42 \)

This result helps us understand the acceleration of change within the function.
The second derivative is crucial for analyzing the behavior of the function, especially in determining points of inflection and optimizing its behavior.