Problem 4 - University Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The first derivative is \( w' = 21z^6 - 21z^2 + 42z \), and the second derivative is \( w'' = 126z^5 - 42z + 42 \).

Step 1: Understand the Problem

We need to find the first and second derivatives of the function \( w = 3z^7 - 7z^3 + 21z^2 \) with respect to \( z \). This involves applying the power rule of differentiation.

Step 2: Apply the Power Rule for the First Derivative

The power rule states that the derivative of \( z^n \) is \( nz^{n-1} \). Applying this to each term of \( w = 3z^7 - 7z^3 + 21z^2 \), we find:- For the term \( 3z^7 \), the derivative is \( 3 \times 7z^{7-1} = 21z^6 \).- For the term \( -7z^3 \), the derivative is \( -7 \times 3z^{3-1} = -21z^2 \).- For the term \( 21z^2 \), the derivative is \( 21 \times 2z^{2-1} = 42z \).Thus, the first derivative \( w' \) is \( w' = 21z^6 - 21z^2 + 42z \).

Step 3: Differentiate Again for the Second Derivative

To find the second derivative, differentiate \( w' = 21z^6 - 21z^2 + 42z \) using the power rule:- For the term \( 21z^6 \), the derivative is \( 21 \times 6z^{6-1} = 126z^5 \).- For the term \( -21z^2 \), the derivative is \( -21 \times 2z^{2-1} = -42z \).- For the term \( 42z \), the derivative is \( 42 \times 1z^{1-1} = 42 \).Thus, the second derivative \( w'' \) is \( w'' = 126z^5 - 42z + 42 \).

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus, particularly when dealing with differentiation. It's a simple yet powerful rule that applies to finding the derivative of a function that is a power of a variable. The rule states that if you have a function of the form \( z^n \), its derivative is \( nz^{n-1} \).

• This means you multiply the exponent by the coefficient of the term.
• Next, you decrease the exponent by one.

Let's apply this to an example: the term \( 3z^7 \) in our function. By the power rule:

• The derivative is \( 3 \times 7z^{7-1} \), simplifying to \( 21z^6 \).

Remember, the power rule makes finding derivatives straightforward, enabling us to tackle each term of the polynomial separately.

First Derivative

The first derivative of a function tells us about its rate of change. When you compute the first derivative using the power rule, you essentially identify how the function's output changes as the input varies.
For the function \( w = 3z^7 - 7z^3 + 21z^2 \), we apply the power rule individually to each term:

• \( 3z^7 \) becomes \( 21z^6 \)
• \( -7z^3 \) turns into \( -21z^2 \)
• \( 21z^2 \) results in \( 42z \)

Thus, the first derivative \( w' \) is \( w' = 21z^6 - 21z^2 + 42z \). This tells us how the function \( w \) changes with respect to \( z \). It's a critical step for analyzing the behavior of the function.

Second Derivative

The second derivative is essential for understanding the concavity and acceleration of a function, showing how the first derivative itself changes. It offers insights into the function's curvature and helps in determining points of inflection.
To find the second derivative of \( w = 3z^7 - 7z^3 + 21z^2 \), we differentiate the first derivative \( w' = 21z^6 - 21z^2 + 42z \) using the power rule again:

• \( 21z^6 \) changes to \( 126z^5 \)
• \( -21z^2 \) becomes \( -42z \)
• \( 42z \) simplifies to \( 42 \)

Thus, the second derivative \( w'' \) is \( w'' = 126z^5 - 42z + 42 \). This derivative assists in analyzing the curvature or the geometric shape of the function graph.

Polynomial Function

Polynomial functions are foundational in algebra and calculus. These functions consist of terms that are sums of constants and variables raised to non-negative integer exponents.
The given function \( w = 3z^7 - 7z^3 + 21z^2 \) is a classic example of a polynomial.

• Here, each term is a form of \( cz^n \), where \( c \) is a constant coefficient, and \( n \) is a non-negative integer.
• Polynomials are generally easy to work with because they are continuous and smooth, which makes them ideal for differentiation.

The structure of polynomial functions allows us to easily apply rules like the power rule, making them straightforward to differentiate whether we are finding the first or second derivative.