Problem 14 - Applied Calculus

Question

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=8 t^{3}-4 t^{2}+12 t-3$$

Step-by-Step Solution

Answer

The derivative of the function is \(24t^2 - 8t + 12\).

Step 1: Apply Power Rule

To find the derivative of the polynomial function \(y = 8t^3 - 4t^2 + 12t - 3\), we will apply the power rule of differentiation. The power rule states that the derivative of \(t^n\) with respect to \(t\) is \(nt^{n-1}\).

Step 2: Differentiate Each Term

Differentiate each term separately using the power rule:- The first term is \(8t^3\). Applying the power rule, the derivative is \(3 \cdot 8t^{3-1} = 24t^2\).- The second term is \(-4t^2\). The derivative is \(2 \cdot (-4)t^{2-1} = -8t\).- The third term is \(12t\). The derivative is \(1 \cdot 12t^{1-1} = 12\).- The fourth term is a constant \(-3\). The derivative of a constant is \(0\).

Step 3: Combine the Derivatives

Combine all the derived terms to form the complete derivative of the original function. Thus,\[ \frac{dy}{dt} = 24t^2 - 8t + 12.\]

Key Concepts

Derivative

A derivative is a fundamental concept in calculus, representing the rate at which a function changes at a certain point. In simpler terms, it's like determining how fast or slow something is changing. When you hear about speed in physics, that's essentially a derivative, as it calculates how much distance changes over time. The derivative provides crucial insights.

• It helps in understanding how variables in a function are interrelated.
• It's widely used in fields like physics, engineering, and economics for optimization problems.

By computing derivatives, you can find rates of change and analyze the behavior of complex systems under varying conditions.

Polynomial differentiation

Polynomial differentiation involves taking the derivative of functions that are polynomials. A polynomial function is an expression that comprises terms of variables raised to whole-number exponents, multiplied by coefficients. For instance, a polynomial function can look like this: \( y = 8t^3 - 4t^2 + 12t - 3 \).
The process of differentiating a polynomial is straightforward and relies on applying the power rule repeatedly to each term of the polynomial. Each term is handled separately, making polynomial differentiation simple once the power rule is understood.

• Always start by identifying each term of the polynomial separately.
• Apply the differentiation rules, especially focusing on the power rule (to be explained in the next section).
• Combine all derived terms to obtain the derivative of the original polynomial.

This method allows for easy handling of polynomials, no matter how many terms or what the exponents are. Think of it as a step-by-step construction, leading to the overall derivative of the polynomial.

Power rule

The power rule is a simple, yet incredibly powerful tool in calculus. It helps us find derivatives of functions swiftly. The power rule states that if you have a function of the form \( t^n \), the derivative is found by multiplying the exponent \( n \) with the coefficient and then reducing the exponent by one, resulting in: \( nt^{n-1} \).
By applying the power rule, differentiating becomes much less complex. For example, in the function \( y = 8t^3 - 4t^2 + 12t - 3 \), each term of the polynomial is differentiated using the power rule:

• The derivative of \( 8t^3 \) is \( 24t^2 \), as we multiply 3 by 8 and reduce the exponent by 1.
• The derivative of \( -4t^2 \) is \( -8t \), following the same rule.
• For \( 12t \), the derivative is simply 12 because \( t^1 \) becomes \( t^0 \), simplifying to 1.
• Any constant term, like \( -3 \), has a derivative of 0, since constants do not change.

Mastery of the power rule is fundamental for tackling polynomial differentiation efficiently, making it a staple technique in calculus.