Problem 48 - Calculus and its applications

Question

Differentiate each function. \(f(t)=\frac{3 t^{2}+2 t-1}{-t^{2}+4 t+1}\)

Step-by-Step Solution

Answer

The derivative of the function is \( f'(t) = \frac{14t^2 + 10t}{(-t^2 + 4t + 1)^2} \).

Step 1: Identify the differentiation method

The function given is a quotient of two polynomials. This indicates that we should use the Quotient Rule for differentiation. The Quotient Rule states that if you have a function \( f(t) = \frac{u(t)}{v(t)} \), its derivative \( f'(t) \) is \( \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2} \).

Step 2: Define the functions u(t) and v(t)

For the given function, let \( u(t) = 3t^2 + 2t - 1 \) and \( v(t) = -t^2 + 4t + 1 \). These will be used in the Quotient Rule to find the derivative.

Step 3: Differentiate u(t) and v(t)

Calculate the derivatives of both functions.For \( u(t) \):The derivative \( u'(t) = \frac{d}{dt}(3t^2 + 2t - 1) = 6t + 2 \).For \( v(t) \):The derivative \( v'(t) = \frac{d}{dt}(-t^2 + 4t + 1) = -2t + 4 \).

Step 4: Apply the Quotient Rule

Substitute \( u(t) \), \( v(t) \), \( u'(t) \), and \( v'(t) \) into the Quotient Rule formula:\[f'(t) = \frac{(6t + 2)(-t^2 + 4t + 1) - (3t^2 + 2t - 1)(-2t + 4)}{(-t^2 + 4t + 1)^2}\]

Step 5: Simplify the numerator

Expand and simplify the terms in the numerator:The first term: \((6t + 2)(-t^2 + 4t + 1) = -6t^3 + 24t^2 + 6t - 2t^2 + 8t + 2 = -6t^3 + 22t^2 + 14t + 2.\)The second term: \( (3t^2 + 2t - 1)(-2t + 4) = -6t^3 + 12t^2 - 4t - 4t^2 + 8t + 2 = -6t^3 + 8t^2 + 4t + 2.\)Combine the two expressions in the numerator:\( -6t^3 + 22t^2 + 14t + 2 + 6t^3 - 8t^2 - 4t - 2 = 14t^2 + 10t. \)

Step 6: Finalize the derivative

Now that you have simplified the numerator, you can write the expression as:\[f'(t) = \frac{14t^2 + 10t}{(-t^2 + 4t + 1)^2}\]This is the derivative of the function \( f(t) \).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that explores how functions change. More precisely, it measures the rate at which a function's value changes as its input changes. At its core, differentiation transforms a function into its derivative, highlighting the function's behavior and slope at any given point.

To understand this better, think of a car traveling on a winding road. The speed of the car at any given moment is analogous to the derivative of its position. Just as speed can tell us how fast the car is moving along the road, differentiation tells us how swiftly the function's output is changing for each incremental change in the input. It's a bit like understanding how steep or flat the path is.

When a function is peaceful, like a straight road, its derivative is constant. Conversely, in more hilly regions with curves and turns, the derivative will vary, indicating changes in the function's behavior. Mastering differentiation enables the prediction and understanding of these behaviors, making it a crucial tool for problem-solving in various fields such as physics, engineering, and economics.

Polynomial Functions

Polynomial functions form the backbone of many mathematical applications. Defined as expressions consisting of variables and coefficients, these functions involve operations of addition, subtraction, multiplication, and non-negative integer exponents.

For example, consider the polynomial function \(f(x) = ax^n + bx^{n-1} +
3 + k\). Here, \(a, b, k,\) and others are coefficients, while the powers of \(x\) must be whole numbers. These functions can take many shapes, from simple lines to complex curves, and they are pivotal in both algebra and calculus.

• Straightforward and structured, they are among the easiest functions to manipulate mathematically.
• They naturally arise in scenarios involving motion, growth, and patterns, making them universally relevant.
• Understanding their properties, like intercepts and end behavior, helps in solving a broad range of problems.

When working with polynomials, especially in differentiation, we can leverage their structured nature to easily calculate derivatives, setting the foundation for more complex calculations.

Derivative

The derivative of a function gives a complete picture of its rates of change at any point. More formally, the derivative of a function at any point is the slope of the tangent line to the graph of the function at that specific point.

When differentiating polynomial functions, each term is processed individually according to the power rule. For a term like \( ax^n \), its derivative will be \( nax^{n-1} \). This rule is powerful in its simplicity, allowing for easy transformation of functions into derivatives.

• The derivative tells us whether the function is increasing or decreasing at a given point.
• Derivatives can also identify maxima and minima, which are critical for optimization problems.
• They provide insights into acceleration and other real-world dynamic systems.

For our example, differentiating a quotient of two polynomial functions requires the use of the Quotient Rule. The rule integrates the derivatives of both the numerator and the denominator, giving us a holistic view of how the entire function behaves. This systematic approach empowers us to solve complex real-world problems by understanding the intricacies and patterns of change in various scenarios.