Problem 13 - Calculus an Applied Approach

Question

Find the derivative of the function. $$ f(t)=-3 t^{2}+2 t-4 $$

Step-by-Step Solution

Answer

The derivative of the function \(f(t)=-3 t^{2}+2 t-4\) is \(f'(t)= -6t + 2\).

Step 1: Identify the function

Identify the function that needs to be differentiated. The function is \(f(t)=-3 t^{2}+2 t-4\).

Step 2: Apply the power rule

The power rule will be applied individually to each term. The power rule states that the derivative of \(x^n\), where n is any real number, is \(nx^{n-1}\). This will be applied to each term individually. Firstly, differentiating \(-3t^2\), the derivative is \(-6t\). Secondly, differentiating \(2t\), the derivative is \(2\). And lastly, differentiating \(-4\), being a constant, the derivative is \(0\).

Step 3: Write down the derivative

Combine the results from step 2 to get the derivative of the function. The derivative of \(f(t)=-3 t^{2}+2 t-4\) is \(f'(t)= -6t + 2\).

Key Concepts

Derivative

The concept of a derivative is vital in calculus, providing a way to measure how a function changes as its input changes. In simpler terms, the derivative tells us the rate at which a function is changing at any given point. When we say we are finding the derivative of a function, we are essentially looking for a formula that gives us this rate of change.
For a function like \(f(t) = -3t^2 + 2t - 4\), the derivative will give us an insight into the behavior of the graph of the function, telling us whether it is increasing or decreasing at a particular point. This is extremely useful in various fields such as physics, engineering, and economics, where understanding change is crucial.

Power Rule

The power rule is a foundational technique in calculus used to find the derivative of functions that involve powers of variables. This rule states that if you have a term in the form of \(x^n\), its derivative will be \(nx^{n-1}\).
Let's break it down with an example for clarification. If you have a function term like \(-3t^2\), applying the power rule involves:

• Taking the exponent (2) and multiplying it by the coefficient (-3), giving \(-6\).
• Subtracting one from the original exponent (2), giving you 1.

Thus, the derivative of \(-3t^2\) using the power rule is \(-6t\). This approach makes it quick and straightforward to differentiate polynomial functions.

Differentiation

Differentiation is the process of finding the derivative of a function. It involves applying rules like the power rule to each term of the function to determine the overall derivative.
In the function \(f(t) = -3t^2 + 2t - 4\), differentiation requires that we process each term separately:

• \(-3t^2\) becomes \(-6t\) after differentiation.
• \(2t\) becomes \(2\), as the derivative of \(t\) is 1.
• The constant \(-4\) becomes \(0\) because the derivative of a constant is always zero.

Finally, we combine all the differentiated terms: \(f'(t) = -6t + 2\). Differentiation tells us precisely how fast the original function \(f(t)\) changes with respect to \(t\).

Functions

Functions are mathematical expressions that relate inputs to outputs, denoted typically as \(f(t)\), \(g(x)\), etc. They form the core of calculus, modeling real-world phenomena by defining how one quantity depends on another.
In calculus, examining how functions change is central to understanding their behavior, which is precisely what differentiation helps accomplish.

• For example, the function \(f(t) = -3t^2 + 2t - 4\) combines various terms involving the variable \(t\).
• This function expresses a dependency of \(f\) on \(t\), allowing us to compute specific values of \(f\) for any given \(t\).

By differentiating, we dissect this relationship, exposing how \(f(t)\) tangibly evolves with changes in \(t\). Understanding functions and their derivatives unlocks insights into much broader systems and processes.