Problem 2 - Applied Calculus

Question

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(P=3 t^{3}+2 e^{t}\)

Step-by-Step Solution

Answer

The derivative of \( P \) is \( 9t^2 + 2e^t \).

Step 1: Identify the Function to Differentiate

The given function is \( P = 3t^3 + 2e^t \). We need to find its derivative \( \frac{dP}{dt} \).

Step 2: Differentiate Each Term Separately

Differentiate the first term \( 3t^3 \). Use the power rule, which states \( \frac{d}{dt} [t^n] = nt^{n-1} \).\[ \frac{d}{dt} [3t^3] = 3 \cdot 3t^{3-1} = 9t^2 \]

Step 3: Differentiate the Exponential Term

Now differentiate the second term, \( 2e^t \). The derivative of \( e^t \) is itself \( e^t \), so:\[ \frac{d}{dt} [2e^t] = 2e^t \]

Step 4: Combine Derivatives

Now that we have the derivatives of both terms, combine them to get the derivative of the entire function:\[ \frac{dP}{dt} = 9t^2 + 2e^t \]

Key Concepts

Power Rule

The Power Rule is a fundamental concept in calculus used to differentiate functions of the form \( t^n \), where \( n \) is any real number. It's one of the first rules you learn when tackling differentiation. This rule simplifies the process by providing a straightforward formula: \( \frac{d}{dt} [t^n] = nt^{n-1} \).

• This means, if you have a term like \( t^3 \), you bring down the exponent (which is 3) in front of the term and then reduce the exponent by one.
• For example, applying the Power Rule to \( 3t^3 \) gives us \( 3 \times 3t^{3-1} = 9t^2 \).

The Power Rule is quick and reliable, especially useful for polynomials. It turns potentially complex calculations into manageable steps.
Practicing this rule helps you gain fluency in working through polynomial derivatives efficiently.

Exponential Functions

Exponential functions, especially those involving \( e^t \), are important in calculus because they appear in many natural phenomena. The base \( e \) is an irrational number approximately equal to 2.71828, crucial in understanding growth processes.

• For the function \( e^t \), its remarkable property is that its derivative is the same as the function itself: \( \frac{d}{dt} [e^t] = e^t \).
• This means exponential functions grow at a rate proportional to their current value, which is why they model unchecked growth effectively, such as populations or investments.
• If you have a function like \( 2e^t \), the derivative is simple: it remains \( 2e^t \).

Understanding this characteristic allows you to quickly differentiate like functions, making exponential rules a breeze in derivative calculations.

Derivative Calculation

Derivative calculation is the method of finding the rate at which a function changes. It involves combining rules like the Power Rule and the derivative of exponential functions.

• Starting with a complex function, you break it into simpler parts, differentiating each independently.
• For example, with \( P = 3t^3 + 2e^t \), you apply the Power Rule to \( 3t^3 \) and the exponential derivative rule to \( 2e^t \), giving you derivatives \( 9t^2 \) and \( 2e^t \) respectively.
• The final step is combining these results: \( \frac{dP}{dt} = 9t^2 + 2e^t \).

Practicing derivative calculations enhances your problem-solving skills, allowing you to tackle not only academic but real-life rate of change problems. It’s essential for understanding how quantities respond to changes over time, which is foundational to much of science and engineering.