Problem 17 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Let \(P(t)=-3+5 t-2 t^{2}\). Cite the formulas that justify steps \((i)-(v i)\) below: $$\begin{aligned} P^{\prime}(t) &=\left[-3+5 t+(-2) t^{2}\right]^{\prime} \\ &=[-3]^{\prime}+[5 t]^{\prime}+\left[(-2) t^{2}\right]^{\prime} \\ &=0+[5 \times t]^{\prime}+\left[(-2) t^{2}\right]^{\prime} \\ &=0+5[t]^{\prime}+(-2)\left[t^{2}\right]^{\prime} \\ &=0+5 \times 1+(-2)\left[t^{2}\right]^{\prime} \\ &=0+5 \times 1+(-2) \times 2 t \\ &=5-4 t\end{aligned}$$

Step-by-Step Solution

Answer

The derivative of the function is \( P^{\prime}(t) = 5 - 4t \).

Step 1: Differentiate the Expression

Start by differentiating the polynomial function, noting that for any constant \( c \), the derivative is zero: \[ P^{\prime}(t) = \left[-3+5t + (-2)t^2\right]^{\prime} \]

Step 2: Apply Linearity of Differentiation

Use the linearity property of differentiation, which states that the derivative of the sum is the sum of the derivatives: \[ P^{\prime}(t) = [-3]^{\prime} + [5t]^{\prime} + [(-2) t^2]^{\prime} \]

Step 3: Differentiate the Constant Term

The derivative of a constant term is zero: \[ [-3]^{\prime} = 0 \]

Step 4: Differentiate the Linear Term

For the linear term \( 5t \), use the power rule which states that \((at)^n \rightarrow a \cdot n \cdot t^{n-1}\) where \(n=1\): \[ [5t]^{\prime} = 5[t]^{\prime} \]

Step 5: Differentiate the Linear Term (Final Result)

Since \( [t]^{\prime} = 1 \), we get: \[ 5[t]^{\prime} = 5 \times 1 = 5 \]

Step 6: Differentiate the Quadratic Term using Power Rule

For the quadratic term \((-2)t^2\), apply the power rule \((at^n)^{\prime} = a \cdot n \cdot t^{n-1}\): \[ [(-2)t^2]^{\prime} = (-2) \times 2 \times t = -4t \]

Step 7: Combine the Results

Combine all the differentiated terms to get the derivative of the function:\[ P^{\prime}(t) = 0 + 5 - 4t = 5 - 4t \]

Key Concepts

Power Rule

The Power Rule is a fundamental technique in calculus used to differentiate polynomials. It simplifies finding derivatives of terms like \( at^n \). Here's how it works:
- For a term \( at^n \), its derivative \((a \, t^n)^{\prime}\) is given by multiplying the exponent \( n \) by the coefficient \( a \), and decreasing the exponent by one. The formula is \( a \cdot n \cdot t^{n-1} \).This rule is especially useful for polynomial functions where you have terms with variables raised to a power. It simplifies the process of differentiation considerably by providing a direct method to handle variable terms.
Let's see it in action with an example: For the term \((-2)t^2\), applying the power rule, we get:- Multiply the coefficient \(-2\) by the exponent \(2\).- Then multiply by \( t \) raised to the power of \( 2-1\), which gives \(-4t\).
This demonstrates the ease with which derivatives of polynomial terms can be determined using the Power Rule.

Linearity of Differentiation

The Linearity of Differentiation is a useful property that assists in breaking down complex expressions into simpler terms. It states that the derivative of a sum is the sum of the derivatives. This means when you have multiple terms added together, as in a polynomial, you can differentiate each term individually, and then add the results to get the overall derivative.

This property is especially beneficial when dealing with polynomials like \(P(t) = -3 + 5t - 2t^2\). Let's break it down:

• Identify each term in the polynomial: \(-3\), \(5t\), and \(-2t^2\).
• Differentiate each term separately: the derivative of a constant (\(-3\)) is \(0\), for \(5t\) use the Power Rule to get \(5\), and for \(-2t^2\), apply the Power Rule to get \(-4t\).
• Combine the results: \(0 + 5 - 4t\).

The power of this rule lies in its ability to simplify the differentiation process by treating each component term as an individual task. It allows us to focus on one piece at a time, then reassemble the puzzle to find the complete derivative.

Polynomial Derivatives

Polynomial derivatives involve finding the derivative of expressions where variables are raised to various powers. These expressions are known as polynomials, and the process of differentiation can reveal information about their rate of change. Knowing how derivatives affect polynomial expression is crucial, as polynomials form the backbone of many mathematical models.
Let’s consider the specific polynomial \(P(t) = -3 + 5t - 2t^2\):

• First, notice this has three terms of decreasing degrees: a constant, a linear term, and a quadratic term.


• Using the techniques of Linearity and Power Rule, each term can be differentiated: - The constant term's derivative is \(0\) because it does not change. - The linear term \(5t\) differentiates to \(5\) (a constant) because, as per Power Rule, the derivative of \(t\) is \(1\). - The quadratic term \(-2t^2\) becomes \(-4t\) upon differentiation.

Putting these results together gives the overall derivative \(5 - 4t\). Understanding polynomial derivatives also helps in analyzing the nature of the graph of a function, identifying turning points, and discovering optimization points in calculus.