Problem 3 - Thomas Calculus

Question

In Exercises 1–12, find the first and second derivatives. $$ s=5 t^{3}-3 t^{5} $$

Step-by-Step Solution

Answer

First derivative: \( 15t^2 - 15t^4 \); Second derivative: \( 30t - 60t^3 \).

Step 1: Identify the Function

The given function is defined as \( s(t) = 5t^3 - 3t^5 \). We need to find its first and second derivatives with respect to \( t \).

Step 2: Apply the Power Rule for the First Derivative

The power rule states that the derivative of \( t^n \) is \( n t^{n-1} \). Apply this to each term in the function:- The derivative of \( 5t^3 \) is \( 3 \times 5 t^{3-1} = 15t^2 \).- The derivative of \( -3t^5 \) is \( 5 \times (-3) t^{5-1} = -15t^4 \).So, the first derivative \( s'(t) \) is \( 15t^2 - 15t^4 \).

Step 3: Simplify the First Derivative (Optional)

The first derivative can be factored if necessary: \( s'(t) = 15t^2(1 - t^2) \). However, this step is only needed for further simplification or factorization.

Step 4: Apply the Power Rule for the Second Derivative

Differentiate \( s'(t) = 15t^2 - 15t^4 \) again using the power rule:- The derivative of \( 15t^2 \) is \( 2 \times 15t^{2-1} = 30t \).- The derivative of \( -15t^4 \) is \( 4 \times (-15)t^{4-1} = -60t^3 \).Thus, the second derivative \( s''(t) \) is \( 30t - 60t^3 \).

Step 5: Simplify the Second Derivative (Optional)

The second derivative can be factored if necessary: \( s''(t) = 30t(1 - 2t^2) \). Again, simplify only if desired for further factorization.

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus that simplifies finding derivatives of functions in polynomial form. It is particularly useful when dealing with terms raised to a power, making it an essential skill for calculus students.
The basic idea behind the power rule is straightforward: if you have a term of the form \( t^n \), where \( n \) is a constant, its derivative is \( n \times t^{n-1} \). Essentially, you bring the exponent down as a coefficient and subtract one from the exponent.

• For example, the derivative of \( 5t^3 \) becomes \( 3 \times 5t^{3-1} = 15t^2 \).
• Similarly, for \( -3t^5 \), it becomes \( 5 \times (-3)t^{5-1} = -15t^4 \).

The power rule applies to each term independently, making it efficient to use. If you're working with multiple terms, as often found in polynomials, you simply apply the rule to each term separately to find their respective derivatives.

First Derivative

The first derivative of a function represents the rate of change or slope of the function at any given point. It is essentially the ‘velocity’ of change for the variable.
In the context of the given function \( s(t) = 5t^3 - 3t^5 \), applying the power rule gives us the first derivative as \( s'(t) = 15t^2 - 15t^4 \). This equation tells us how the original function \( s(t) \) changes with respect to \( t \).

• The term \( 15t^2 \) indicates the change contributed by the original \( 5t^3 \).
• The term \( -15t^4 \) represents the change from \( -3t^5 \).

The first derivative can often be simplified for ease. For instance, factoring \( s'(t) \) gives \( 15t^2(1 - t^2) \), making it easier to analyze or solve specific problems, such as finding critical points or local extrema.

Second Derivative

The second derivative of a function describes how the rate of change itself is changing, often referred to as ‘acceleration’ in kinematic terms, but here it assesses the concavity of the graph.
Once the first derivative \( s'(t) = 15t^2 - 15t^4 \) is obtained, the power rule is again used to find the second derivative:

• The derivative of \( 15t^2 \) becomes \( 30t \).
• The derivative of \( -15t^4 \) is \( -60t^3 \).

These calculations give us the second derivative \( s''(t) = 30t - 60t^3 \). By examining \( s''(t) \), we can understand how the slope of \( s(t) \) itself changes and determine the concavity of the graph. Factorization yields \( 30t(1 - 2t^2) \), simplifying further analysis and visualization.Understanding the second derivative is crucial for identifying points of inflection, where the concavity changes, an important aspect of curve analysis.