Problem 26 - Calculus

Question

If \(f(t)=t-t^{2}\) find \(v(t)\) and \(f(3 t)\). Does the slope of \(f(3 t)\) equal \(v(3 t)\) or \(3 v(t)\) or \(3 v(3 t) ?\)

Step-by-Step Solution

Answer

The slope of \( f(3t) \) is equal to \( 3v(3t) \).

Step 1: Understanding the Function

The function provided is \( f(t) = t - t^2 \). It is a quadratic function, which is important for identifying the behavior of its slope, often referred to as velocity \( v(t) \).

Step 2: Calculating the Derivative to Find v(t)

To find \( v(t) \), which represents the slope of \( f(t) \), we need to find the derivative \( f'(t) \). Differentiating the function, \( f(t) = t - t^2 \), we get: \[ v(t) = f'(t) = 1 - 2t. \]

Step 3: Substituting into the Function for f(3t)

Now, calculate \( f(3t) \) by substituting \( 3t \) into the function \( f(t) \): \[ f(3t) = 3t - (3t)^2 = 3t - 9t^2. \]

Step 4: Finding the Derivative of f(3t)

To determine the slope of \( f(3t) \), find its derivative: \( f'(3t) \). Differentiating \( f(3t) = 3t - 9t^2 \) gives: \[ rac{d}{dt}[f(3t)] = 3 - 18t. \]

Step 5: Comparing Slopes: v(3t) and 3v(t)

First, calculate \( v(3t) \) by substituting \( 3t \) into \( v(t) \): \[ v(3t) = 1 - 2(3t) = 1 - 6t. \] Now, find \( 3v(t) \): \[ 3v(t) = 3(1 - 2t) = 3 - 6t. \]

Step 6: Analyzing the Comparison

Compare the derivatives calculated. The derivative of \( f(3t) \) is \( 3 - 18t \). While \( v(3t) = 1 - 6t \), and \( 3v(t) = 3 - 6t \). \( f'(3t) = 3 - 18t \) equals \( 3v(3t) = 3(1 - 6t) = 3 - 18t \), hence \( f'(3t) = 3v(3t) \).

Key Concepts

Derivative

In calculus, a derivative refers to the rate at which a function changes as its input changes. It’s an essential concept when analyzing the behavior and the slope of functions. For any function, its derivative represents the slope of the tangent line to the curve at any point.
To find the derivative of a function, you differentiate it. For example, if you have a function like \(f(t) = t - t^2\), you differentiate each term to find \(f'(t)\), which gives you the slope of the function at any point \(t\). The first derivative of this function is \(v(t) = 1 - 2t\), indicating how the slope of the curve changes with different values of \(t\).

• Constant Rule: The derivative of a constant is 0.
• Power Rule: If \(f(t) = t^n\), then \(f'(t) = n \cdot t^{n-1}\).

Mastering these rules helps in efficiently finding derivatives of various functions.

Quadratic Function

A quadratic function is a type of polynomial function that can be expressed in the form \(f(t) = at^2 + bt + c\). In our example, the function \(f(t) = t - t^2\) can be rewritten as \(-t^2 + t\).
Quadratic functions have a characteristic U-shaped graph known as a parabola. The coefficients \(a, b,\) and \(c\) determine the direction and shape of the parabola.

• Vertex: The turning point of the parabola, which can be found using \(t = -\frac{b}{2a}\).
• Direction: If \(a > 0\), the parabola opens upwards. If \(a < 0\), it opens downwards.
• Intercepts: Calculated by setting \(f(t) = 0\) and solving for \(t\), these determine where the graph crosses the axes.

Quadratic functions often appear in problems that involve optimization and physics scenarios like projectile motion or areas.

Function Substitution

Function substitution is a method where a specific value or expression replaces the variable within a function. This technique is particularly useful when working with composite or nested functions.
In our exercise, we substitute \(3t\) into the function \(f(t)\) to find \(f(3t)\). This means \(f(3t) = 3t - (3t)^2\), which simplifies to \(3t - 9t^2\).

• Purpose: Helps in simplifying functions to evaluate them under different scenarios.
• Common Substitutions: Adjusting the function for scaling or shifting, such as \(t\) replaced by \(kt\) for scaling by \(k\).

Substitution is a handy tool for manipulating functions to see how they behave under transformations or specific conditions.

Chain Rule

The chain rule is a fundamental differentiation rule in calculus used to differentiate composite functions. A composite function is a function formed by combining two functions, such that the output of one function becomes the input for another.
This rule states that if you have a function \(y = g(h(t))\), then the derivative \(\frac{dy}{dt}\) can be found by \(\frac{dy}{dh} \times \frac{dh}{dt}\).
In our example, to differentiate \(f(3t)\), you would multiply the derivative of the outer function by the derivative of the inner function. Hence, for \(f(3t)\), which is \(3t - 9t^2\), differentiation gives a result of \(3 - 18t\), following the chain rule.

• Purpose: Simplifies the process of finding the derivative of complex functions.
• Application: Widely used in higher calculus for more complex function analysis.

Mastering the chain rule enables seamless differentiation of nested functions, making it a powerful tool for tackling complex calculus problems.