Problem 52 - Calculus

Question

For \(f(t)=v t+C\) find the formulas and slopes of a) \(3 f(t)+1\) (b) \(f(3 t+1)\) (c) \(2 f(4 t)\) d) \(f(-t)\) (e) \(f(t)-f(0)\) (f) \(f(f(t))\)

Step-by-Step Solution

Answer

(a) 3v; (b) 3v; (c) 8v; (d) -v; (e) v; (f) v^2.

Step 1: Analyze the original function

The function given is \(f(t) = v t + C\). Here, \(v\) represents the slope of the function, as it is the coefficient of \(t\), and \(C\) is the y-intercept.

Step 2: Solve for (a) 3f(t) + 1

The expression is \(3f(t) + 1 = 3(v t + C) + 1 = 3v t + 3C + 1\). The slope of this function is \(3v\).

Step 3: Solve for (b) f(3t + 1)

Substitute \(3t + 1\) into \(f(t)\): \(f(3t+1) = v(3t+1) + C = 3vt + v + C\). The slope of this function is \(3v\).

Step 4: Solve for (c) 2f(4t)

Substitute \(4t\) into \(f(t)\) and scale by 2: \(2f(4t) = 2(v(4t) + C) = 8vt + 2C\). The slope of this function is \(8v\).

Step 5: Solve for (d) f(-t)

Substitute \(-t\) into \(f(t)\): \(f(-t) = v(-t) + C = -vt + C\). The slope of this function is \(-v\).

Step 6: Solve for (e) f(t) - f(0)

Calculate \(f(0)\): \(f(0) = v(0) + C = C\). Then, \(f(t) - f(0) = vt + C - C = vt\). The slope of this function is \(v\).

Step 7: Solve for (f) f(f(t))

Substitute \(f(t)\) into \(f(t)\): \(f(f(t)) = f(vt + C) = v(vt + C) + C = v^2t + vC + C\). The slope of this function is \(v^2\).

Key Concepts

Function Transformations

Understanding function transformations is key when working with linear functions like \(f(t) = vt + C\). Transformation involves altering the function's appearance by changing certain parameters. Typical transformations include:

• **Vertical Scaling**: Changing the coefficient directly multiplying a function, like \(3f(t)\), results in vertical stretching or compressing depending on whether the scalar is greater or less than 1.
• **Horizontal Scaling**: Altering the input variable, like \(f(3t)\), compresses or extends the function along the t-axis.
• **Translation**: Adding or subtracting a constant, such as in \(f(t) + C\), shifts the function vertically or horizontally.
• **Reflection**: Using a negative sign in the input, such as in \(f(-t)\), reflects the function across an axis.

These transformations are simple yet powerful tools in modifying linear functions to achieve desired properties.

Slope

The slope is a fundamental component of any linear function, defining its steepness. In the linear equation \(f(t) = vt + C\), \(v\) is the slope.Understanding slope is crucial, as it:

• Indicates how fast or slow the function changes with \(t\).
• Reflects whether the function is rising or falling. A positive \(v\) means rising from left to right, while a negative value indicates a falling function.
• Helps in comparing different linear functions, making it easy to see which function increases or decreases more rapidly.

By manipulating the slope through transformations, you can adjust the linear function to fit differing scenarios, such as increasing the steepness or reversing its direction.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis, which happens when \(t = 0\). For the linear function \(f(t) = vt + C\), the y-intercept is \(C\).In contexts such as graphing or real-world problem solving, the y-intercept provides:

• A starting value of the function when the independent variable is zero.
• The initial condition of a situation modeled by the equation.
• A fixed point that helps determine the entire line along with the slope.

Adjusting the y-intercept through transformations, such as adding a constant to the function, shifts the graph up or down, affecting how the function is positioned on a graph.

Composition of Functions

The composition of functions involves inserting one function into another. For instance, finding \(f(f(t))\) means replacing \(t\) in the function \(f(t) = vt + C\) with \(f(t)\) itself.This mathematical operation gives rise to:

• New functions that can help solve complex problems by building on simpler components.
• Insight into sequences of operations, where the output of one function becomes the input of another.
• Opportunities to explore limits and continuity in advanced mathematical contexts.

In our example, the composition yields \(f(f(t)) = v^2t + vC + C\), illustrating how the slope and intercept are modified when functions are composed.