Problem 64 - Calculus and its applications

Question

A company determines that the value of an investment after \(t\) years is \(V\), in millions of dollars, where \(V\) is given by $$V(t)=5 t^{3}-30 t^{2}+45 t+5 \sqrt{t}$$ Note: Calculators often use only the variables \(y\) and \(x,\) so you may need to change the variables. a) Graph \(V\) over the interval [0,5] . b) Find the equation of the secant line passing through the points \((1, V(1))\) and \((5, V(5))\). Then graph this secant line using the same axes as in part (a). c) Find the average rate of change of the investment between year 1 and year 5 d) Repeat parts (b) and (c) for the following pairs of points: \((1, V(1))\) and \((4, V(4)) ;(1, V(1))\) and \((3, V(3)) ;(1, V(1))\) and \((1.5, V(1.5))\) e) What appears to be the slope of the tangent line to the graph at the point \((1, V(1)) ?\) f) Approximate the rate at which the value of the investment is changing at \(t=1\) yr.

Step-by-Step Solution

Answer

Graph the function and secant lines, calculate average and instantaneous changes using slopes; the instantaneous rate at \(t=1\) is approximately given by \(V'(1)\).

Step 1: Graph V(t) over the interval [0,5]

To graph the function, substitute different values of \(t\) into the equation \(V(t) = 5t^3 - 30t^2 + 45t + 5\sqrt{t}\) for \( t \in [0,5] \) and plot the points. A visual tool or graphing calculator may help in plotting the curve accurately over this interval.

Step 2: Equation of the Secant Line

To find the equation of the secant line passing through points \((1, V(1))\) and \((5, V(5))\), first calculate \(V(1)\) and \(V(5)\) by substituting into the expression for \(V(t)\). Then, the slope \(m\) of the secant line is given by \(m = \frac{V(5) - V(1)}{5 - 1}\). The equation is \(y = m(x - 1) + V(1)\). Graph this line along with the function from Step 1.

Step 3: Average Rate of Change from Year 1 to Year 5

The average rate of change is simply the slope calculated in Step 2, which was \(m = \frac{V(5) - V(1)}{5 - 1}\).

Step 4: Repeat Calculations for Different Pairs

For each specified pair of points, calculate \(V(t_1)\) and \(V(t_2)\), then find the slope \(m = \frac{V(t_2) - V(t_1)}{t_2 - t_1}\), and write the equation of the secant line as \(y = m(x - t_1) + V(t_1)\). Pairs to consider are \((1, V(1))\) and \((4, V(4))\), \((1, V(1))\) and \((3, V(3))\), and \((1, V(1))\) and \((1.5, V(1.5))\). Graph these lines over the curve for visual comparison.

Step 5: Determine Slope of Tangent Line at t=1

To approximate the slope of the tangent line at \(t = 1\), look at the secant line slopes from Step 4 as \(t_2\) gets closer to 1. These approximate the derivative \(V'(1)\). If done manually, we find \(V'(1)\) by differentiating \(V(t)\) and evaluating at \(t = 1\).

Step 6: Approximate Rate of Change at t=1

The approximate rate of change, effectively \(V'(1)\), is derived from Step 5. It matches the theoretical instantaneous rate of change, which is calculated using calculus derivative techniques, confirming the tangent slope value at \(t = 1\).

Key Concepts

Graphing Functions

When graphing functions, particularly complex ones like the investment value function given by \( V(t) = 5t^3 - 30t^2 + 45t + 5\sqrt{t} \), it's important to accurately represent the behavior of the function over a specific interval. In our case, this interval is \([0, 5]\).

• To start, substitute different values of \( t \) into the function to calculate corresponding \( V(t) \) values. This involves computing the values for both polynomial and square root components.
• Once values are obtained, plot these calculated points on a graph. Join points to trace the curve of the function.
• Using a graphing calculator or software may improve accuracy, as it can smoothly plot complex curves and resolve intricate changes in direction or concavity.

These aids help identify key features such as where the function increases or decreases, and any local minima or maxima within the plotted interval. Graphing lets us visually analyze how the investment value fluctuates over time.

Secant Line Equation

The secant line provides an average slope between two points on a curve. For the investment function, the secant line between two given points, such as \((1, V(1))\) and \((5, V(5))\), represents the average change in investment value over that interval. To determine the equation:

• Calculate \( V(1) \) and \( V(5) \) by substituting these respective \( t \) values into the function \( V(t) \).
• The slope \( m \) of the secant line is expressed as \( m = \frac{V(5) - V(1)}{5 - 1} \).
• Using this slope, the equation of the line becomes \( y = m(x - 1) + V(1) \), which is simply the point-slope form of the line equation.

Graphing this secant line alongside \( V(t) \) helps visualize how it averages the rate of change between these points, enabling a deeper understanding of growth trends over the time period.

Average Rate of Change

The average rate of change of a function over an interval provides insight into how a quantity transitions between two points in time. For our function, this concept is crucial to understand the investment's behavior. Conceptually, the average rate from year 1 to year 5 is articulated as the slope of the secant line connecting \((1, V(1))\) and \((5, V(5))\):

• Compute it using the formula \( \text{Average Rate} = \frac{V(5) - V(1)}{5 - 1} \).
• This rate signifies the mean change per year in investment value from the start of year 1 to end of year 5.

Averaging provides a simplified view of change, effectively smoothing out any nuanced fluctuations in the value function, yet giving a robust indicator of overall performance.

Tangent Line Slope

Approximating the slope of a tangent line at a specific point on a curve is akin to finding the instant rate of change at that point. It provides valuable insight into exactly how steep the curve is, indicating the immediate pace at which the investment's value is changing.For point \((1, V(1))\), this involves:

• Looking at the slopes of secant lines as they come increasingly close to touching the graph at \(t = 1\). As intervals between points narrow (e.g., moving from \((1, V(1))\) to \((1.5, V(1.5))\)), the secant slopes better approximate the tangent slope.
• To pinpoint the exact tangent slope, we would use calculus techniques, namely, differentiating \(V(t)\) to get \(V'(t)\) and substituting \(t = 1\) into \(V'(t)\).

The tangent slope at \(t=1\) provides the rate at which the investment's value is instantaneously changing, crucial for understanding short-term behavior.