Problem 37 - Calculus

Question

The value of a car, \(V=f(a),\) in thousands of dollars, is a function of the age of the car, \(a\), in years. (a) Interpret the statement \(f(5)=6\) (b) Sketch a possible graph of \(V\) against \(a\). Is \(f\) an increasing or decreasing function? Explain. (c) Explain the significance of the horizontal and vertical intercepts in terms of the value of the car.

Step-by-Step Solution

Answer

(a) The car is worth $6,000 at 5 years old. (b) Decreasing function; value decreases with age. (c) Horizontal intercept: when value is $0. Vertical intercept: initial car value.

Step 1: Interpret the Function Value

The statement \(f(5) = 6\) means that when the car is 5 years old, its value is \(6,000 (since the value is given in thousands of dollars). In other words, a 5-year-old car is estimated to be worth \)6,000.

Step 2: Sketching the Graph

To sketch the graph of the value function \(V = f(a)\), plot the age \(a\) on the x-axis and the value \(V\) on the y-axis. The point \((5, 6)\) indicates that at 5 years, the car's value is $6,000. Generally, for most cars, their value decreases as they get older, so the graph should slope downwards, showing a decreasing function.

Step 3: Analyze Increasing or Decreasing Nature of Function

The function \(f(a)\) is decreasing if the value of the car declines as the age \(a\) increases. Since cars usually depreciate over time, \(f(a)\) is expected to be a decreasing function, which means as cars get older, their values drop.

Step 4: Identify Horizontal Intercept

The horizontal intercept occurs where the value \(V\) reaches zero, which means the car has no monetary value left. This would happen if the car's age \(a\) were so high or the car's condition deteriorated to the point it has no resale value. However, this intercept's exact location requires more information.

Step 5: Identify Vertical Intercept

The vertical intercept occurs where the age \(a\) is zero, representing the car's initial value when new. This point \((0, V_0)\), where \(V_0\) is a positive value, shows the value of the car when it is brand new.

Key Concepts

Decreasing Function

A decreasing function in mathematics means that as you move from left to right along the x-axis on a graph, the y-values decline. For the problem at hand, the function describes the relationship between the age of a car, denoted by \(a\), and its value, \(V=f(a)\). Here, the car is worth less as it ages, which is typical because vehicles depreciate over time. When you interpret this scenario, a decreasing function shows that:

• As \(a\) increases, \(f(a)\) decreases, meaning the older the car, the less it is worth.
• The graph will slope downwards from left to right.

This trend is evident in the statement \(f(5) = 6\), indicating a 5-year-old car is valued at thousands less than when it was new. Understanding decreasing functions helps in many real-world situations where quantity declines over time, like depreciation.

Graph Interpretation

Graphs are visual tools that help us understand relationships between variables. In this context, the graph depicts the car's age on the x-axis and its value on the y-axis. As you look at the graph:

• Plot the initial point at (0, \(V_0\)) where \(V_0\) is the car's brand-new value.
• With \(f(5) = 6\), another significant point is (5, 6), marking the car's value five years later.
• Expect the graph to slope downwards, portraying the decreasing function typical of a declining car value.

By observing the graph, one can easily understand at which ages the car depreciates most, providing valuable insight into the depreciation timeline. It's a practical tool to predict future values and make informed choices regarding asset management.

Horizontal and Vertical Intercepts

Intercepts are crucial points on a graph that provide specific insights about the function. For the car value function \(V=f(a)\):

• The *horizontal intercept* occurs when \(V=0\). This means the car has no monetary value left. The location of this intercept would tell us when the car becomes valueless, but requires specific data to pinpoint.
• The *vertical intercept* is found where \(a=0\), indicating the value of the car when it was brand new. This point, (0, \(V_0\)), shows the maximum initial value. It's vital for understanding the starting point of depreciation over time.

Understanding these intercepts helps in scenarios like assessing the longevity of a car's value and determining initial investments. It's essential for both projecting future values and knowing the time frame in which the car retains worth.