Problem 91 - Calculus Volume 1

Question

Describe what the two expressions represent in terms of each of the given situations. Be sure to include units. a. \(\frac{f(x+h)-f(x)}{h}\) b. \(f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) \(C(x)\) denotes the total amount of money (in thousands of dollars) spent on concessions by \(x\) customers at an amusement park.

Step-by-Step Solution

Answer

a: Average change in spending per customer; b: Instantaneous change in spending per customer.

Step 1: Understanding Expression a: Difference Quotient

The expression \( \frac{f(x+h)-f(x)}{h} \) is known as the difference quotient and it represents the average rate of change of the function \( f(x) \) over the interval from \( x \) to \( x + h \). This can be interpreted as the average change in concession spending per additional customer when you increase the number of customers from \( x \) to \( x + h \). The units for this expression would be thousands of dollars per customer.

Step 2: Understanding Expression b: Derivative as a Limit

The expression \( f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \) defines the derivative of the function \( f(x) \) at the point \( x \). In the context of the concession function \( C(x) \), this represents the instantaneous rate of change in the amount of money spent on concessions with respect to the number of customers at exactly \( x \) customers. The units here are also thousands of dollars per customer.

Key Concepts

Difference Quotient

The difference quotient, expressed as \( \frac{f(x+h)-f(x)}{h} \), plays a crucial role in calculus. It is essentially used to find the average rate at which a function changes over a specific interval. To conceptualize this, imagine you are looking at the function \( f(x) \) between two points: \( x \) and \( x + h \).

• This formula computes how much the function \( f(x) \) changes as you move from \( x \) to \( x + h \).
• This gives you the average change rate over this interval.

In our exercise, this applies to the total amount of money, \( C(x) \), spent on concessions by a number of customers in thousands of dollars. Hence, the units are thousands of dollars per customer.

Average Rate of Change

The concept of the average rate of change often appears when studying functions over an interval. The formula \( \frac{f(x+h)-f(x)}{h} \) describes this rate. This helps measure how a function's value alters on average over a specified spectrum.

• It defines an average measure, akin to calculating the average speed using total distance and total time.
• In our given problem, it reflects how much spending changes per customer within that interval.

Understanding this gives students insight into how functions behave in incremental steps, foundational in calculus learning.

Derivative

The derivative is a powerful tool in calculus, defined as \( f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \). It paints a picture of the function's behavior at an exact point by taking the limit as the interval approaches zero.

• In everyday terms, the derivative describes the rate of change of a function at a particular moment, much like determining speed at an exact time.
• When applied to the concession function \( C(x) \), it specifies how spending increases per customer exactly at \( x \) customers.

The derivative thus turns the average rate into an instantaneous one by narrowing down the interval to the smallest conceivable size.

Instantaneous Rate of Change

The instantaneous rate of change bridges the gap between average rate and derivative, highlighting how rapidly a function shifts at a specific point. Think of it as pinpointing the exact speed of a car at a precise second rather than over a journey.

• This rate directly relates to the derivative of a function, showing the exact rate of change at an individual point.
• In the exercise context, it determines how swiftly the concession cost moves per customer at that precise count.

Such immediate insight is a gateway to understanding more complex motion and growth theories in calculus, making it essential for mastering change calculations.