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Chapter 9: Problem 68

SELLING PRICE OF DVD RECORDERS The rise of digital music and the improvement to the DVD format are some of the reasons why the average selling price of standalone DVD recorders will drop in the coming years. The function $$ A(t)=\frac{699}{(t+1)^{0.94}} \quad(0 \leq t \leq 5) $$ gives the projected average selling price (in dollars) of stand-alone DVD recorders in year \(t\), where \(t=0\) corresponds to the beginning of 2002 . How fast was the average selling price of standalone DVD recorders falling at the beginning of \(2002 ?\) How fast was it falling at the beginning of \(2006 ?\)

Short Answer

Expert verified
In summary, at the beginning of 2002, the average selling price of standalone DVD recorders was falling at a rate of approximately \(656.06\) dollars per year; while at the beginning of 2006, it was falling at a rate of approximately \(336.79\) dollars per year.

Step by step solution

01

Find the first derivative of A(t) with respect to t

We need to find the first derivative of the function \(A(t) = \frac{699}{(t+1)^{0.94}}\). To do this, we can apply the power rule along with the chain rule: $$ \frac{dA}{dt} = -0.94 \cdot \frac{699}{(t+1)^{1.94}} $$
02

Evaluate the first derivative at t=0

Now, we shall find the rate of change of average selling price at the beginning of 2002, which corresponds to \(t=0\). Plug in the value \(t=0\) to the first derivative expression we found in Step 1: $$ \frac{dA}{dt}(0) = -0.94 \cdot \frac{699}{(0+1)^{1.94}} = -0.94 \cdot 699 \approx -656.06 $$ So, at the beginning of 2002, the average selling price of standalone DVD recorders was falling at a rate of approximately \(656.06\) dollars per year.
03

Evaluate the first derivative at t=4

Similarly, find the rate of change of average selling price at the beginning of 2006, which corresponds to \(t=4\). Plug in the value \(t=4\) to the first derivative expression: $$ \frac{dA}{dt}(4) = -0.94 \cdot \frac{699}{(4+1)^{1.94}} = -0.94 \cdot \frac{699}{5^{1.94}} \approx -336.79 $$ So, at the beginning of 2006, the average selling price of standalone DVD recorders was falling at a rate of approximately \(336.79\) dollars per year. To sum up, the average selling price was falling at a rate of about \(656.06\) dollars per year at the beginning of 2002, and it was falling at a rate of about \(336.79\) dollars per year at the beginning of 2006.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivatives in Economics
In economics, understanding how variables change over time is essential. The concept of a derivative is a powerful tool to measure this rate of change. Put simply, a derivative in economics can tell us how a quantity, such as the selling price of a product, changes in response to a change in another variable, such as time.

Think of a derivative as the economic heartbeat: it measures the pulse of how fast or slow prices, costs, or revenues are moving. In our exercise, the derivative gives us the rate at which the average selling price of DVD recorders is falling yearly. It's like a snapshot that captures the speed of price decline at a specific moment, providing businesses with crucial information to make strategic decisions, like adjusting production or planning for future financial expectations.
Rate of Change Calculation
The rate of change in a given context refers to the speed at which a value is increasing or decreasing. Calculating it can reveal the dynamics behind the scene. For example, when a business is assessing its product pricing strategies, knowing how quickly prices are falling could lead to adjustments in marketing or sales techniques.

To calculate the rate of change, we use derivatives. In our example, the price of DVD recorders is given as a function of time, and we find the rate of change by taking the first derivative of that function. Economics often deals with non-linear relations, and derivatives shine here, providing precise information even when changes aren't constant. When we say the average selling price is dropping at approximately $656.06 per year in 2002, it's an actionable insight. The business can plan for a drop in revenue or adjust pricing strategies accordingly.
Application of Power Rule
The power rule is a basic yet invaluable tool in calculus, widely used across various fields, including economics. It simplifies the process of finding the derivative of functions that are power functions鈥攚here the variable has an exponent. The rule states that if you have a function like \( x^n \), its derivative is \( nx^{n-1} \).

For instance, with the DVD recorders鈥� price function \( A(t) = \frac{699}{(t+1)^{0.94}} \), using the power rule makes finding the rate of change straightforward. The negative sign indicates the price is decreasing over time. Adjusting a product's pricing strategy involves not just understanding that a price is dropping but also how rapidly that decrease is happening. The power rule helps us discern this quickly and accurately which is crucial for making timely business decisions.

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Most popular questions from this chapter

TELEVISION VIEWING The number of viewers of a television series introduced several years ago is approximated by the function $$ N(t)=(60+2 t)^{2 / 3} \quad(1 \leq t \leq 26) $$ where \(N(t)\) (measured in millions) denotes the number of weekly viewers of the series in the \(t\) heek. Find the rate of increase of the weekly audience at the end of week 2 and at the end of week \(12 .\) How many viewers were there in week 2 ? In week 24 ?

Find the derivative of each function. \(f(t)=\left(5 t^{3}+2 t^{2}-t+4\right)^{-3}\)

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Find \(\frac{d y}{d u^{\prime}} \frac{d u}{d x^{\prime}}\) and \(\frac{d y}{d x}\). \(y=2 u^{2}+1\) and \(u=x^{2}+1\)

CRUISE SHIP BookINGS The management of Cruise World, operators of Caribbean luxury cruises, expects that the percentage of young adults booking passage on their cruises in the years ahead will rise dramatically. They have constructed the following model, which gives the percentage of young adult passengers in year \(t\) : $$ p=f(t)=50\left(\frac{t^{2}+2 t+4}{t^{2}+4 t+8}\right) \quad(0 \leq t \leq 5) $$ Young adults normally pick shorter cruises and generally spend less on their passage. The following model gives an approximation of the average amount of money \(R\) (in dollars) spent per passenger on a cruise when the percentage of young adults is \(p\) : $$ R(p)=1000\left(\frac{p+4}{p+2}\right) $$ Find the rate at which the price of the average passage will be changing 2 yr from now.
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