Question

Credit Card Rate The average annual rate \(r\) (in percent form) for commercial bank credit cards from 2000 through 2005 can be modeled by \(r=\sqrt{-1.7409 t^{4}+18.070 t^{3}-52.68 t^{2}+10.9 t+249}\) where \(t\) represents the year, with \(t=0\) corresponding to 2000\. (a) Find the derivative of this model. Which differentiation rule(s) did you use? (b) Use a graphing utility to graph the derivative on the interval \(0 \leq t \leq 5\). (c) Use the trace feature to find the years during which the finance rate was changing the most. (d) Use the trace feature to find the years during which the finance rate was changing the least.

Short answer

The derivative of the model is \[r\ '(t)= \frac{1}{2}(-1.7409 t^{4}+18.070 t^{3}-52.68 t^{2}+10.9 t+249)^{-\frac{1}{2}}*(-6.9636 t^{3}+54.21 t^{2}-105.36 t+10.9)\] The graph of this derivative gives the years during which the finance rate was changing the most and least.

Step-by-step solution

Find the derivative

Differentiation of the function \(r(t)= \sqrt{-1.7409 t^{4}+18.070 t^{3}-52.68 t^{2}+10.9 t+249}\) involves using chain rule and power rule. Using chain rule on the outer function first \(f(g(x))=g'(x)f(g(x))\), then power rule \(nx^{n-1}\) for the inner function, we get: \[r\ '(t)= \frac{1}{2}(-1.7409 t^{4}+18.070 t^{3}-52.68 t^{2}+10.9 t+249)^{-\frac{1}{2}}*(-6.9636 t^{3}+54.21 t^{2}-105.36 t+10.9)\]

Graph the derivative

Using a graphing utility (like Desmos or a graphing calculator), graph the derivative function \[r\ '(t)= \frac{1}{2}(-1.7409 t^{4}+18.070 t^{3}-52.68 t^{2}+10.9 t+249)^{-\frac{1}{2}}*(-6.9636 t^{3}+54.21 t^{2}-105.36 t+10.9)\] on the interval \(0 \leq t \leq 5\). This will help visually represent how the finance rate changes over time.

Find years of maximum change

Using the trace feature of the graphing tool, hover over the highest and lowest points on the graph (These are points where the derivative function will achieve its maximum and minimum values). The respective x-values correspond to the years (with 0 corresponding to 2000) during which the finance rate was changing the most.

Find years of minimum change

Using the trace feature of the graphing tool, hover over the flat parts of the graph. The respective x-values correspond to the years (with 0 corresponding to 2000) during which the finance rate was changing the least.

Key concepts

Derivative of a Function

The derivative of a function is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. In simple terms, it tells us the rate at which one quantity is changing with respect to another. For instance, in a credit card rate problem, if the annual rate is given by a function in terms of time, the derivative of that function would represent the rate of change of the credit card's interest rate with respect to time.

In the given exercise, the derivative of the rate function provides valuable financial insights. It can indicate periods when the interest rates are increasing or decreasing rapidly, which could influence consumer decisions. To find the derivative of a complex function like this, we apply rules of differentiation such as the chain rule and power rule.

In practice, the derivative is also used for optimization problems, where businesses might want to find the maximum profit or minimum cost by determining where the derivative equals zero. Understanding the derivative is thus not only important for mathematics but also for making informed decisions in finance and economics.

Chain Rule

The chain rule is a powerful tool in calculus used to differentiate compositions of functions. It's akin to a machine with two gears, where the rotation of one gear causes the second gear to rotate as well. When we have a function inside another function, just like having a gear inside another, we apply the chain rule to find the derivative.

For a composition of two functions, denoted as \(f(g(x))\), the chain rule states that the derivative is the derivative of the outer function \(f\) with respect to its inner function \(g\), multiplied by the derivative of the inner function \(g\) with respect to \(x\). In mathematical terms, this is expressed as \(f'(g(x))\cdot g'(x)\). The exercise provided is a classic example where we use the chain rule to differentiate the square root of a polynomial.

In real-world scenarios, the chain rule allows us to model and analyze situations where one action affects another in a sequence, such as the impact of changing interest rates on loan payments or the effect of supply changes on pricing.

Graphing Derivatives

Graphing derivatives serves as a visual interpretation of the derivative's behavior, giving insight into the function's rate of change at any given point. It's similar to looking at a speedometer that shows how quickly a car's speed changes over time.

In our credit card rate model, graphing \(r'(t)\) helps us see not just numerical values but also trends and patterns in the changing finance rates. Graphs can show the points where the rate is increasing or decreasing fastest, indicating critical moments that might need further analysis or action. In the educational context, using graphing utilities like Desmos or a graphing calculator can make the concepts more tangible, improving student understanding and engagement.

Critical thinking can be enhanced by analyzing the graph for intervals of increase and decrease, curvature, and understanding concavity, which ties back to key concepts in calculus such as inflection points and optimization.

Finance Rate Change

When graphed, the derivative of the finance rate function allows us to identify periods of significant change in the finance rate. These changes can be critical for financial forecasting and decision-making.

With the rates modeled by a function of time, the steepest parts of the derivative's graph represent the years with the greatest rate of change. A steep upward slope indicates a rapid increase in rates, which could signal an aggressive monetary policy or a response to inflation. Conversely, a steep downward slope might suggest a potential easing of rates, creating different investment opportunities.

Understanding these rate changes is crucial for consumers considering new credit or for businesses planning their financial strategies. By interpreting the derivative graph and using features like the trace function, we can pinpoint exact periods of maximum and minimum rate change, giving individuals and organizations a quantitative basis for their financial decisions.