In the wake of the 2008 financial crisis, bank failures...

In the wake of the 2008 financial crisis, bank failures started spiraling upward. The number of bank failures peaked in 2010 at 157 . Thereafter, the number of bank failures began to fall sharply. The number of bank failures from 2010 through 2012 is described by the function $$f(t)=157 e^{-0.55 t} \quad(0 \leq t \leq 2)$$ where $t$ is measured in years, with $t=0$ corresponding to 2010 . a. How fast was the number of bank failures dropping in $2011 ?$ b. If the trend continued, how many bank failures were there in $2013 ?$

In the wake of the 2008 financial crisis, bank failures started spiraling upward. The number of bank failures peaked in 2010 at 157 . Thereafter, the number of bank failures began to fall sharply. The number of bank failures from 2010 through 2012 is described by the function $$f(t)=157 e^{-0.55 t} \quad(0 \leq t \leq 2)$$ where $t$ is measured in years, with $t=0$ corresponding to 2010 .
a. How fast was the number of bank failures dropping in $2011 ?$
b. If the trend continued, how many bank failures were there in $2013 ?$

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Key Concepts

Exponential Decay

Exponential decay describes processes where the rate of decrease of a quantity is proportional to its current value. In modeling, this concept is represented by functions of the form f(t) = A·e^(kt) with k being a negative constant, which captures the idea that the decline is steep initially and then slows down over time.

Instantaneous Rate of Change

The instantaneous rate of change is the derivative of a function at a specific point, indicating how fast the function’s value is changing at that exact moment. This concept is central to calculus and is used to determine precise rates of decrease or increase in dynamic systems.

Chain Rule

The chain rule is a fundamental tool in differentiation used for computing the derivative of a composite function. It allows one to differentiate functions where one function is nested within another, and it is particularly useful when working with exponential decay models that involve the exponential of a linear term.

Extrapolation in Mathematical Modeling

Extrapolation refers to extending a model beyond its original domain to make predictions about future or unobserved values. This concept assumes that the underlying trend continues outside the range of the observed data, making it possible to estimate outcomes at points not directly measured.

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