A county realty group estimates that the number of housing...

A county realty group estimates that the number of housing starts per year over the next three years will be
$$H(r)=\frac{300}{1+0.03 r^{2}}$$
where $r$ is the mortgage rate (in percent).
a. Where is $H(r)$ increasing?
b. Where is $H(r)$ decreasing?

Key Concepts

Derivative and Rate of Change

The derivative is a fundamental concept in calculus that measures the instantaneous rate of change of a function. By analyzing the sign of the derivative, one can determine where the function increases (positive derivative) or decreases (negative derivative). This concept is used to identify intervals of monotonic behavior in functions.

Monotonicity

Monotonicity refers to the property of a function being entirely non-decreasing or non-increasing over an interval. Using the first derivative test, which involves checking whether the derivative is positive or negative, we can determine the intervals over which the function is monotonically increasing or decreasing.

Application of Calculus to Function Analysis

Calculus provides tools such as differentiation to analyze and predict the behavior of functions in various contexts. By applying these techniques, one can draw conclusions about relationships in mathematical models, which is essential for solving problems in real-world scenarios.

Differentiation of Rational Functions

Rational functions, which are expressed as the quotient of two polynomials, often require the use of differentiation rules such as the quotient rule and the chain rule. Mastery of these techniques is crucial for effectively analyzing the behavior of rational functions with respect to changes in their variables.

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