Problem 38 - Calculus for Biology and Medicine

Question

Let \(n\) and \(m\) be two positive integers with \(m \leq n .\) Answer the following questions about \(y=x^{-n}\) and \(y=x^{-m}\) for \(x>0\) : Where do the curves intersect? Which function is greater for small values of \(x ?\) for large values of \(x ?\)

Step-by-Step Solution

Answer

The curves intersect when \(m = n\). For small \(x\), \(x^{-m} > x^{-n}\) and for large \(x\), \(x^{-m} > x^{-n}\).

Step 1: Equate the Functions for Intersection Points

To determine where the curves intersect, set the two equations equal to each other. This means solving \(x^{-n} = x^{-m}\). This equation holds true when the exponents are equal, i.e., \(-n = -m\). Since \(m \leq n\), this implies that \(m = n\). Thus, the two curves intersect when \(m = n\). If \(m eq n\), the curves do not intersect.

Step 2: Analyze Behavior for Small Values of x

For small values of \(x\) (approaching zero), we need to determine which function is greater. Since both exponents are negative, a smaller \(x\) will generate a larger result. Given \(m \leq n\), for small \(x\), \(x^{-n} = x^{-|n|}\) produces larger values (since \(n\) is larger), thus \(x^{-m} > x^{-n}\).

Step 3: Analyze Behavior for Large Values of x

For large values of \(x\), both exponents being negative means the expressions are approaching zero. For large \(x\), a larger negative exponent results in a smaller value. Since \(m \leq n\), for large \(x\), \(x^{-n}\) is smaller than \(x^{-m}\). Hence, \(x^{-m} > x^{-n}\).

Key Concepts

Intersection Points

To find where two curves intersect, you need to equate their function expressions and solve for the variable. Here, we are dealing with the functions \(y = x^{-n}\) and \(y = x^{-m}\), where \(x > 0\). For the curves to intersect, their expressions must be equal, meaning \(x^{-n} = x^{-m}\).

This boils down to the exponents being the same; in mathematical terms, this is when \(-n = -m\). Given \(m \leq n\), the exponents equal each other only when \(m = n\). This comparison reveals that the curves intersect only when the values of \(m\) and \(n\) are identical.

If \(m eq n\), the curves will not intersect at any point because their rate of change will never match. This understanding simplifies graphs' analysis since knowing the curves do not intersect can often rule out certain configurations or simplify graph predictions.

Exponent Comparison

Analyzing functions with different exponents involves comparing the values these exponents produce under various circumstances. Let’s look at the behavior of \(x^{-n}\) and \(x^{-m}\) as \(x\) changes.

For small values of \(x\) (as \(x\) approaches zero), the role of the exponents is crucial. Since we are dealing with negative exponents, smaller values of \(x\) lead to larger outcomes because

• the negative exponents "flip" the relationship between size and magnitude,
• a more negative exponent results in a higher value.

Thus, for \( m \leq n \), \(x^{-m} > x^{-n}\) at small values of \(x\) as the contribution of the larger exponent \(-n\) is less dominating than \(-m\).

Consider this behavior as a key insight for varifying which function provides a higher result when \(x\) tends towards zero. It's a handy method to quickly judge function behaviors at specific points.

Curve Analysis

Understanding the shape and slope of curves, particularly when defined by exponents, is vital in calculus and graph analysis. The behavior of curves like \(y = x^{-n}\) and \(y = x^{-m}\) can noticeably change based on the value of \(x\).

When \(x\) takes on larger values, both expressions with negative exponents tend to zero. However, differences between \(x^{-n}\) and \(x^{-m}\) become significant based on the exponent:

• A larger negative exponent \(-n\) causes \(x^{-n}\) to decrease faster than \(x^{-m}\).
• Therefore, for large \(x\), \(x^{-m} > x^{-n}\).

This behavior illustrates how changing the exponents can modify the steepness and reach of a curve.

Analyzing these curve dynamics is instrumental in understanding function behavior over different intervals. Such insights not only allow you to build intuition around calculus functions but also facilitate their graphical interpretation, aiding in visualizing complex math relations.