Problem 53 - Calculus Single Variable

Question

Let \(k\) and \(c\) be positive constants. Suppose that a yeast population has mass \(m\) that satisfies the inequalities \(0

Step-by-Step Solution

Answer

The ordinate of the point of inflection is \( y = c \). The graph is concave up for \( m(t) < c \) and concave down for \( m(t) > c \).

Step 1: Write the Differential Equation

The given differential equation is \( m'(t) = k \cdot m(t) \cdot (2c - m(t)) \). We need to rewrite this in a form that allows us to analyze the concavity and the points of inflection of the graph.

Step 2: Complete the Square

The expression \( m(t) \cdot (2c - m(t)) \) is equivalent to \( - (m(t) - c)^2 + c^2 \), once completed as a square. This can be seen if you expand \( -(m(t) - c)^2 + c^2 = - (m^2(t) - 2cm(t) + c^2) + c^2 = -m^2(t) + 2c m(t) \).

Step 3: Rewrite Differential Equation

Substitute the completed square into the differential equation: \[ m'(t) = k \cdot \left(- (m(t) - c)^2 + c^2\right) \] or equivalently, \[ m'(t) = k \cdot (c^2 - (m(t) - c)^2) \] which shows the symmetry around \(m(t)=c\).

Step 4: Find the Inflection Point

For an inflection point, the second derivative must change sign. The maximum value of \(- (m(t) - c)^2 + c^2\) occurs at \(m(t)=c\) and this is where the concavity changes. Thus, the the ordinate of the point of inflection is \(y=c\).

Step 5: Determine the Concavity

Now, examine the expression \(- (m(t) - c)^2 + c^2\) to determine the concavity. For \(m(t) < c\), the expression is positive, indicating upward concavity (concave up). For \(m(t) > c\), the expression is negative, indicating downward concavity (concave down). Therefore, at \(m(t)=c\), the graph changes from concave up to concave down, proving it is an inflection point.

Key Concepts

Concavity

Concavity is a measure of how a curve bends. It tells us the direction in which a curve "opens". In the context of functions, a graph can be concave up or concave down.

• If a portion of the graph is concave up, it resembles the shape of a cup (\("\cup"\)), where the slope of the tangent line is increasing.
• Conversely, if it's concave down, it resembles an upside-down cup (\("\cap"\)), where the slope of the tangent line is decreasing.

To identify concavity on a graph of a function like \(y=m(t)\), we often look at the second derivative. Here, the second derivative of \(m(t)\), noted as \(m''(t)\), helps us see whether the function is bending upwards or downwards.In this exercise, once we completed the square, the concavity analysis tells us:

• For \(m(t) < c\), the function is concave up because the expression becomes positive. The tangents are getting steeper as \(t\) increases.
• For \(m(t) > c\), the function is concave down. The tangents decrease, and the curve is frowning.

Spotting where the concavity changes is crucial—it leads us to an important feature of the graph known as an inflection point.

Inflection Point

An inflection point is found where the curve changes from being concave up to concave down, or vice versa. It is a point on the graph where the concavity "switches" direction. Often, this means where the second derivative, \(m''(t)\), changes sign.To determine the inflection point for the function \(y = m(t)\), we used completing the square. This method revealed a symmetry in the graph around a specific value, \(m(t) = c\).

The Important Relationship

At \(m(t) = c\), the expression \(- (m(t) - c)^2 + c^2\) reaches its turning point. This is not just a coincidence; it signifies where the rate of change of the slope itself changes. Thus, \(m(t) = c\) becomes the ordinate of our inflection point.Why does this matter? Understanding where the concavity changes helps in sketching and analyzing the behavior of solutions to differential equations, which in turn informs us about real-world phenomena modeled by these equations.To recapitulate:

• For \(m(t) < c\), \(y = m(t)\) is concave up.
• For \(m(t) > c\), \(y = m(t)\) is concave down.
• The change at \(m(t) = c\) provides the exact "inflection point" where this transition occurs.

Completing the Square

Completing the square is a powerful algebraic technique used to simplify quadratic expressions, making it easier to analyze graphs. This method often illuminates important properties of the function.

Why Complete the Square?

By transforming the quadratic expression inside the differential equation, \(m(t) \cdot (2c - m(t))\), into the completed square form \(- (m(t) - c)^2 + c^2\), we clearly see its symmetry around \(m(t) = c\). This transformation helps us understand concavity and locate the inflection point.

Key Steps in Completing the Square

• Start with the standard quadratic form: \(ax^2 + bx + c\).
• Factor the quadratic term, and then adjust linear and constant term to form a perfect square trinomial.
• In our case, this was \(- (m(t) - c)^2 + c^2\), after reformatting \(m(t) \cdot (2c - m(t))\).

Here's why it simplifies understanding: The completed square easily shows us the maximum or minimum points of the curve. In differential equations, knowing this can be very advantageous for predicting behavior over time.Using this algebraic "trick," we peeled back layers of complexity to read more information off the equation, which then fed into our analysis of concavity and inflection points. It's a great example of how algebra aids in navigating calculus nuances.