Math 121 - Calculus for Biology I Spring Semester, 2004 Quotient Rule - Examples
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1. Differentiation - Quotient Rule
2. Graphing Rational Function
3. Genetic Control by Repression

Differentiation - Quotient Rule

Below are two functions for using the rules of differentiation. The second function relates to the continuous version of the logistic growth model.

Example 1: Differentiate the following functions:

Solution: The quotient rule is applied to each of these functions.

Graphing a Rational Function

Example 2: Consider the function:

Differentiate this function. Find all intercepts, asymptotes, and extrema. Graph the function.

Solution:

• The quotient rule for differentiation is applied to f(x) yielding

• The y-intercept is y = f(0) = -9/2
• The x-intercept is found by solving f(x) = 0
• This is solved by setting the numerator equal to zero

• The x-intercept is x = 3

Asymptotes

• The vertical asymptotes are found finding when the denominator is zero, so a vertical asymptote occurs at x = 2
• There are no horizontal asymptotes as the power of the numerator exceeds the power of the denominator.

Critical points

• Set the derivative equal to zero
• Set the numerator equal to zero

• The critical points are x_c = 1 and x_c = 3
• Evaluating the function f(x) at these critical points
  • Local maximum at (1, -4)
  • Local minimum at (3, 0)

Example 3:

• In 1960, Jacob and Monod won a Nobel prize for their theory of induction and repression in genetic control
• Many metabolic pathways in cells use endproduct repression of the gene or negative feedback to control important biochemical substances
• The biochemical kinetics of repression of a substance x satisfies a rate function

• Consider the specific rate function

• Differentiate this rate function
• Sketch a graph of this rate function and its derivative
• Find all intercepts, any asymptotes, and any extrema for the rate function and its derivative
• When is the rate function decreasing most rapidly?

Solution:

• The rate function has an R-intercept, R(0) = 90/27 = 10/3
• There is a horizontal asymptote of R = 0
• From the quotient rule, the derivative is

• For x > 0, the derivative of the rate function is negative (decreasing)
• There is clearly a maximum at x = 0
• A rate function doesn't make sense for x < 0
• Below is a graph of the rate function

• The derivative has an intercept at (0, 0)
• A horizontal asymptote R' = 0
• The derivative can be written

• The second derivative is

• This second derivative is zero when x = 3 (x = -3 is outside the domain)
• Thus, R' (x) has a minimum at (3, -5/12)
• Below is a graph of the derivative of the repression rate function.

• The original rate function is decreasing most rapidly at x = 3