Polynomial Derivative Sign Analysis
Consider the function \( f(x)= x^3 – 3x^2 + 2x \). Determine the intervals on which the function is increasing and decreasing. Provide your answer using inequalities derived from the sign of \( f'(x) \).
Graphical Comparison of Function Growth Rates
Consider the functions \(f(x) = 4x + 2\) and \(g(x) = x^2 + 1\). Using graphing technology, determine the x-value at which the instantaneous rate of change (slope) of \(g(x)\) equals the constant rate of change of \(f(x)\). Explain briefly how you would use the graphical intersection of slopes to verify your result.
Inferring Differential Equations from Slope Field Patterns
A slope field is described as follows: At every point (x, y), the small line segment indicates a slope that depends solely on the y-coordinate. In other words, regardless of the value of x, the slope is determined entirely by y, and appears to be directly proportional to y. Based on this description, write a […]