Tasks: Graphical Calculus Mastery – Mathematics for Business

1 Problem 1: Polynomial Function (x)

Given the graph of \(f(x)\) below, sketch the graph of \(f'(x)\).

Given the graph of \(f(x)\), sketch \(f'(x)\) and identify all critical points.

Sketch \(f'(x)\) for the piecewise linear function shown. Where does \(f'(x)\) not exist?

Sketch \(f'(x)\) for \(f(x) = |x - 2|\) on the interval \([-1, 5]\).

Sketch \(f'(x)\) and \(f''(x)\) for the quartic function shown.

For each function graph below, sketch \(f'(x)\). Identify critical points.

Given the graph of \(f'(x)\) below:

Given \(f'(x)\) shown below:

Given the step function \(f'(x)\) below, sketch \(f(x)\).

For each derivative graph, answer: Where is \(f\) increasing? Where does \(f\) have local extrema?

Given \(f(x) = x^4 - 4x^3 + 4x^2\):

Find \(f'(x)\) and \(f''(x)\).
Find all critical points and classify them.
Find all inflection points.
Determine intervals where \(f\) is increasing/decreasing and concave up/down.
Sketch the graphs of \(f(x)\), \(f'(x)\), and \(f''(x)\) on the same set of axes.

A company’s revenue over 10 months is modeled by: \[R(t) = -t^3 + 9t^2 - 15t + 50\]

where \(t\) is months and \(R\) is in thousands €.

Consider the piecewise function:

\[f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 3 - x & \text{if } x \geq 1 \end{cases}\]