2. Curve Sketching
  3. Curve Sketching and the First Derivative

Exercises


  1. Given the graph of \( y = f(x) \), state the intervals on which \( f(x) \) is increasing and the intervals on which \( f(x) \) is decreasing.

    1. Graph of quartic opening down with turning points at x=-2,-1,0

    2. Graph of cubic with negative leading coefficient with triple root at x=1

    3. Graph of -1/(x + 1)

    4. Piecewise function; parabola opening down with vertex at x=-0.5 for x less than 1.5; linear with negative slope elsewhere
  2. For each of the following, find all values of \( x \) for which \( f'(x) = 0 \).
    1. \( f(x) = \tfrac{3}{2}x^2 - 2x \)
    2. \( f(x) = x^3 + 3x^2 - 9x + 2 \)
    3. \( f(x) = (x^2 - 3)(x + 4) \)
    4. \( f(x) = \dfrac{x^2 - 1}{\sqrt{x - 1}} \)
  3. Suppose that \( g'(x) = x^2 - x - 6 \). Find the interval(s) where \( g(x) \) is increasing, and where \( g(x) \) is decreasing.
  4. Find the interval(s) where \( f(x) = -x^3 + 18x^2 - 105x + 4 \) is increasing.
  5. Use graphing technology to graph the function \( f(x) = x^3 - 3x + 1 \).  Use the graph to determine the intervals of increase and decrease of \( f(x) \). Verify your answer using algebraic methods.
  6. Let \( f(x) = \dfrac{x + 4}{x + 7} \). Find the intervals over which the function is increasing and decreasing.
  7. Find the intervals on which the function \( f(t) = t^4 - 10t^2 + 9 \) is decreasing.
    1. Find the largest value of \( A \) such that the function \( g(s) = s^3 - 3s^2 - 24s + 1 \) is increasing on the interval \( (-5, A) \).
    2. Find the largest value of \( A \) such that \( h(s) = \dfrac{1}{(s - 9)^4} \) is increasing for all \( s \) in the interval \( (-\infty, A) \).
  9. Suppose \( \dfrac{du}{dx} = (x^2 + 1)(x - 3)(x - 1)(x + 5) \). Find all intervals where \( u(x) \) is increasing.
  10. Find the intervals on which the function \( f(x) = (x^2 - 9)^{\frac{2}{3}} \) is increasing and the intervals on which it is decreasing.
  11. Sketch a graph of a differentiable function \( f(x) \) that has the following properties:
    • \( f'(-2) = f'(3) = 0 \)
    • \( f'(x) \gt 0 \) for \( -2 \lt x \lt 3 \) and \( x \gt 3 \)
    • \( f'(x) \lt 0 \) for \( x \lt -2 \)
    • The points \( (-2, -1) \) and \( (3, 2) \) lie on the graph of \( f \)
  12. Find an equation of a cubic polynomial \( g(x) \) (degree \( 3 \)) that is increasing on the intervals \( ( -\infty, -1 ) \) and \( ( 2, \infty ) \), decreasing on the interval \( ( -1, 2 ) \), and satisfies \( g(-1) = 13 \) and \( g(2) = -14 \). Verify your answer using graphing technology.