Derivatives and the Shape of a Graph

  • The first derivative test
  1. $\rhd$ The first derivative test (5:24) A discussion on finding relative extrema using derivatives.
  2. $\rhd$ Finding relative extrema (7:48) Find the relative maximum point of $g(x)=x^4-x^5$.
  3. $\rhd$ Analyzing mistakes when finding extrema I (4:32) Pamela was asked to find where $h(x) = x^3-6x^2+12x$ has a relative extremum.  Is her work correct?
  4. $\rhd$ Analyzing mistakes when finding extrema II (4:41) Erin was asked to find if $f(x)=(x^-1)^{2/3}$ has a relative maximum.  Is her work correct?
  5. * Finding relative extrema A text with detailed examples and questions.
  6. * Practice: Finding relative extrema.
  7. * Relative minima and maxima review A text with detailed examples and questions.
  • Determining concavity of intervals and inflection points: graphical
  1. $\rhd$ Concavity introduction (9:53) Finding where the function is concave up/down using derivatives.
  2. $\rhd$ Analyzing concavity graphically (2:22) A function $f(x)$ is plotted. Highlight were $f'(x)>0$ and $f”(x)<0$.
  3. * Practice: Concavity intro. (4 problems)
  4. $\rhd$ Inflection points (2:33)  A discussion based on a graph of $f(x)$, $f'(x)$ and $f”(x)$.
  5. $\rhd$ Inflection points (graphical) (3:20) The graph of a differentiable function $g(x)$ over $[-4,4]$ is given. How many inflection points does the graph of $g$ have?
  6. * Practice: Inflection points. (4 problems)
  • Determining concavity of intervals and inflection points: algebraic
  1. $\rhd$ Analyzing concavity (9:15) Find the intervals where $g(x)=-x^4+6x^2-2x-3$ is concave up/down.
  2. $\rhd$ Inflection points (5:34) Find the points of inflection of $g(x)=\dfrac{1}{4}x^4-4x^3+24x^2$.
  3. $\rhd$ Mistakes when finding inflection points (6:10) Robert was asked to find where $g(x)=\sqrt[3]x$ has inflection points. Is his work correct?
  4. $\rhd$ Mistakes when finding inflection points (4:01) Olga was asked to find where $g(x)=(x-2)^4$ has inflection points. Is her work correct?
  5. * Analyzing the second derivative to find inflection points. A review with detailed examples and questions.
  6. * Practice: Analyze concavity. (4 problems)
  7. * Practice: Find inflection points. (4 problems)
  • Using the second derivative test to find extrema
  1. $\rhd$ The second derivative test (6:12) A discussion on the second derivative to find relative extrema.
  2. * Practice: The second derivative test. (4 problems)