Learning Outcomes

  1. Explain how the sign of the first derivative affects the shape of a function’s graph.
  2. Use the first derivative test for local extrema.
  3. Explain how the sign of the second derivative affects the shape of a function’s graph.
  4. Find the inflection points of a function.
  5. Use the second derivative test for local extrema.

Textbook

  • Chapter 4.5 ¬†Derivatives and the Shape of a Graph ¬†

Textbook Assignment

  • p. 403: ¬† 223-226 all, 229

WeBWork Assignment

  • Application-Monotonicity
  • Application-Shape of Polynomials

Exit problems of the session 

  1. For the functions f(x)=2x^3-3x^2-2:

(a). Determine the intervals of increase and decrease

(b). Determine the local maxima and minima

(c). Determine the intervals of concavity

(d). Determine the points of inflection 

(e). Sketch the graph with the above information indicated on the graph

 

Key Concepts

  • First derivative and the shape of a function:
    • If f'(x)>0 over an interval I, then f(x) increases over I.¬†
    • If f'(x)<0 over an interval I, then f(x) decreases over I.¬†
  • First Derivative Test for local extrema:
    • If c is a critical point of f(x) and f'(x)>0 for x<c and f'(x)<0 for x>c, then f(x) has a local maxima at c.¬†
    • If c is a critical point of f(x) and f'(x)<0 for x<c and f'(x)>0 for x>c, then f(x) has a local minima at c.¬†
  • Second derivative and the shape of a function:
    • If f''(x)>0 over an interval I, then f(x) is concave up over I.¬†
    • If f''(x)<0 over an interval I, then f(x) is concave down over I.¬†
  • Inflection point: if f(x) is continuous at c and f(x) changes concavity at c, the point (c, f(c)) is an inflection point of f(x).
  • Second Derivative Test for local extrema:
    • If f'(c)=0 and f''(c)>0, then f(x) has a local minima at c.
    • If f'(c)=0 and f''(c)<0, then f(x) has a local maxima at c.
 

Videos and Practice Problems of Selected Topics

  • 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)