AD - Curve Sketching Lesson
Curve Sketching
When sketching the graph of a function, it is rarely possible to show the entire graph. Deciding which part of the graph to display is crucial, whether the sketch is done by hand or with a grapher. The tools of differential calculus enable us to make an informed decision about which viewing window best reflects all of the interesting features of a function's graph.
View the presentation on graphing using derivatives.
The following guidelines outline steps for identifying key features of the graph of y = f(x).
1. Identify the domain and range of y = f(x).
2. Determine intercepts, symmetry, and asymptotes that may exist.
3. Find the derivatives f ' and f '' and locate the x-values for which f ' and f '' either are zero or do not exist.
4. Find critical numbers of f, if any, and identify the function's behavior relative to local and global extrema at each one.
5. Determine intervals of increasing and decreasing.
6. Find points of inflection, if any, and determine the concavity of the curve.
7. Sketch the curve that is consistent with the accumulated information.
Curve Sketching Practice
1. Find the x value(s) of any inflection point(s) for the cubic curve y = ax3 + bx2 + cx + d, a≠0.
Solution: x=−b3a
2. Describe the concavity and any extrema associated with the curve f(x) = 3x 1/2 - 1.
Solution: Always concave down and absolute (global) minimum at (0,-1)
The Graph of the Derivative
The graph of the first derivative, f'(x), on the interval [-2, 3] is provided below. Determine each of the following based on the graph.
1. State the x-values of the critical numbers for f(x).
Solution: x=-1, x=2; Critical values for f(x) will appear as x-intercepts on f'(x)
2. Determine if the critical values of f(x) are a relative maximum, relative minimum, or neither.
Solution: x=-1 is a relative min, x-2 is neither; f'(x) changes from negative to positive at x=-1 so it is a relative minimum. f'(x) is positive on both sides of x=2 so it is a critical number but not an extremum.
3. Determine the interval(s) where f(x) is increasing.
Solution: (-1,2) and (2,3); f'(x) is positive in the intervals (-1,2) and (2,3) which indicates that f(x) is increasing
4. State the x-values of the inflection points for f(x).
Solution: x=0, x=2; The inflection points of f(x) are the locations of the relative extrema for f'(x) or where the slope of a tangent line = 0.
5. Determine the interval(s) where f(x) is concave down.
Solution: (0,2); f'(x) is decreasing on (0,2) which indicates that f(x) is concave down.
Curve Sketching: Even More Problems!
Even More Problems! Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of curve sketching.
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