Calculus Section 3.6 A Summary of Curve Sketching
-Determine the important values to find and use while analyzing a graph and its derivatives.
Domain
What the _______________ can be for a function.
Range
What the _______________ can be for a function.
X-values of discontinuity
Where the function is undefined, ___________ in the graph.
X-value of nondifferentiability
__________ turns, ___________ tangent lines, points of discontinuity.
X- and Y- intercepts
x-intercept is where ____________. y-intercept is where _____________.
Vertical Asymptotes
lim f ( x)  _______, where the denominator equals ___________.
x c
Horizontal Asymptotes
lim f ( x)  L
x 
Critical Numbers
Where f '( x) = ________ or undefined.
Increasing/Decreasing
Increasing when f '( x) =___________, decreasing when f '( x) = ____________.
Relative Extrema
Can only occur at __________________________. Relative maximum when you switch from ______________
to _______________. Relative minimum when you switch from _________________ to ________________.
Alternatively, you can say that a minimum occurs when the graph of f '( x) goes from negative to positive. A
maximum occurs when the graph of f '( x) goes from positive to negative (1st derivative test).
Also, a maximum occurs when f ''( x) of a critical number is ______________. A minimum occurs when f ''( x)
of a critical number is ________________ (2nd derivative test).
Concavity
The curviness of a line. A function is ________________ if f '' is positive, and a function is ________________
if f '' is negative. If the graph of f '( x) is increasing, then f(x) is concave __________. If the graph of f '( x) is
decreasing, then f(x) is concave __________.
Inflection Points
Where the ___________________ of a function changes. Found by setting ___________ equal to _________.
Can be found on the graph of f ( x) where f ( x) switches curviness from concave up to concave down (or vice
versa). Inflection points are found on the graph of f '( x) at its _______________________________.
The graph of a function f(x) is given. For f(x), what are the critical points, intervals of increasing/decreasing,
what are its relative maximums/minimums, its points of inflection, and where is it concave up/down?
The graph of a function f '( x) is given. For f(x), what are the critical points, intervals of increasing/decreasing,
what are its relative maximums/minimums, its points of inflection, and where is it concave up/down?