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- https://math.libretexts.org/Courses/City_University_of_New_York/Calculus_I_(CUNY)/04%3A_Applications_of_Derivatives/4.06%3A_Derivatives_and_the_Shape_of_a_GraphUsing the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the sec...Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
- https://math.libretexts.org/Courses/Mission_College/Math_3A%3A_Calculus_I_(Reed)/04%3A_Applications_of_Derivatives/4.04%3A_Derivatives_and_the_Shape_of_a_GraphUsing the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the sec...Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
- https://math.libretexts.org/Courses/Penn_State_University_Greater_Allegheny/Math_140%3A_Calculus_1_(Gaydos)/04%3A_Applications_of_Derivatives/4.04%3A_Derivatives_and_the_Shape_of_a_GraphUsing the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the sec...Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
- https://math.libretexts.org/Courses/Laney_College/Math_3A%3A_Calculus_1_(Fall_2022)/04%3A_Applications_of_Derivatives/4.06%3A_Derivatives_and_the_Shape_of_a_GraphUsing the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the sec...Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
- https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Tran)/04%3A_Applications_of_Derivatives/4.05%3A_Derivatives_and_the_Shape_of_a_GraphUsing the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the sec...Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_400%3A_Calculus_I_-_Differential_Calculus/04%3A_Appropriate_Applications/4.03%3A_Graphing_Using_Calculus_-_Shaping_the_CurveThis section covers techniques for graphing functions by analyzing their shapes using calculus. It explains how the first and second derivatives indicate increasing/decreasing intervals, concavity, an...This section covers techniques for graphing functions by analyzing their shapes using calculus. It explains how the first and second derivatives indicate increasing/decreasing intervals, concavity, and inflection points. By applying these concepts, you can accurately sketch the function’s curve and identify key features such as peaks, valleys, and changes in direction.
- https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/04%3A_Applications_of_Derivatives/4.04%3A_The_Second_Derivative_Test_and_Curve_SketchingTo determine concavity, we need to find the second derivative f″(x). The first derivative is f′(x)=3x2−12x+9, so the second derivative is f″(x)=6x−12. If the function changes concavity,...To determine concavity, we need to find the second derivative f″(x). The first derivative is f′(x)=3x2−12x+9, so the second derivative is f″(x)=6x−12. If the function changes concavity, it occurs either when f″(x)=0 or f″(x) is undefined.
- https://math.libretexts.org/Courses/Chabot_College/MTH_1%3A_Calculus_I/04%3A_Applications_of_Derivatives/4.03%3A_Graphing_FunctionsUsing the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the sec...Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
- https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/04%3A_Applications_of_Derivatives/4.03%3A_The_First_Derivative_TestSince f′ is a continuous function, to determine the sign of f′(x) over each subinterval, it suffices to choose a point over each of the intervals (−∞,−1),(−1,3) and (3,∞) and determine...Since f′ is a continuous function, to determine the sign of f′(x) over each subinterval, it suffices to choose a point over each of the intervals (−∞,−1),(−1,3) and (3,∞) and determine the sign of f′ at each of these points.
- https://math.libretexts.org/Courses/Prince_Georges_Community_College/MAT_2410%3A_Calculus_(Open_Stax)_Novick/04%3A_Applications_of_Derivatives/4.06%3A_Derivatives_and_the_Shape_of_a_GraphUsing the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the sec...Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
- https://math.libretexts.org/Courses/Southwestern_College/Business_Calculus/03%3A_Unit_3_-_Derivatives/3.08%3A_Derivatives_and_the_Shape_of_a_GraphUsing the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the sec...Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
