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4.6: Derivatives and the Shape of a Graph

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_Graph

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 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.

4.3: Derivatives and the Shape of a Graph

https://math.libretexts.org/Under_Construction/Purgatory/Remixer_University/Username%3A_hdagnew@ucdavis.edu/Courses%2F%2FRemixer_University%2F%2FUsername%3A_hdagnew@ucdavis.edu%2F%2FMonroe2/Courses%2F%2FRemixer_University%2F%2FUsername%3A_hdagnew@ucdavis.edu%2F%2FMonroe2%2F%2FChapter_4%3A_Applications_of_Derivatives/Courses%2F%2FRemixer_University%2F%2FUsername%3A_hdagnew@ucdavis.edu%2F%2FMonroe2%2F%2FChapter_4%3A_Applications_of_Derivatives%2F%2F4.3%3A_Derivatives_and_the_Shape_of_a_Graph

let

$f$ be a continuous function over an interval

$I$ containing a critical point

$c$ such that

$f$ is differentiable over

$I$ except possibly at

$c$ ; if

$f′$ changes sign from positive ...let

$f$ be a continuous function over an interval

$I$ containing a critical point

$c$ such that

$f$ is differentiable over

$I$ except possibly at

$c$ ; if

$f′$ changes sign from positive to negative as

$x$ increases through

$c$ , then

$f$ has a local maximum at

$c$ ; if

$f′$ changes sign from negative to positive as

$x$ increases through

$c$ , then

$f$ has a local minimum at

$c$ ; if

$f′$ does not change sign as

$x$ increases through

$c$ , then f does not have a …

4.4: Derivatives and the Shape of a Graph

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_Graph

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 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.

1.6: The Second Derivative

https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/01%3A_Understanding_the_Derivative/1.06%3A_The_Second_Derivative

A differentiable function f is increasing at a point or on an interval whenever its first derivative is positive, and decreasing whenever its first derivative is negative. By taking the derivative of ...A differentiable function f is increasing at a point or on an interval whenever its first derivative is positive, and decreasing whenever its first derivative is negative. By taking the derivative of the derivative of a function f', we arrive at the second derivative, f''. The second derivative measures the instantaneous rate of change of the first derivative, and thus the sign of the second derivative tells us whether or not the slope of the tangent line to f is increasing or decreasing.

3.4: Concavity and the Second Derivative

https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/03%3A_The_Graphical_Behavior_of_Functions/3.04%3A_Concavity_and_the_Second_Derivative

We have been learning how the first and second derivatives of a function relate information about the graph of that function. We have found intervals of increasing and decreasing, intervals where the ...We have been learning how the first and second derivatives of a function relate information about the graph of that function. We have found intervals of increasing and decreasing, intervals where the graph is concave up and down, along with the locations of relative extrema and inflection points.

4.4: Derivatives and the Shape of a Graph

https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT301_Calculus_I/04%3A_Applications_of_Derivatives/4.04%3A_Derivatives_and_the_Shape_of_a_Graph

let

$f$ be a continuous function over an interval

$I$ containing a critical point

$c$ such that

$f$ is differentiable over

$I$ except possibly at

$c$ ; if

$f′$ changes sign from positive ...let

$f$ be a continuous function over an interval

$I$ containing a critical point

$c$ such that

$f$ is differentiable over

$I$ except possibly at

$c$ ; if

$f′$ changes sign from positive to negative as

$x$ increases through

$c$ , then

$f$ has a local maximum at

$c$ ; if

$f′$ changes sign from negative to positive as

$x$ increases through

$c$ , then

$f$ has a local minimum at

$c$ ; if

$f′$ does not change sign as

$x$ increases through

$c$ , then f does not have a …

4.4: Concavity and Curve Sketching

https://math.libretexts.org/Courses/University_of_California_Davis/UCD_Mat_21A%3A_Differential_Calculus/4%3A_Applications_of_Definite_Integrals/4.4%3A_Concavity_and_Curve_Sketching

We know that the sign of the derivative tells us whether a function is increasing or decreasing; for example, when f′(x)>0, f(x) is increasing. The sign of the second derivative f′′(x) tells us whe...We know that the sign of the derivative tells us whether a function is increasing or decreasing; for example, when f′(x)>0, f(x) is increasing. The sign of the second derivative f′′(x) tells us whether f′ is increasing or decreasing; we have seen that if f′ is zero and increasing at a point then there is a local minimum at the point, and if f′ is zero and decreasing at a point then there is a local maximum at the point. We extracted information about f from information about f′′.

4.4: Derivatives and the Shape of a Graph

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_Graph

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 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.

4.6: Derivatives and the Shape of a Graph

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_Graph

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 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.

4.5: Derivatives and the Shape of a Graph

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_Graph

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 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.

4.3: Graphing Using Calculus - Shaping the Curve

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_Curve

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, 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.

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