Search
- Filter Results
- Location
- Classification
- Include attachments
- https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/03%3A_Using_Derivatives/3.02%3A_Using_Derivatives_to_Describe_Families_of_FunctionsGiven a family of functions that depends on one or more parameters, by investigating how critical numbers and locations where the second derivative is zero depend on the values of these parameters, we...Given a family of functions that depends on one or more parameters, by investigating how critical numbers and locations where the second derivative is zero depend on the values of these parameters, we can often accurately describe the shape of the function in terms of the parameters. In particular, just as we can created first and second derivative sign charts for a single function, we often can do so for entire families of functions.
- https://math.libretexts.org/Courses/Hope_College/Math_126_-_Calculus_with_Review_II/02%3A_Using_Derivatives/2.03%3A_Using_Derivatives_to_Describe_Families_of_FunctionsGiven a family of functions that depends on one or more parameters, by investigating how critical numbers and locations where the second derivative is zero depend on the values of these parameters, we...Given a family of functions that depends on one or more parameters, by investigating how critical numbers and locations where the second derivative is zero depend on the values of these parameters, we can often accurately describe the shape of the function in terms of the parameters. In particular, just as we can created first and second derivative sign charts for a single function, we often can do so for entire families of functions.
- https://math.libretexts.org/Bookshelves/Calculus/Differential_Calculus_for_the_Life_Sciences_(Edelstein-Keshet)/06%3A_Sketching_the_Graph_of_a_Function_using_Calculus_Tools/6.01%3A_Overall_Shape_of_the_Graph_of_a_FunctionSee video summarizing the connection between the shape of the graph of \(y = f(x)\), and the derivatives of the function, \(f^{\prime}(x)\), \(f^{\prime \prime}(x)\) of the function. \[\begin{aligned}...See video summarizing the connection between the shape of the graph of \(y = f(x)\), and the derivatives of the function, \(f^{\prime}(x)\), \(f^{\prime \prime}(x)\) of the function. \[\begin{aligned} f^{\prime}(x) & =(12 / 5) x^{5}-4 x^{3}+1 \\ f^{\prime \prime}(x) & =12 x^{4}-14 x^{2}=12 x^{2}\left(x^{2}-1\right)=12 x^{2}(x+1)(x-1) \end{aligned} \nonumber \]
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Math_For_Liberal_Art_Students_2e_(Diaz)/11%3A_Normal_Distribution/11.02%3A_The_Density_Curve_of_a_Normal_DistributionIn this section, we will continue our investigation of normal distributions to include density curves and learn various methods for calculating probabilities from the normal density curve. A density c...In this section, we will continue our investigation of normal distributions to include density curves and learn various methods for calculating probabilities from the normal density curve. A density curve is an idealized representation of a distribution in which the area under the curve is defined to be 1. Density curves need not be normal, but the normal density curve will be the most useful to us.
- https://math.libretexts.org/Workbench/Calculus_I%3A_Differential_Calculus/04%3A_Applications_of_Derivatives/4.06%3A_Limits_at_Infinity_and_Asymptotes/4.6E%3A_Exercises_for_Section_4.6This page offers exercises on identifying and evaluating vertical and horizontal asymptotes in mathematical functions and limits. It provides graphs showcasing asymptotic behavior, calculations for de...This page offers exercises on identifying and evaluating vertical and horizontal asymptotes in mathematical functions and limits. It provides graphs showcasing asymptotic behavior, calculations for determining the presence of asymptotes, and evaluations of limits as \(x\) approaches extreme values. The text discusses properties of various functions, including local maxima, minima, and inflection points, while outlining conditions for asymptote definitions.
- https://math.libretexts.org/Courses/Chabot_College/Math_in_Society_(Zhang)/11%3A_Normal_Distribution/11.02%3A_The_Density_Curve_of_a_Normal_DistributionIn this section, we will continue our investigation of normal distributions to include density curves and learn various methods for calculating probabilities from the normal density curve. A density c...In this section, we will continue our investigation of normal distributions to include density curves and learn various methods for calculating probabilities from the normal density curve. A density curve is an idealized representation of a distribution in which the area under the curve is defined to be 1. Density curves need not be normal, but the normal density curve will be the most useful to us.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/05%3A_Curve_Sketching/5.04%3A_Concavity_and_Inflection_PointsIf we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which...If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points.
- https://math.libretexts.org/Workbench/Calculus_I%3A_Differential_Calculus/04%3A_Applications_of_Derivatives/4.05%3A_Derivatives_and_the_Shape_of_a_Graph/4.5E%3A_Exercises_for_Section_4.5This page covers calculus concepts including critical and inflection points, function behavior, and derivative tests. It explains critical points as where the derivative is zero or undefined, signalin...This page covers calculus concepts including critical and inflection points, function behavior, and derivative tests. It explains critical points as where the derivative is zero or undefined, signaling potential local extrema or inflection points, with a focus on increasing/decreasing intervals and concavity. Mathematical exercises analyze specific functions for these properties, helping students apply their knowledge.
- https://math.libretexts.org/Under_Construction/Purgatory/Book%3A_Active_Calculus_(Boelkins_et_al.)/03%3A_Using_Derivatives/3.02%3A_Using_Derivatives_to_Describe_Families_of_FunctionsGiven a family of functions that depends on one or more parameters, by investigating how critical numbers and locations where the second derivative is zero depend on the values of these parameters, we...Given a family of functions that depends on one or more parameters, by investigating how critical numbers and locations where the second derivative is zero depend on the values of these parameters, we can often accurately describe the shape of the function in terms of the parameters. In particular, just as we can created first and second derivative sign charts for a single function, we often can do so for entire families of functions.
