Search
- Filter Results
- Location
- Classification
- Include attachments
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_Variables/1.03%3A_Partial_Derivatives/1.3E%3A_Exercises_for_Section_1.3This page discusses exercises on calculating partial and higher-order derivatives of functions, including limit definitions, surface plot analysis, and practical applications like volume and area. It ...This page discusses exercises on calculating partial and higher-order derivatives of functions, including limit definitions, surface plot analysis, and practical applications like volume and area. It also covers problems in differential calculus, such as finding points where partial derivatives equal zero, verifying Laplace's and heat equations, and analyzing rates of change in contexts like dimensions and productivity functions. Answers to the various exercises are provided throughout.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_Variables/1.05%3A_The_Chain_Rule_for_Multivariable_Functions/1.5E%3A_Exercises_for_Section_1.5Figure \(\PageIndex{1}\): The graph illustrates the volume of a frustum of a cone is given by the formula \( V=\frac{1}{3}πz(x^2+y^2+xy),\) where \( x\) is the radius of the smaller circle, \( y\) is ...Figure \(\PageIndex{1}\): The graph illustrates the volume of a frustum of a cone is given by the formula \( V=\frac{1}{3}πz(x^2+y^2+xy),\) where \( x\) is the radius of the smaller circle, \( y\) is the radius of the larger circle, and \( z\) is the height of the frustum
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_Variables/1.07%3A_Maxima_Minima_Problems/1.7E%3A_Exercises_for_Section_1.7Since \( x^2y^2>0\) for all \( x\) and \( y\) different from zero, and \( x^2y^2=0\) when either \( x\) or \( y\) equals zero (or both), then the absolute minimum of \(0\) occurs at all points on the ...Since \( x^2y^2>0\) for all \( x\) and \( y\) different from zero, and \( x^2y^2=0\) when either \( x\) or \( y\) equals zero (or both), then the absolute minimum of \(0\) occurs at all points on the \(x\)- or \(y\)-axes, that is, for all points on the lines \( x = 0 \) and \( y = 0\).
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_Variables/1.02%3A_Limits_and_ContinuityWe have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a func...We have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a function of more than one variable to be continuous at a point in its domain. It turns out these concepts have aspects that just don’t occur with functions of one variable.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_Variables/1.08%3A_Lagrange_Multipliers/1.8E%3A_Exercises_for_Section_1.8Subject to the given constraint, \(f\) has a relative minimum of \(-\frac{2\sqrt{3}}{3}\) at \( \left( \sqrt{2},\, 1,\, -\frac{\sqrt{6}}{3} \right),\; \left( \sqrt{2},\, -1,\, \frac{\sqrt{6}}{3} \righ...Subject to the given constraint, \(f\) has a relative minimum of \(-\frac{2\sqrt{3}}{3}\) at \( \left( \sqrt{2},\, 1,\, -\frac{\sqrt{6}}{3} \right),\; \left( \sqrt{2},\, -1,\, \frac{\sqrt{6}}{3} \right),\; \left( -\sqrt{2},\, 1,\, \frac{\sqrt{6}}{3} \right),\) and \( \left( -\sqrt{2},\, -1,\, -\frac{\sqrt{6}}{3} \right) \) and a relative maximum of \(\frac{2\sqrt{3}}{3}\) at \( \left( \sqrt{2},\, 1,\, \frac{\sqrt{6}}{3} \right),\; \left( \sqrt{2},\, -1,\, -\frac{\sqrt{6}}{3} \right),\; \left( -…
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_Variables/1.08%3A_Lagrange_MultipliersSolving optimization problems for functions of two or more variables can be similar to solving such problems in single-variable calculus. However, techniques for dealing with multiple variables allow ...Solving optimization problems for functions of two or more variables can be similar to solving such problems in single-variable calculus. However, techniques for dealing with multiple variables allow us to solve more varied optimization problems for which we need to deal with additional conditions or constraints. In this section, we examine one of the more common and useful methods for solving optimization problems with constraints.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_VariablesWhen dealing with a function of more than one independent variable, several questions naturally arise. For example, how do we calculate limits of functions of more than one variable? The definition of...When dealing with a function of more than one independent variable, several questions naturally arise. For example, how do we calculate limits of functions of more than one variable? The definition of derivative we used before involved a limit. Does the new definition of derivative involve limits as well? Do the rules of differentiation apply in this context? Can we find relative extrema of functions using derivatives? All these questions are answered in this chapter.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_Variables/1.05%3A_The_Chain_Rule_for_Multivariable_FunctionsIn single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. The same thing is...In single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule. In this section, we study extensions of the chain rule and learn how to take derivatives of compositions of functions of more than one variable.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_Variables/1.06%3A_Directional_Derivatives_and_the_GradientA function \(z=f(x,y)\) has two partial derivatives: \(∂z/∂x\) and \(∂z/∂y\). These derivatives correspond to each of the independent variables and can be interpreted as instantaneous rates of change ...A function \(z=f(x,y)\) has two partial derivatives: \(∂z/∂x\) and \(∂z/∂y\). These derivatives correspond to each of the independent variables and can be interpreted as instantaneous rates of change (that is, as slopes of a tangent line). Similarly, \(∂z/∂y\) represents the slope of the tangent line parallel to the y-axis. Now we consider the possibility of a tangent line parallel to neither axis.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_Variables/1.06%3A_Directional_Derivatives_and_the_Gradient/1.6E%3A_Exercises_for_Section_1.6This page includes exercises on directional derivatives and gradients for functions such as \( f(x,y)=5-2x^2-\frac{1}{2}y^2 \) and \( f(x,y)=y^2\cos(2x) \), evaluated at points like \( P(3,4) \). It a...This page includes exercises on directional derivatives and gradients for functions such as \( f(x,y)=5-2x^2-\frac{1}{2}y^2 \) and \( f(x,y)=y^2\cos(2x) \), evaluated at points like \( P(3,4) \). It also covers gradients in multiple dimensions, tangent planes, and normal lines, alongside problems related to temperature change and electric potential.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_Variables/1.02%3A_Limits_and_Continuity/1.2E%3A_Exercises_for_Section_1.2This page covers limits and continuity of functions in multiple variables, analyzing two and three-variable limits and their dependence on paths. It identifies regions of continuity and discontinuity,...This page covers limits and continuity of functions in multiple variables, analyzing two and three-variable limits and their dependence on paths. It identifies regions of continuity and discontinuity, concluding that some limits do not exist while others are continuous at certain points. It highlights \( f(g(x,y)) \) and \( f(x,y) = x^2 - 4y \) as examples, noting specific continuity conditions. Contributions come from Gilbert Strang and Jed Herman, with a CC-BY-SA-NC 4.0 license.
