You CAN Ace Calculus

 basics of vectors partial derivatives gradients

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

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17calculus > partial derivatives > lagrange multipliers

 Basic Technique Two Constraints Equations That Do Not Use The Gradient Practice

The Method of Lagrange Multipliers is used to find maximums and minimums of a function subject to one or more constraints. We could also say that we want to optimize the function or find the extreme values of the function.
We highly recommend that you download the notes for this topic from Dr Chris Tisdell. Look for the pdf link entitled Extreme values + Lagrange multipliers.

In basic calculus, we learned that finding the critical points gives us information about maximums, minimums (and saddlepoints). We use the same idea here, i.e. locations where the derivative is zero gives us possible locations of maxs/mins. We use this method to integrate the constraints into the equation.

Basic Technique

If we are given a function $$f(x,y,z)$$ that we want to optimize (find maximums, minimums or both) subject to a constraint $$g(x,y,z) = 0$$, we set up the gradient equation $$\nabla f = \lambda \nabla g$$. [ For a version of the equations that do not use the gradient, see below. ]

The variable $$\lambda$$ is just a number (independent of $$x$$, $$y$$ and $$z$$) called a Lagrange multiplier. We introduce $$\lambda$$ during the course of solving this problem but it will not appear in our answer.

Remember that the gradient is a vector. So, the above equation gives us three equations ( one for each variable ) and four unknowns ( $$x$$, $$y$$, $$z$$ and $$\lambda$$ ). Using also the constraint equation $$g(x,y,z) = 0$$, we can now (theoretically) solve for all four unknowns.

This next video clip explains this technique in more detail.

### Dr Chris Tisdell - Lagrange Multipliers (clip 1) [6mins-23secs]

video by Dr Chris Tisdell

So, why does this work? It seems kind of strange that introducing another variable would enable us to optimize a function. Here is a great video clip that explains this. His use of graphs is very good to visualize what is going on.

### Dr Chris Tisdell - Lagrange Multipliers (clip 1) [8mins-17secs]

video by Dr Chris Tisdell

Two Constraints

Okay, so now you know how to handle one constraint, if you are given two constraints, you just add another lagrange multiplier. We usually use the Greek letter mu, $$\mu$$. The equation then looks like $$\nabla f = \lambda \nabla g + \mu \nabla h$$ where the function we want to optimize is $$f(x,y,z)$$ and the constraint equations are $$g(x,y,z) = 0$$ and $$h(x,y,z) = 0$$. Here is a video explaining this in more detail, including an example.

### Dr Chris Tisdell - Lagrange Multipliers: 2 Constraints [14mins-23secs]

video by Dr Chris Tisdell

Equations That Do Not Use The Gradient

An alternate version of the equations using the Lagrange Method that do not use the gradient is given below. We present equations with three variables and two constraints. As you would expect, you can use the same ideas with two variables (drop $$z$$) and one constraint (drop $$h(x,y,z)$$).

Equations And Set Up

Optimize $$f(x,y,z)$$

Constraints $$g(x,y,z)=0$$ and $$h(x,y,z) = 0$$

$$L(x,y,z,\lambda, \mu) = f(x,y,z) - \lambda g(x,y,z) - \mu h(x,y,z)$$

Equations To Solve

$$\partial L / \partial x = 0 ~~~ \to ~~~ \partial f / \partial x - \lambda ~ \partial g / \partial x - \mu ~ \partial h / \partial x = 0$$

$$\partial L / \partial y = 0 ~~~ \to ~~~ \partial f / \partial y - \lambda ~ \partial g / \partial y - \mu ~ \partial h / \partial y = 0$$

$$\partial L / \partial z = 0 ~~~ \to ~~~ \partial f / \partial z - \lambda ~ \partial g / \partial z - \mu ~ \partial h / \partial z = 0$$

$$g(x,y,z) = 0$$

$$h(x,y,z) = 0$$

5 equations and 5 unknowns

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

Instructions - - Unless otherwise instructed, follow these guidelines.
1. Optimize the functions subject to the given constraints.
2. Show whether they are maximums or minimums.
3. If you are not told whether to find maximums or minimums, find all of them.
4. Give all answers in exact form.

Basic Problems

 optimize: $$f(x,y) = x^2+y^2$$ constraint: $$2x+6y=2000$$

Problem Statement

 optimize: $$f(x,y) = x^2+y^2$$ constraint: $$2x+6y=2000$$

Solution

This solution is in two consecutive videos.

### 812 solution video

video by Krista King Math

### 812 solution video

video by Krista King Math

 optimize: $$f(x,y) = x^2+y^2$$ constraint: $$xy=1$$

Problem Statement

 optimize: $$f(x,y) = x^2+y^2$$ constraint: $$xy=1$$

Solution

This solution is in two consecutive videos.

### 813 solution video

video by Krista King Math

### 813 solution video

video by Krista King Math

 minimize: $$C(x,y) = 6x^2 + 12y^2$$ constraint: $$x + y = 90$$

Problem Statement

 minimize: $$C(x,y) = 6x^2 + 12y^2$$ constraint: $$x + y = 90$$

Solution

### 816 solution video

video by PatrickJMT

 optimize: $$f(x,y) = xy$$ constraint: $$x^2 + 2y^2 = 1$$

Problem Statement

 optimize: $$f(x,y) = xy$$ constraint: $$x^2 + 2y^2 = 1$$

Solution

### 818 solution video

video by Dr Chris Tisdell

 minimize: $$f(x,y,z) = x^2 + y^2 + z^2$$ constraint: $$2x + y - z = 1$$

Problem Statement

 minimize: $$f(x,y,z) = x^2 + y^2 + z^2$$ constraint: $$2x + y - z = 1$$

Solution

### 819 solution video

video by Dr Chris Tisdell

 maximize: $$f(x,y,z) = x^2+2y-z^2$$ constraint: $$2x=y$$ and $$y+z=0$$

Problem Statement

 maximize: $$f(x,y,z) = x^2+2y-z^2$$ constraint: $$2x=y$$ and $$y+z=0$$

Solution

### 820 solution video

video by Dr Chris Tisdell

 optimize: $$f(x,y) = 3x+4y$$ constraint: $$x^2+y^2 = 1$$

Problem Statement

 optimize: $$f(x,y) = 3x+4y$$ constraint: $$x^2+y^2 = 1$$

Solution

### 821 solution video

video by Dr Chris Tisdell

Determine the point on the surface of $$xyz = 1$$ that is closest to the origin and satifies $$x > 0$$, $$y > 0$$ and $$z > 0$$.

Problem Statement

Determine the point on the surface of $$xyz = 1$$ that is closest to the origin and satifies $$x > 0$$, $$y > 0$$ and $$z > 0$$.

Solution

### 822 solution video

video by Dr Chris Tisdell

 maximize: $$T(x,y) = 6xy$$ constraint: $$x^2 + y^2 = 8$$

Problem Statement

 maximize: $$T(x,y) = 6xy$$ constraint: $$x^2 + y^2 = 8$$

Solution

### 823 solution video

video by Dr Chris Tisdell

We have a thin metal plate that occupies the region in the xy-plane $$x^2 + y^2 \leq 25$$. If $$f(x,y) = 4x^2 - 4xy + y^2$$ denotes the temperature (in degrees C) at any point on the plate, determine the highest and lowest temperatures on the edge of the plate.

Problem Statement

We have a thin metal plate that occupies the region in the xy-plane $$x^2 + y^2 \leq 25$$. If $$f(x,y) = 4x^2 - 4xy + y^2$$ denotes the temperature (in degrees C) at any point on the plate, determine the highest and lowest temperatures on the edge of the plate.

Solution

### 824 solution video

video by Dr Chris Tisdell

 optimize: $$f(x,y,z) = xyz$$ constraint: $$x^2 + y^2 + z^2 = 3$$

Problem Statement

 optimize: $$f(x,y,z) = xyz$$ constraint: $$x^2 + y^2 + z^2 = 3$$

Solution

### 1825 solution video

video by Krista King Math

Find the maximum and minimum values of the function $$f(x,y) = e^{xy}$$ subject to $$x^3 + y^3 = 16$$.

Problem Statement

Find the maximum and minimum values of the function $$f(x,y) = e^{xy}$$ subject to $$x^3 + y^3 = 16$$.

Solution

### 1924 solution video

video by MIP4U

Find the rectangle with maximum perimeter that can inscribed in the ellipse $$x^2 + 4y^2 = 4$$.

Problem Statement

Find the rectangle with maximum perimeter that can inscribed in the ellipse $$x^2 + 4y^2 = 4$$.

Solution

### 825 solution video

video by MIT OCW

Intermediate Problems

 optimize: $$f(x,y) = (x-1)^2 + (y-2)^2 - 4$$ constraint: $$3x + 5y = 47$$

Problem Statement

 optimize: $$f(x,y) = (x-1)^2 + (y-2)^2 - 4$$ constraint: $$3x + 5y = 47$$

Solution

This solution is in three consecutive videos.

### 814 solution video

video by Krista King Math

### 814 solution video

video by Krista King Math

### 814 solution video

video by Krista King Math

 optimize: $$f(x,y,z) = x^2+y^2+z^2$$ constraints: $$x+y+z = 1$$; $$x+2y+3z = 6$$

Problem Statement

 optimize: $$f(x,y,z) = x^2+y^2+z^2$$ constraints: $$x+y+z = 1$$; $$x+2y+3z = 6$$

Solution

### 815 solution video

video by Krista King Math

 optimize: $$f(x,y,z) = 3x-y-3z$$ constraints: $$x+y-z=0$$; $$x^2+2z^2 = 1$$

Problem Statement

 optimize: $$f(x,y,z) = 3x-y-3z$$ constraints: $$x+y-z=0$$; $$x^2+2z^2 = 1$$

Solution

This solution is presented in two consecutive videos.

### 817 solution video

video by PatrickJMT

### 817 solution video

video by PatrickJMT

 optimize: $$f(x,y,z) = x^2+x+2y^2+3z^2$$ constraint: $$x^2+y^2+z^2 = 1$$

Problem Statement

 optimize: $$f(x,y,z) = x^2+x+2y^2+3z^2$$ constraint: $$x^2+y^2+z^2 = 1$$

Solution

video by MIT OCW