2.E: Functions of Several Variables (Exercises)
2.1: Functions of Two or Three Variables
A
2.1.1. \(f (x, y) = x^ 2 + y^ 2 −1\)
2.1.2. \(f (x, y) = \frac{1}{ x^ 2 + y^ 2}\)
2.1.3. \(f (x, y) = \sqrt{ x^ 2 + y^ 2 −4}\)
2.1.4. \(f (x, y) = \frac{x^ 2 +1}{ y}\)
2.1.5. \(f (x, y, z) = \sin (x yz)\)
2.1.6. \(f (x, y, z) = \sqrt{ (x−1)(yz −1)}\)
For Exercises 7-18, evaluate the given limit.
2.1.7. \(\lim\limits_{(x,y)→(0,0)} \cos (x y)\)
2.1.8. \(\lim\limits_{(x,y)→(0,0)} e^{ x y}\)
2.1.9. \(\lim\limits_{(x,y)→(0,0)} \frac{x^ 2 − y^ 2}{ x^ 2 + y^ 2}\)
2.1.10. \(\lim\limits_{(x,y)→(0,0)} \frac{x y^2}{ x^ 2 + y^ 4}\)
2.1.11. \(\lim\limits_{(x,y)→(1,−1)} \frac{x^ 2 −2x y+ y^ 2}{ x− y}\)
2.1.12. \(\lim\limits_{(x,y)→(0,0)} \frac{x y^2}{ x^ 2 + y^ 2}\)
2.1.13. \(\lim\limits_{(x,y)→(1,1)} \frac{x^ 2 − y^ 2}{ x− y}\)
2.1.14. \(\lim\limits_{(x,y)→(0,0)} \frac{x^ 2 −2x y+ y^ 2}{ x− y}\)
2.1.15. \(\lim\limits_{(x,y)→(0,0)} \frac{y^ 4 \sin (x y)}{ x^ 2 + y^ 2}\)
2.1.16. \(\lim\limits_{(x,y)→(0,0)} (x^ 2 + y^ 2 )\cos \left ( \frac{1}{ x y}\right ) \)
2.1.17. \(\lim\limits_{(x,y)→(0,0)} \frac{x}{ y}\)
2.1.18. \(\lim\limits_{(x,y)→(0,0)} \cos \left ( \frac{1}{ x y}\right ) \)
B
2.1.19. Show that \(f (x, y) = \frac{1}{ 2πσ^2} e^{ −(x^ 2+y^ 2 )/2σ^ 2}\), for \(σ > 0\), is constant on the circle of radius \(r > 0\) centered at the origin. This function is called a Gaussian blur , and is used as a filter in image processing software to produce a “blurred” effect.
2.1.20. Suppose that \(f (x, y) ≤ f (y, x) \text{ for all }(x, y)\) in \(\mathbb{R}^ 2\). Show that \(f (x, y) = f (y, x) \text{ for all }(x, y)\) in \(\mathbb{R}^ 2\).
2.1.21. Use the substitution \(r = \sqrt{ x^ 2 + y^ 2}\) to show that
\[\lim\limits_{(x,y)→(0,0)} \frac{\sin \sqrt{ x^ 2 + y^ 2}}{ \sqrt{ x^ 2 + y^ 2}} = 1 .\]
( Hint: You will need to use L’Hôpital’s Rule for single-variable limits. )
C
2.1.22. Prove Theorem 2.1(a) in the case of addition. ( Hint: Use Definition 2.1 .)
2.1.23. Prove Theorem 2.1(b).
2.2: Partial Derivatives
A
For Exercises 1-16, find \(\frac{∂f}{ ∂x} \text{ and }\frac{∂f}{ ∂y}\).
2.2.1. \(f (x, y) = x^ 2 + y^ 2\)
2.2.2. \(f (x, y) = \cos (x+ y)\)
2.2.3. \(f (x, y) = \sqrt{ x^ 2 + y+4}\)
2.2.4. \(f (x, y) = \frac{x+1}{ y+1}\)
2.2.5. \(f (x, y) = e^{ x y} + x y\)
2.2.6. \(f (x, y) = x^ 2 − y^ 2 +6x y+4x−8y+2\)
2.2.7. \(f (x, y) = x^ 4\)
2.2.8. \(f (x, y) = x+2y\)
2.2.9. \(f (x, y) = \sqrt{ x^ 2 + y^ 2}\)
2.2.10. \(f (x, y) = \sin (x+ y)\)
2.2.11. \(f (x, y) = \sqrt[3]{ x^ 2 + y+4}\)
2.2.12. \(f (x, y) = \frac{x y+1}{ x+ y}\)
2.2.13. \(f (x, y) = e^{ −(x^ 2+y^ 2 )}\)
2.2.14. \(f (x, y) = \ln (x y)\)
2.2.15. \(f (x, y) = \sin (x y)\)
2.2.16. \(f (x, y) = \tan (x+ y)\)
For Exercises 17-26, find \(\frac{∂^ 2 f}{ ∂x^ 2} ,\, \frac{∂^ 2 f}{ ∂y^ 2} \text{ and }\frac{∂^ 2 f}{ ∂y∂x}\) (use Exercises 1-8, 14, 15).
2.2.17. \(f (x, y) = x^ 2 + y^ 2\)
2.2.18. \(f (x, y) = \cos (x+ y)\)
2.2.19. \(f (x, y) = \sqrt{ x^ 2 + y+4}\)
2.2.20. \(f (x, y) = \frac{x+1}{ y+1}\)
2.2.21. \(f (x, y) = e^{ x y} + x y\)
2.2.22. \(f (x, y) = x^ 2 − y^ 2 +6x y+4x−8y+2\)
2.2.23. \(f (x, y) = x^ 4\)
2.2.24 . \(f (x, y) = x+2y\)
2.2.25. \(f (x, y) = \ln (x y)\)
2.2.26. \(f (x, y) = \sin (x y)\)
B
2.2.27. Show that the function \(f (x, y) = \sin (x+ y)+\cos (x− y)\) satisfies the wave equation
\[\frac{∂^ 2 f}{ ∂x^ 2} − \frac{∂^ 2 f}{ ∂y^ 2} = 0 .\]
The wave equation is an example of a partial differential equation .
2.2.28 Let \(u \text{ and }v\) be twice-differentiable functions of a single variable, and let \(c \neq 0\) be a constant. Show that \(f (x, y) = u(x+ c y)+v(x− c y)\) is a solution of the general one-dimensional wave equation
\[\frac{∂^ 2 f}{ ∂x^ 2} − \frac{1}{ c^ 2} \frac{∂^ 2 f}{ ∂y^ 2} = 0\]
2.3: Tangent Plane to a Surface
A
For Exercises 1-6, find the equation of the tangent plane to the surface \(z = f (x, y)\) at the point \(P\).
2.3.1. \(f (x, y) = x^ 2 + y^ 3 ,\, P = (1,1,2)\)
2.3.2. \(f (x, y) = x y,\, P = (1,−1,−1) \)
2.3.3. \(f (x, y) = x^ 2 y,\, P = (−1,1,1)\)
2.3.4. \(f (x, y) = xe^ y ,\, P = (1,0,1)\)
2.3.5. \(f (x, y) = x+2y,\, P = (2,1,4)\)
2.3.6. \(f (x, y) = \sqrt{ x^ 2 + y^ 2},\, P = (3,4,5)\)
For Exercises 7-10, find the equation of the tangent plane to the given surface at the point \(P\).
2.3.7. \(\frac{x^ 2}{ 4} + \frac{y^ 2}{ 9} + \frac{z^ 2}{ 16} = 1,\, P = \left ( 1,2, \frac{2 \sqrt{ 11}}{ 3} \right ) \)
2.3.8. \(x^ 2 + y^ 2 + z^ 2 = 9,\, P = (0,0,3)\)
2.3.9. \(x^ 2 + y^ 2 − z^ 2 = 0,\, P = (3,4,5)\)
2.3.10. \(x^ 2 + y^ 2 = 4,\, P = ( \sqrt{ 3},1,0)\)
2.4: Directional Derivatives and the Gradient
A
For Exercises 1-10, compute the gradient \(∇f\).
2.4.1. \(f (x, y) = x^ 2 + y^ 2 −1\)
2.4.2. \(f (x, y) = \frac{1}{ x^ 2 + y^ 2}\)
2.4.3. \(f (x, y) = \sqrt{ x^ 2 + y^ 2 +4}\)
2.4.4. \(f (x, y) = x^ 2 e^ y\)
2.4.5. \(f (x, y) = \ln (x y)\)
2.4.6. \(f (x, y) = 2x+5y\)
2.4.7. \(f (x, y, z) = \sin (x yz)\)
2.4.8. \(f (x, y, z) = x^ 2 e^{ yz}\)
2.4.9. \(f (x, y, z) = x^ 2 + y^ 2 + z^ 2\)
2.4.10. \(f (x, y, z) = \sqrt{ x^ 2 + y^ 2 + z^ 2}\)
For Exercises 11-14, find the directional derivative of \(f\) at the point \(P\) in the direction of \(v = \left ( \frac{1}{ \sqrt{ 2}} , \frac{1}{ \sqrt{ 2}} \right ) \).
2.4.11. \(f (x, y) = x^ 2 + y^ 2 −1,\, P = (1,1)\)
2.4.12. \(f (x, y) = \frac{1}{ x^ 2 + y^ 2} ,\, P = (1,1)\)
2.4.13. \(f (x, y) = \sqrt{ x^ 2 + y^ 2 +4},\, P = (1,1)\)
2.4.14. \(f (x, y) = x^ 2 e^ y ,\, P = (1,1)\)
For Exercises 15-16, find the directional derivative of \(f\) at the point \(P\) in the direction of \(v = \left ( \frac{1}{ \sqrt{ 3}} , \frac{1}{ \sqrt{ 3}} , \frac{1}{ \sqrt{ 3}} \right ) \).
2.4.15. \(f (x, y, z) = \sin (x yz),\, P = (1,1,1)\)
2.4.16. \(f (x, y, z) = x^ 2 e^{ yz} ,\, P = (1,1,1)\)
2.4.17. Repeat Example 2.16 at the point \((2,3)\).
2.4.18. Repeat Example 2.17 at the point \((3,1,2)\).
B
For Exercises 19-26, let \(f (x, y) \text{ and }g(x, y)\) be continuously differentiable real-valued functions, let \(c\) be a constant, and let \(v\) be a unit vector in \(\mathbb{R}^ 2\). Show that:
2.4.19. \(∇(c f ) = c∇f \)
2.4.20. \(∇(f + g) = ∇f + ∇g\)
2.4.21. \(∇(f g) = f ∇g + g∇f\)
2.4.22. \(∇(f /g) = \frac{g∇f − f ∇g}{ g^ 2}\text{ if }g(x, y) \neq 0\)
2.4.23. \(D_{−v} f = −D_v f\)
2.4.24. \(D_v(c f ) = c D_v f\)
2.4.25. \(D_v(f + g) = D_v f + D_v g\)
2.4.26. \(D_v(f g) = f D_v g + g D_v f\)
2.4.27. The function \(r(x, y) = \sqrt{ x^ 2 + y^ 2}\) is the length of the position vector \(\textbf{r} = x\textbf{i} + y\textbf{j}\) for each point \((x, y)\) in \(\mathbb{R}^ 2\). Show that \(∇r = \frac{1}{ r} \textbf{r}\) when \((x, y) \neq (0,0)\), and that \(∇(r^ 2 ) = 2\textbf{r}\).
2.5: Maxima and Minima
A
For Exercises 1-10, find all local maxima and minima of the function \(f (x, y)\).
2.5.1. \(f (x, y) = x^ 3 −3x+ y^ 2\)
2.5.2. \(f (x, y) = x^ 3 −12x+ y^ 2 +8y\)
2.5.3. \(f (x, y) = x^ 3 −3x+ y^ 3 −3y\)
2.5.4. \(f (x, y) = x^ 3 +3x^ 2 + y^ 3 −3y^ 2\)
2.5.5. \(f (x, y) = 2x^ 3 +6x y+3y^ 2\)
2.5.6. \(f (x, y) = 2x^ 3 −6x y+ y^ 2\)
2.5.7. \(f (x, y) = \sqrt{ x^ 2 + y^ 2}\)
2.5.8. \(f (x, y) = x+2y\)
2.5.9. \(f (x, y) = 4x^ 2 −4x y+2y^ 2 +10x−6y\)
2.5.10. \(f (x, y) = −4x^ 2 +4x y−2y^ 2 +16x−12y\)
B
2.5.11. For a rectangular solid of volume 1000 cubic meters, find the dimensions that will minimize the surface area. ( Hint: Use the volume condition to write the surface area as a function of just two variables. )
2.5.12. Prove that if \((a,b)\) is a local maximum or local minimum point for a smooth function \(f (x, y)\), then the tangent plane to the surface \(z = f (x, y)\) at the point \((a,b, f (a,b))\) is parallel to the \(x y\)-plane. ( Hint: Use Theorem 2.5 .)
C
2.5.13. Find three positive numbers \(x, y, z\) whose sum is 10 such that \(x^ 2 y^ 2 z\) is a maximum.
2.6: Unconstrained Optimization: Numerical Methods
C
2.6.1. Recall Example 2.21 from the previous section, where we showed that the point \((2,1)\) was a global minimum for the function \(f (x, y) = (x −2)^4 +(x −2y)^ 2\). Notice that our computer program can be modified fairly easily to use this function (just change the return values in the fx, fy, fxx, fyy and fxy function definitions to use the appropriate partial derivative). Either modify that program or write one of your own in a programming language of your choice to show that Newton’s algorithm does lead to the point \((2,1)\). First use the initial point \((0,3)\), then use the initial point \((3,2)\), and compare the results. Make sure that your program attempts to do 100 iterations of the algorithm. Did anything strange happen when your program ran? If so, how do you explain it? ( Hint: Something strange should happen. )
2.6.2. There is a version of Newton’s algorithm for solving a system of two equations
\[f_1(x, y) = 0 \quad \text{ and }\quad f_2(x, y) = 0 ,\]
where \(f_1(x, y) \text{ and }f_2(x, y)\) are smooth real-valued functions:
Pick an initial point \((x_0 , y_0)\). For \(n = 0,1,2,3,...,\) define:
\[x_{n+1} = x_n - \frac{\begin{vmatrix} f_1(x_n, y_n) & f_2(x_n, y_n) \\ \frac{∂f_1}{ ∂y} (x_n, y_n) & \frac{∂f_2}{ ∂y} (x_n, y_n) \\ \end{vmatrix}}{D(x_n, y_n)},\quad y_{n+1} = y_n + \frac{\begin{vmatrix} f_1(x_n, y_n) & f_2(x_n, y_n) \\ \frac{∂f_1}{ ∂x} (x_n, y_n) & \frac{∂f_2}{ ∂x} (x_n, y_n) \\ \end{vmatrix}}{D(x_n, y_n)},\text{ where} \]
\[D(x_n, y_n) = \frac{∂f_1}{ ∂x} (x_n, y_n) \frac{∂f_2}{ ∂y} (x_n, y_n)− \frac{∂f_1}{ ∂y} (x_n, y_n) \frac{∂f_2}{ ∂x} (x_n, y_n) .\]
Then the sequence of points \((x_n, y_n)_{n=1}^{\infty}\) converges to a solution. Write a computer program that uses this algorithm to find approximate solutions to the system of equations
\[f_1(x, y) = \sin (x y)− x− y = 0 \quad \text{ and }\quad f_2(x, y) = e^{ 2x} −2x+3y = 0 .\]
Show that you get two different solutions when using \((0,0) \text{ and }(1,1)\) for the initial point \((x_0 , y_0)\).
2.7: Constrained Optimization: Lagrange Multipliers
A
2.7.1. Find the constrained maxima and minima of \(f (x, y) = 2x+ y\) given that \(x^ 2 + y^ 2 = 4\).
2.7.2. Find the constrained maxima and minima of \(f (x, y) = x y\) given that \(x^ 2 +3y^ 2 = 6\).
2.7.3. Find the points on the circle \(x^ 2+ y^ 2 = 100\) which are closest to and farthest from the point \((2,3)\).
B
2.7.4. Find the constrained maxima and minima of \(f (x, y, z) = x + y^ 2 +2z\) given that \(4x^ 2 +9y^ 2 − 36z^ 2 = 36\).
2.7.5. Find the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid
\[\frac{x^ 2}{ a^ 2} + \frac{y^ 2}{ b^ 2} + \frac{z^ 2}{ c^ 2} = 1 .\]