X2 4. g(x. y.2) = y +: (1, 1, 1)

Linear Algebra: A Modern Introduction
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Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...
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EXERCISES 12.3
In Exercises 1-10, find all the first partial derivatives of the
function specified, and evaluate them at the given point.
1. f(x, y) = x - y + 2, (3,2)
2. f(x, y) = xy + x², (2,0)
3. f(x. v.2) = x³v*z*, (0,-1,–1)
18. f(x, y) = y e¬*² at (0, 1)
19. f(x, y) = In(x² + y²) at (1, –2)
2ху
x² + y²
20. f(х, у) —
at (0, 2)
21. f(x, y) = tan-' (y/x) at (1, –1)
22. f(x, y) = /1+ x³y² at (2, 1)
23. Find the coordinates of all points on the surface with equation
2 = x* - 4xy' + 6y² – 2 where the surface has a horizontal
tangent plane.
X2
4. g(x, y, z) =
(1,1, 1)
y +:
5. z = tan- (2). (-1,1)
6. w = In(1 + e*>yz), (2,0,–1)
7. f(x, y) = sin(x /). (.4)
24. Find all horizontal planes that are tangent to the surface with
equation z = xye-(x²+y³)/2, At what points are they
tangent?
8. f(x, y) =-
(-3, 4)
In Exercises 25–31, show that the given function satisfies the given
partinl differentialLoquation.
E3 25. z = x e", x
9. w = xO m2), (e,2,e)
X1 - x
X3 + x
In Exercises 11-12, calculate the first partial derivatives of the
given functions at (0, 0). You will have to use Definition 4.
2x- y3
az az
dy
az
+ y
10. g(x1, x2, X3, X4)
(3, 1, -1,-2)
ax
E3 26. 2
x + y
az
az
E3 27. z = Vx2 + y?, x
+ y
11. f(x. y) = {x² + 3y '
if (x, y) # (0,0)
ax
ay
0,
if (x, y) = (0,0).
E3 28. w = x? + yz, x
+z
= 2w
y
ax
ay
az
x² – 2y²
if x # y
E3 29. w
dw
+ z
az
12. /(х. у) —
+ y
-2w
X-y
0,
ax
ay
if x = y.
In Exercises 13-22, find equations of the tangent plane and normal
E3 30. z = f(x² + y²), where f is any differentiable function of
one variable,
line to the graph of the given function at the point with specified
values of x and y.
az
az
= 0.
dy
13. f(x, y) = x² – y² at (-2, 1)
y
ax
14. f(x, y) =
at (1, 1)
x + y
E3 31. :=
f(x² – y²), where f is any differentiable function of
15. f(x. v) = cos(x/y) at (7, 4)
16. f(x, y) = e*y at (2,0)
one variable,
Transcribed Image Text:EXERCISES 12.3 In Exercises 1-10, find all the first partial derivatives of the function specified, and evaluate them at the given point. 1. f(x, y) = x - y + 2, (3,2) 2. f(x, y) = xy + x², (2,0) 3. f(x. v.2) = x³v*z*, (0,-1,–1) 18. f(x, y) = y e¬*² at (0, 1) 19. f(x, y) = In(x² + y²) at (1, –2) 2ху x² + y² 20. f(х, у) — at (0, 2) 21. f(x, y) = tan-' (y/x) at (1, –1) 22. f(x, y) = /1+ x³y² at (2, 1) 23. Find the coordinates of all points on the surface with equation 2 = x* - 4xy' + 6y² – 2 where the surface has a horizontal tangent plane. X2 4. g(x, y, z) = (1,1, 1) y +: 5. z = tan- (2). (-1,1) 6. w = In(1 + e*>yz), (2,0,–1) 7. f(x, y) = sin(x /). (.4) 24. Find all horizontal planes that are tangent to the surface with equation z = xye-(x²+y³)/2, At what points are they tangent? 8. f(x, y) =- (-3, 4) In Exercises 25–31, show that the given function satisfies the given partinl differentialLoquation. E3 25. z = x e", x 9. w = xO m2), (e,2,e) X1 - x X3 + x In Exercises 11-12, calculate the first partial derivatives of the given functions at (0, 0). You will have to use Definition 4. 2x- y3 az az dy az + y 10. g(x1, x2, X3, X4) (3, 1, -1,-2) ax E3 26. 2 x + y az az E3 27. z = Vx2 + y?, x + y 11. f(x. y) = {x² + 3y ' if (x, y) # (0,0) ax ay 0, if (x, y) = (0,0). E3 28. w = x? + yz, x +z = 2w y ax ay az x² – 2y² if x # y E3 29. w dw + z az 12. /(х. у) — + y -2w X-y 0, ax ay if x = y. In Exercises 13-22, find equations of the tangent plane and normal E3 30. z = f(x² + y²), where f is any differentiable function of one variable, line to the graph of the given function at the point with specified values of x and y. az az = 0. dy 13. f(x, y) = x² – y² at (-2, 1) y ax 14. f(x, y) = at (1, 1) x + y E3 31. := f(x² – y²), where f is any differentiable function of 15. f(x. v) = cos(x/y) at (7, 4) 16. f(x, y) = e*y at (2,0) one variable,
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