. You are told that there is a function f whose partial deriva- tives are f(x, y) = x + 4y and fy(x, y) = 3x - y. Should you believe it? raboloid
. You are told that there is a function f whose partial deriva- tives are f(x, y) = x + 4y and fy(x, y) = 3x - y. Should you believe it? raboloid
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
97
![domain of
94. The average energy E (in kcal) needed for a lizard to
dwalk or run a distance of 1 km has been modeled by
the equation
12.8
=
al of zomil nog E(m, v):
2.65m0.66 +
Saniq
306 Ingv
where m is the body mass of the lizard (in grams) and vis
its speed (in km/h). Calculate Em (400, 8) and E,(400, 8)
and interpret your answers. ai
0116 Source: C. Robbins, Wildlife Feeding and Nutrition, 2d ed. (San Diego:
PomiAcademic Press, 1993).
deuomit eas
3.5m 0.75
95. The kinetic energy of a body with mass m and velocity v is
(K=mv². Show that mak
siq vos sri
ƏK ²K
0am dv²
əm
= K
=
(ox - x)8 + (px
96.
If a, b, c are the sides of a triangle and A, B, C are the
opposite angles, find A/da, dA/ab, A/ac by implicit
differentiation of the Law of Cosines.
97. You are told that there is a function f whose partial deriva-
tives are f(x, y) = x + 4y and fy(x, y) = 3x - y. Should
(= suves a
you believe it?
ud
25vig
98. The paraboloid z = 6 - x - x² - 2y² intersects the plane
x = 1 in a parabola. Find parametric equations for the
tangent line to this parabola at the point (1, 2, −4). Use a
computer to graph the paraboloid, the parabola, and the
tangent line on the same screen.
900
an (II)
(1-7)S
99. The ellipsoid 4x² + 2y² + z² = 16 intersects the plane
y = 2 in an ellipse. Find parametric equations for the tan-
gent line to this ellipse at the point (1, 2, 2).
where w=
2π/365 and A is a positive constant.
(a) Find aT/ax. What is its physical significance?
O
sib is
mo
100. In a study of frost penetration it was found that the temper-
ature T at time t (measured in days) at a depth x (measured
in feet) can be modeled by the function
T(x, t) = To + Tex sin(wt - Ax) (0%) ==
Jrigg
28
CAS
Inspn
(b) Find aT/at. What is its physical significance?
(c) Show that I satisfies the heat equation T, = kTxx for a
certain constant k.
0, and T₁
(d) If X 0.2, To
graph T(x, t).
bill
(e) What is the physical significance of the term -λx in
the expression sin(wt - Ax)?
101. Use Clairaut's Theorem to show that if the third-order
partial derivatives of f are continuous, then
fxyy=fyxy = fyyx
qn vlogola
102. (a) How many nth-order partial derivatives does a func-
to abution of two variables have?
BAR
103. If
104. If f(x, y) = √√x³ + y³, find f(0, 0).
908
osing
105. Let
mont
16 intersects the plane-guinug.
(ox ox) =
Islimia
onil indgas f(x, y) =
Oint
is this
14.4 Tangent Planes and Linear Approximations
S = (1 02
=
(b) If these partial derivatives are all continuous, how
many of them can be distinct?
pu
(c) Answer the question in part (a) for a function of three
2
Tonil to find fx(1, 0). [Hint: Instead of finding fx(x, y) first, note
that it's easier to use Equation 1 or Equation 2.]
10, use a computer to
variables.
Inter
FFP
f(x, y) = x(x² + y²)-3/2 sin(x²y)
opdinarallel limes, which
3
x³y - xy³
x² + y²
-
if (x, y) = (0, 0)
if (x, y) = (0, 0)
2
(a) Use a computer to graph f.
(b) Find f(x, y) and fy(x, y) when (x, y) = (0, 0).
bra anglo
(c) Find f(0, 0) and fy(0, 0) using Equations 2 and 3.
(d) Show that fry (0, 0) -1 and fyx(0, 0) = 1.
(e) Does the result of part (d) contradict Clairaut's
Theorem? Use graphs of fxy and fyx to illustrate your
answer.
vinalimia
imia
eri
art
ato
to noil
(1.1);
One of the most important ideas in single-variable calculus is that as we zoom in toward
a point on the graph of a differentiable function, the graph becomes indistinguishable
from its tangent line and we can approximate the function by a linear function. (See Sec-
tion 3.10.) Here we develop similar ideas in three dimensions. As we zoom in toward a
point on a surface that is the graph of a differentiable function of two variables, the sur-
face looks more and more like a plane (its tangent plane) and we can approximate the
wted (E11) 16 sasiq insgast 21 function by a linear function of two variables. We also extend the idea of a differential to
y yd (E.I.1) Inioq sdj buswol functions of two or more variables.
the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F89e107ee-bcda-4baf-97d6-3bdf5aeb3272%2F5412cd9e-9961-4a60-93d2-d243ea1b3d1a%2Fpsj9ll_processed.jpeg&w=3840&q=75)
Transcribed Image Text:domain of
94. The average energy E (in kcal) needed for a lizard to
dwalk or run a distance of 1 km has been modeled by
the equation
12.8
=
al of zomil nog E(m, v):
2.65m0.66 +
Saniq
306 Ingv
where m is the body mass of the lizard (in grams) and vis
its speed (in km/h). Calculate Em (400, 8) and E,(400, 8)
and interpret your answers. ai
0116 Source: C. Robbins, Wildlife Feeding and Nutrition, 2d ed. (San Diego:
PomiAcademic Press, 1993).
deuomit eas
3.5m 0.75
95. The kinetic energy of a body with mass m and velocity v is
(K=mv². Show that mak
siq vos sri
ƏK ²K
0am dv²
əm
= K
=
(ox - x)8 + (px
96.
If a, b, c are the sides of a triangle and A, B, C are the
opposite angles, find A/da, dA/ab, A/ac by implicit
differentiation of the Law of Cosines.
97. You are told that there is a function f whose partial deriva-
tives are f(x, y) = x + 4y and fy(x, y) = 3x - y. Should
(= suves a
you believe it?
ud
25vig
98. The paraboloid z = 6 - x - x² - 2y² intersects the plane
x = 1 in a parabola. Find parametric equations for the
tangent line to this parabola at the point (1, 2, −4). Use a
computer to graph the paraboloid, the parabola, and the
tangent line on the same screen.
900
an (II)
(1-7)S
99. The ellipsoid 4x² + 2y² + z² = 16 intersects the plane
y = 2 in an ellipse. Find parametric equations for the tan-
gent line to this ellipse at the point (1, 2, 2).
where w=
2π/365 and A is a positive constant.
(a) Find aT/ax. What is its physical significance?
O
sib is
mo
100. In a study of frost penetration it was found that the temper-
ature T at time t (measured in days) at a depth x (measured
in feet) can be modeled by the function
T(x, t) = To + Tex sin(wt - Ax) (0%) ==
Jrigg
28
CAS
Inspn
(b) Find aT/at. What is its physical significance?
(c) Show that I satisfies the heat equation T, = kTxx for a
certain constant k.
0, and T₁
(d) If X 0.2, To
graph T(x, t).
bill
(e) What is the physical significance of the term -λx in
the expression sin(wt - Ax)?
101. Use Clairaut's Theorem to show that if the third-order
partial derivatives of f are continuous, then
fxyy=fyxy = fyyx
qn vlogola
102. (a) How many nth-order partial derivatives does a func-
to abution of two variables have?
BAR
103. If
104. If f(x, y) = √√x³ + y³, find f(0, 0).
908
osing
105. Let
mont
16 intersects the plane-guinug.
(ox ox) =
Islimia
onil indgas f(x, y) =
Oint
is this
14.4 Tangent Planes and Linear Approximations
S = (1 02
=
(b) If these partial derivatives are all continuous, how
many of them can be distinct?
pu
(c) Answer the question in part (a) for a function of three
2
Tonil to find fx(1, 0). [Hint: Instead of finding fx(x, y) first, note
that it's easier to use Equation 1 or Equation 2.]
10, use a computer to
variables.
Inter
FFP
f(x, y) = x(x² + y²)-3/2 sin(x²y)
opdinarallel limes, which
3
x³y - xy³
x² + y²
-
if (x, y) = (0, 0)
if (x, y) = (0, 0)
2
(a) Use a computer to graph f.
(b) Find f(x, y) and fy(x, y) when (x, y) = (0, 0).
bra anglo
(c) Find f(0, 0) and fy(0, 0) using Equations 2 and 3.
(d) Show that fry (0, 0) -1 and fyx(0, 0) = 1.
(e) Does the result of part (d) contradict Clairaut's
Theorem? Use graphs of fxy and fyx to illustrate your
answer.
vinalimia
imia
eri
art
ato
to noil
(1.1);
One of the most important ideas in single-variable calculus is that as we zoom in toward
a point on the graph of a differentiable function, the graph becomes indistinguishable
from its tangent line and we can approximate the function by a linear function. (See Sec-
tion 3.10.) Here we develop similar ideas in three dimensions. As we zoom in toward a
point on a surface that is the graph of a differentiable function of two variables, the sur-
face looks more and more like a plane (its tangent plane) and we can approximate the
wted (E11) 16 sasiq insgast 21 function by a linear function of two variables. We also extend the idea of a differential to
y yd (E.I.1) Inioq sdj buswol functions of two or more variables.
the
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