Given a role \(f\) of two contained variables \(x\) and \(y\text{,}\) how are the 2d-lodge partial derivatives of \(f\) defined?
What do the second-society fractional derivatives \(f_{xx}\text{,}\) \(f_{yy}\text{,}\) \(f_{xy}\text{,}\) and \(f_{yx}\) of a function \(f\) tell us virtually the function's behavior?
Call up that for a single-variable function \(f\text{,}\) the 2nd derivative of \(f\) is divers to be the derivative of the first derivative. That is, \(f''(x) = \frac{d}{dx}[f'(x)]\text{,}\) which can exist stated in terms of the limit definition of the derivative past writing
In what follows, we begin exploring the four different second-lodge partial derivatives of a function of two variables and seek to understand what these diverse derivatives tell us nigh the function'southward behavior.
Preview Activity10.three.ane.
Once more, let'due south consider the function \(f\) divers by \(f(x,y) = \frac{ten^2\sin(2y)}{32}\) that measures a projectile'southward range every bit a function of its initial speed \(10\) and launch angle \(y\text{.}\) The graph of this function, including traces with \(x=150\) and \(y=0.half dozen\text{,}\) is shown in Figure 10.3.1.
Effigy10.3.1.The distance role with traces \(10=150\) and \(y=0.half dozen\text{.}\)
Compute the partial derivative \(f_x\text{.}\) Discover that \(f_x\) itself is a new office of \(x\) and \(y\text{,}\) so nosotros may now compute the partial derivatives of \(f_x\text{.}\) Observe the partial derivative \(f_{xx} = (f_x)_x\) and show that \(f_{20}(150,0.6) \approx 0.058\text{.}\)
Figure 10.iii.2 shows the trace of \(f\) with \(y=0.half-dozen\) with three tangent lines included. Explain how your result from part (a) of this preview activity is reflected in this effigy.
Figure10.3.ii.The trace with \(y=0.6\text{.}\)
Determine the fractional derivative \(f_y\text{,}\) and so find the fractional derivative \(f_{yy}=(f_y)_y\text{.}\) Evaluate \(f_{yy}(150, 0.half dozen)\text{.}\)
Figurex.3.3.More than traces of the range function.
Figure 10.3.3 shows the trace \(f(150, y)\) and includes three tangent lines. Explain how the value of \(f_{yy}(150,0.vi)\) is reflected in this figure.
Because \(f_x\) and \(f_y\) are each functions of both \(ten\) and \(y\text{,}\) they each have two fractional derivatives. Not simply can we compute \(f_{xx} = (f_x)_x\text{,}\) only likewise \(f_{xy} = (f_x)_y\text{;}\) as well, in add-on to \(f_{yy} = (f_y)_y\text{,}\) but also \(f_{yx} = (f_y)_x\text{.}\) For the range function \(f(10,y) = \frac{10^2\sin(2y)}{32}\text{,}\) use your earlier computations of \(f_x\) and \(f_y\) to now determine \(f_{xy}\) and \(f_{yx}\text{.}\) Write i sentence to explain how you calculated these "mixed" partial derivatives.
Subsection10.3.12nd-Order Partial Derivatives
A function \(f\) of two independent variables \(x\) and \(y\) has two first club partial derivatives, \(f_x\) and \(f_y\text{.}\) As we saw in Preview Activity 10.3.1, each of these showtime-society partial derivatives has ii fractional derivatives, giving a total of four second-order partial derivatives:
means that we showtime differentiate with respect to \(x\) and then with respect to \(y\text{;}\) this tin can be expressed in the alternating notation \(f_{xy} = (f_x)_y\text{.}\) However, to find the second fractional derivative
Be certain to note carefully the difference between Leibniz note and subscript notation and the order in which \(ten\) and \(y\) appear in each. In improver, remember that anytime we compute a partial derivative, we agree constant the variable(s) other than the one we are differentiating with respect to.
Activity10.3.2.
Find all 2nd society partial derivatives of the following functions. For each partial derivative you calculate, state explicitly which variable is being held constant.
\(\displaystyle f(x,y) = x^2y^3\)
\(\displaystyle f(x,y) = y\cos(x)\)
\(\displaystyle g(s,t) = st^3 + southward^4\)
How many second order partial derivatives does the office \(h\) defined by \(h(x,y,z) = 9x^9z-xyz^ix + 9\) accept? Find \(h_{xz}\) and \(h_{zx}\) (you do not need to detect the other second order partial derivatives).
In Preview Action 10.iii.one and Activity 10.three.2, you may accept noticed that the mixed 2d-order fractional derivatives are equal. This observation holds generally and is known as Clairaut'south Theorem.
Clairaut'south Theorem.
Let \(f\) be a function of several variables for which the partial derivatives \(f_{xy}\) and \(f_{yx}\) are continuous near the bespeak \((a,b)\text{.}\) And so
Subsectionten.iii.2Interpreting the Second-Order Partial Derivatives
Think from unmarried variable calculus that the second derivative measures the instantaneous charge per unit of change of the derivative. This observation is the key to understanding the meaning of the second-order fractional derivatives.
Figure10.three.four.The tangent lines to a trace with increasing \(x\text{.}\)
Furthermore, nosotros recall that the second derivative of a role at a point provides us with information near the concavity of the function at that bespeak. Since the unmixed 2d-guild partial derivative \(f_{xx}\) requires us to hold \(y\) constant and differentiate twice with respect to \(x\text{,}\) we may but view \(f_{twenty}\) as the second derivative of a trace of \(f\) where \(y\) is fixed. As such, \(f_{xx}\) will measure the concavity of this trace.
Consider, for example, \(f(x,y) = \sin(ten) e^{-y}\text{.}\) Figure 10.3.4 shows the graph of this function along with the trace given past \(y=-1.5\text{.}\) Also shown are three tangent lines to this trace, with increasing \(x\)-values from left to right among the three plots in Figure x.3.four.
That the slope of the tangent line is decreasing as \(ten\) increases is reflected, every bit it is in i-variable calculus, in the fact that the trace is concave downwards. Indeed, we run into that \(f_x(ten,y)=\cos(x)e^{-y}\) then \(f_{20}(x,y)=-\sin(x)due east^{-y} \lt 0\text{,}\) since \(e^{-y} > 0\) for all values of \(y\text{,}\) including \(y = -1.5\text{.}\)
In the post-obit action, we further explore what second-guild fractional derivatives tell us about the geometric behavior of a surface.
Actionx.iii.3.
We continue to consider the part \(f\) divers by \(f(x,y) = \sin(x) e^{-y}\text{.}\)
In Figure 10.3.5, we see the trace of \(f(x,y) = \sin(10) e^{-y}\) that has \(10\) held abiding with \(x = 1.75\text{.}\) We also see 3 different lines that are tangent to the trace of \(f\) in the \(y\) direction at values of \(y\) that are increasing from left to right in the figure. Write a couple of sentences that describe whether the gradient of the tangent lines to this bend increase or decrease equally \(y\) increases, and, after computing \(f_{yy}(ten,y)\text{,}\) explain how this observation is related to the value of \(f_{yy}(1.75,y)\text{.}\) Be sure to accost the notion of concavity in your response.(You need to exist careful about the directions in which \(x\) and \(y\) are increasing.)
Effigy10.3.v.The tangent lines to a trace with increasing \(y\text{.}\)
In Figure ten.3.6, nosotros start to recollect near the mixed partial derivative, \(f_{xy}\text{.}\) Hither, nosotros offset hold \(y\) constant to generate the get-go-gild partial derivative \(f_x\text{,}\) and then we concord \(10\) constant to compute \(f_{xy}\text{.}\) This leads to get-go thinking nearly a trace with \(x\) being abiding, followed by slopes of tangent lines in the \(x\)-direction that slide forth the original trace. You might think of sliding your pencil down the trace with \(x\) constant in a way that its slope indicates \((f_x)_y\) in order to farther breathing the 3 snapshots shown in the effigy.
Figure10.iii.6.The trace of \(z = f(ten,y) = \sin(x)e^{-y}\) with \(ten = ane.75\text{,}\) along with tangent lines in the \(y\)-direction at three dissimilar points. Based on Figure ten.iii.six, is \(f_{xy}(1.75, -1.5)\) positive or negative? Why?
Determine the formula for \(f_{xy}(x,y)\text{,}\) and hence evaluate \(f_{xy}(1.75, -1.v)\text{.}\) How does this value compare with your observations in (b)?
We know that \(f_{20}(one.75, -i.5)\) measures the concavity of the \(y = -1.five\) trace, and that \(f_{yy}(ane.75, -ane.five)\) measures the concavity of the \(x = 1.75\) trace. What do yous think the quantity \(f_{xy}(one.75, -1.5)\) measures?
On Figure ten.3.6, sketch the trace with \(y = -1.5\text{,}\) and sketch iii tangent lines whose slopes correspond to the value of \(f_{yx}(x,-ane.5)\) for three different values of \(ten\text{,}\) the middle of which is \(x = -1.5\text{.}\) Is \(f_{yx}(one.75, -1.5)\) positive or negative? Why? What does \(f_{yx}(1.75, -1.5)\) measure?
Merely as with the first-social club partial derivatives, nosotros can approximate second-gild partial derivatives in the state of affairs where we have only partial data about the function.
Activity10.3.four.
As nosotros saw in Activity ten.2.5, the wind chill \(w(v,T)\text{,}\) in degrees Fahrenheit, is a function of the wind speed, in miles per hour, and the air temperature, in degrees Fahrenheit. Some values of the wind chill are recorded in Table 10.three.7.
Table10.iii.vii.Wind chill every bit a function of wind speed and temperature.
\(v \backslash T\)
-30
-25
-20
-15
-x
-five
0
5
10
15
20
5
-46
-twoscore
-34
-28
-22
-sixteen
-11
-v
1
7
xiii
x
-53
-47
-41
-35
-28
-22
-16
-10
-4
3
9
15
-58
-51
-45
-39
-32
-26
-19
-13
-seven
0
6
20
-61
-55
-48
-42
-35
-29
-22
-15
-nine
-2
four
25
-64
-58
-51
-44
-37
-31
-24
-17
-11
-4
iii
30
-67
-lx
-53
-46
-39
-33
-26
-xix
-12
-v
one
35
-69
-62
-55
-48
-41
-34
-27
-21
-14
-7
0
40
-71
-64
-57
-50
-43
-36
-29
-22
-xv
-8
-ane
Estimate the partial derivatives \(w_{T}(20,-15)\text{,}\) \(w_{T}(20,-10)\text{,}\) and \(w_T(xx,-five)\text{.}\) Employ these results to guess the 2nd-club partial \(w_{TT}(20, -ten)\text{.}\)
In a similar manner, estimate the 2d-order partial \(w_{vv}(20,-10)\text{.}\)
Estimate the fractional derivatives \(w_T(xx,-10)\text{,}\) \(w_T(25,-10)\text{,}\) and \(w_T(fifteen,-10)\text{,}\) and utilise your results to estimate the partial \(w_{Tv}(twenty,-10)\text{.}\)
In a similar way, guess the partial derivative \(w_{vT}(20,-10)\text{.}\)
Write several sentences that explain what the values \(w_{TT}(20, -10)\text{,}\) \(w_{vv}(20,-x)\text{,}\) and \(w_{Tv}(xx,-10)\) betoken regarding the behavior of \(w(v,T)\text{.}\)
Every bit we accept found in Activities 10.three.3 and Activeness 10.3.iv, we may think of \(f_{xy}\) as measuring the "twist" of the graph every bit we increase \(y\) along a item trace where \(x\) is held constant. In the same way, \(f_{yx}\) measures how the graph twists as we increase \(x\text{.}\) If we think that Clairaut's theorem tells us that \(f_{xy} = f_{yx}\text{,}\) nosotros see that the amount of twisting is the aforementioned in both directions. This twisting is perhaps more easily seen in Figure 10.3.8, which shows the graph of \(f(x,y) = -xy\text{,}\) for which \(f_{xy} = -1\text{.}\)
Figureten.3.8.The graph of \(f(x,y) = -xy\text{.}\)
Subsection10.3.3Summary
In that location are four 2d-order partial derivatives of a function \(f\) of 2 independent variables \(10\) and \(y\text{:}\)
The unmixed second-order partial derivatives, \(f_{20}\) and \(f_{yy}\text{,}\) tell the states about the concavity of the traces. The mixed 2nd-order partial derivatives, \(f_{xy}\) and \(f_{yx}\text{,}\) tell us how the graph of \(f\) twists.
Exercises10.3.4Exercises
one.
Calculate all iv 2nd-order partial derivatives of \(\displaystyle f(ten,y) = 4x^{2}y+8xy^{3}\text{.}\)
\(f_{xx} \, (10,y) =\)
\(f_{xy} \, (x,y) =\)
\(f_{yx} \, (x,y) =\)
\(f_{yy} \, (x,y) =\)
2.
Find all the first and 2d social club fractional derivatives of \(f(ten, y) = iii\sin(2x+y) - 4\cos(x-y)\text{.}\)
A. \(\frac{\partial f}{\partial x} = f_x =\)
B. \(\frac{\partial f}{\fractional y} = f_y =\)
C. \(\frac{{\fractional^2}f}{\partial x^2} = f_x{}_x =\)
D. \(\frac{{\partial^2}f}{\partial y^2} = f_y{}_y =\)
East. \(\frac{{\partial^two}f}{\fractional x \partial y} = f_y{}_x =\)
F. \(\frac{{\partial^two}f}{\fractional y \fractional x} = f_x{}_y =\)
Given \(F(r,south,t)=r\!\left(9s^{4}-t^{5}\correct)\text{,}\) compute:
\(F_{rst}=\)
6.
Calculate all four second-gild fractional derivatives and bank check that \(f_{xy}=f_{yx}\text{.}\) Presume the variables are restricted to a domain on which the function is defined.
If \(z_{xy} = 5 y\) and all of the second club partial derivatives of \(z\) are continuous, then
(a) \(z_{yx} =\)
(b) \(z_{xyx} =\)
(c) \(z_{xyy} =\)
10.
If \(z = f(10) + y g(x)\text{,}\) what tin can nosotros say near \(z_{yy}\text{?}\)
\(\displaystyle z_{yy} = 0\)
\(\displaystyle z_{yy} = y\)
\(\displaystyle z_{yy} = z_{20}\)
\(\displaystyle z_{yy} = chiliad(x)\)
We cannot say anything
11.
Shown in Figure 10.3.9 is a contour plot of a office \(f\) with the values of \(f\) labeled on the contours. The point \((2,1)\) is highlighted in red.
Figure10.3.9.A contour plot of \(f(x,y)\text{.}\)
Gauge the partial derivatives \(f_x(2,i)\) and \(f_y(2,1)\text{.}\)
Determine whether the second-guild partial derivative \(f_{xx}(2,ane)\) is positive or negative, and explain your thinking.
Make up one's mind whether the second-gild partial derivative \(f_{yy}(2,1)\) is positive or negative, and explain your thinking.
Determine whether the second-guild partial derivative \(f_{xy}(ii,ane)\) is positive or negative, and explain your thinking.
Determine whether the second-order partial derivative \(f_{yx}(ii,i)\) is positive or negative, and explicate your thinking.
Consider a function \(grand\) of the variables \(x\) and \(y\) for which \(g_x(ii,2) > 0\) and \(g_{20}(2,two) \lt 0\text{.}\) Sketch possible behavior of some contours around \((2,ii)\) on the left axes in Figure x.iii.10.
Figurex.3.10.Plots for contours of \(g\) and \(h\text{.}\)
Consider a function \(h\) of the variables \(ten\) and \(y\) for which \(h_x(2,2) > 0\) and \(h_{xy}(two,ii) \lt 0\text{.}\) Sketch possible behavior of some contour lines around \((two,2)\) on the correct axes in Figure 10.3.10.
12.
The Heat Index, \(I\text{,}\) (measured in apparent degrees F) is a part of the actual temperature \(T\) outside (in degrees F) and the relative humidity \(H\) (measured as a percent). A portion of the table which gives values for this part, \(I(T,H)\text{,}\) is reproduced in Tabular array x.3.11.
Table10.three.11.Heat index.
T \(\downarrow \backslash\) H \(\rightarrow\)
70
75
80
85
90
106
109
112
115
92
112
115
119
123
94
118
122
127
132
96
125
130
135
141
State the limit definition of the value \(I_{TT}(94,75)\text{.}\) Then, estimate \(I_{TT}(94,75)\text{,}\) and write one consummate sentence that carefully explains the meaning of this value, including units.
Country the limit definition of the value \(I_{HH}(94,75)\text{.}\) And so, guess \(I_{HH}(94,75)\text{,}\) and write one consummate sentence that advisedly explains the significant of this value, including units.
Finally, do likewise to estimate \(I_{HT}(94,75)\text{,}\) and write a sentence to explicate the meaning of the value you constitute.
13.
The temperature on a heated metallic plate positioned in the showtime quadrant of the \(xy\)-plane is given by
Presume that temperature is measured in degrees Celsius and that \(x\) and \(y\) are each measured in inches.
Determine \(C_{xx}(x,y)\) and \(C_{yy}(x,y)\text{.}\) Do not practice any additional work to algebraically simplify your results.
Summate \(C_{xx}(1.1, 1.two)\text{.}\) Suppose that an ant is walking past the point \((1.1, 1.2)\) along the line \(y = 1.2\text{.}\) Write a sentence to explain the meaning of the value of \(C_{xx}(1.1, 1.2)\text{,}\) including units.
Calculate \(C_{yy}(ane.1, 1.2)\text{.}\) Suppose instead that an ant is walking past the point \((i.i, 1.2)\) along the line \(x = 1.1\text{.}\) Write a sentence to explain the meaning of the value of \(C_{yy}(one.1, 1.2)\text{,}\) including units.
Determine \(C_{xy}(x,y)\) and hence compute \(C_{xy}(1.i, i.2)\text{.}\) What is the meaning of this value? Explain, in terms of an ant walking on the heated metal plate.
14.
Let \(f(x,y) = 8 - ten^2 - y^ii\) and \(1000(10,y) = 8 - 10^2 + 4xy - y^2\text{.}\)
Determine \(f_x\text{,}\) \(f_y\text{,}\) \(f_{20}\text{,}\) \(f_{yy}\text{,}\) \(f_{xy}\text{,}\) and \(f_{yx}\text{.}\)
Evaluate each of the fractional derivatives in (a) at the point \((0,0)\text{.}\)
What practice the values in (b) propose most the behavior of \(f\) almost \((0,0)\text{?}\) Plot a graph of \(f\) and compare what you see visually to what the values advise.
Decide \(g_x\text{,}\) \(g_y\text{,}\) \(g_{xx}\text{,}\) \(g_{yy}\text{,}\) \(g_{xy}\text{,}\) and \(g_{yx}\text{.}\)
Evaluate each of the partial derivatives in (d) at the point \((0,0)\text{.}\)
What exercise the values in (e) suggest almost the beliefs of \(g\) near \((0,0)\text{?}\) Plot a graph of \(g\) and compare what you lot come across visually to what the values suggest.
What practice the functions \(f\) and \(thou\) accept in common at \((0,0)\text{?}\) What is dissimilar? What do your observations tell you regarding the importance of a certain second-order partial derivative?
fifteen.
Let \(f(x,y) = \frac{1}{2}xy^2\) represent the kinetic energy in Joules of an object of mass \(x\) in kilograms with velocity \(y\) in meters per second. Let \((a,b)\) exist the bespeak \((4,v)\) in the domain of \(f\text{.}\)
Calculate \(\frac{ \partial^2 f}{\partial x^2}\) at the betoken \((a,b)\text{.}\) Then explain as best you lot can what this second order partial derivative tells us about kinetic free energy.
Calculate \(\frac{ \partial^2 f}{\fractional y^2}\) at the point \((a,b)\text{.}\) So explain every bit best you tin can what this second order fractional derivative tells us nigh kinetic energy.
Summate \(\frac{ \fractional^ii f}{\partial y \partial x}\) at the bespeak \((a,b)\text{.}\) And so explain as best you tin what this second social club partial derivative tells us about kinetic free energy.
Summate \(\frac{ \partial^2 f}{\partial 10 \partial y}\) at the point \((a,b)\text{.}\) And then explicate as best you tin can what this second guild fractional derivative tells u.s. most kinetic energy.
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