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Mathematics LibreTexts

3.2.E: Directional Derivatives and the Gradient (Exercises)

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Exercise 3.2.E.1

Suppose f:R2R is defined by

f(x,y)=3x2+2y2.

Let

u=15(1,2).

Find Duf(3,1) directly from the definition (3.2.2).

Answer

Duf(3,1)=265

Exercise 3.2.E.2

For each of the following functions, find the partial derivatives with respect to each variable.

(a) f(x,y)=4xx2+y2

(b) g(x,y)=4xy2ey2

(c) f(x,y,z)=3x2y3z413x2y

(d) h(x,y,z)=4xze1x2+y2+z2

(e) g(w,x,y,z)=sin(w2+x2+2y2+3z2)

Answer

(a) fx(x,y)=4y24x2(x2+y2)2;fy(x,y)=8xy(x2+y2)2

(c) fx(x,y,z)=6xy3z426xyfy(x,y,z)=9x2y2z413x2fz(x,y,z)=12x2y3z3

(e) gw(w,x,y,z)=wcos(w2+x2+2y2+3z2)w2+x2+2y2+3z2

gx(w,x,y,z)=xcos(w2+x2+2y2+3z2)w2+x2+2y2+3z2

gy(w,x,y,z)=2ycos(w2+x2+2y2+3z2)w2+x2+2y2+3z2

gz(w,x,y,z)=3zcos(w2+x2+2y2+3z2)w2+x2+2y2+3z2

Exercise 3.2.E.3

Find the gradient of each of the following functions.

(a) f(x,y,z)=x2+y2+z2

(b) g(x,y,z)=1x2+y2+z2

(c) f(w,x,y,z)=tan1(4w+3x+5y+z)

Answer

(a) f(x,y,z)=1x2+y2+z2(x,y,z)

(c) f(w,x,y,z)=11+(4w+3x+5y+z)2(4,3,5,1)

Exercise 3.2.E.4

Find Duf(c) for each of the following.

(a) f(x,y)=3x2+5y2,u=113(3,2),c=(2,1)

(b) f(x,y)=x22y2,u=15(1,2),c=(2,3)

(c) f(x,y,z)=1x2+y2+z2,u=16(1,2,1),c=(2,2,1)

Answer

(a) Du(2,1)=5613

(c) Du(2,2,1)=196

Exercise 3.2.E.5

For each of the following, find the directional derivative of f at the point c in the direction of the specified vector w.

(a) f(x,y)=3x2y,w=(2,3),c=(2,1)

(b) f(x,y,z)=log(x2+2y2+z2),w=(1,2,3),c=(2,1,1)

(c) f(t,x,y,z)=tx2yz2,w=(1,1,2,3),c=(2,1,1,2)

Answer

(a) Duf(2,1)=1213, where u=113(2,3)

(c) Duf(2,1,1,2)=415, where u=115(1,1,2,3)

Exercise 3.2.E.6

A metal plate is heated so that its temperature at a point (x,y) is

T(x,y)=50y2e15(x2+y2).

A bug is placed at the point (2,1).

(a) The bug heads toward the point (1,−2). What is the rate of change of temperature in this direction?

(b) In what direction should the bug head in order to warm up at the fastest rate? What is the rate of change of temperature in this direction?

(c) In what direction should the bug head in order to cool off at the fastest rate? What is the rate of change of temperature in this direction?

(d) Make a plot of the gradient vectors and discuss what it tells you about the temperatures on the plate.

Answer

(a) 2010e1

(b) Direction: 15(1,2) ; Rate of change: 405e1

(c) Direction: 15(1,2) ; Rate of change: 405e1

Exercise 3.2.E.7

A heat-seeking bug is a bug that always moves in the direction of the greatest increasein heat. Discuss the behavior of a heat seeking bug placed on a metal plate heated so that the temperature at (x,y) is given by

T(x,y)=10040xye110(x2+y2).

Exercise 3.2.E.8

Suppose g:R2R is defined by

g(x,y)={xyx2+y2, if (x,y)(0,0),0, if (x,y)=(0,0).

We saw above that both partial derivatives of g exist at (0,0), although g is not continuous at (0,0).

(a) Show that neither gx nor gy is continuous at (0,0).

(b) Let

u=12(1,1).

Show that Dug(0,0) does not exist. In particular, Dug(0,0)g(0,0)u.

Exercise 3.2.E.9

Suppose the price of a certain commodity, call it commodity A, is x dollars per unit and the price of another commodity, B, is y dollars per unit. Moreover, suppose that dA(x,y) represents the number of units of A that will be sold at these prices and dB(x,y) represents the number of units of B that will be sold at these prices. These functions are known as the demand functions for A and B.

(a) Explain why it is reasonable to assume that xdA(x,y)<0

and

ydB(x,y)<0

for all (x,y).

(b) Suppose the two commodities are competitive. For example, they might be two different brands of the same product. In this case, what would be reasonable assumptions for the signs of

ydA(x,y)

and

xdB(x,y)?

(c) Suppose the two commodities complement each other. For example, commodity A might be a computer and commodity B a type of software. In this case, what would be reasonable assumptions for the signs of

ydA(x,y)

and

xdB(x,y)?

Answer

(b) ydA(x,y)>0,xdB(x,y)>0

(c) ydA(x,y)<0,xdB(x,y)<0

Exercise 3.2.E.10

Suppose P(x1,x2,,xn) represents the total production per week of a certain factory as a function of x1, the number of workers, and other variables, such as the size of the supply inventory, the number of hours the assembly lines run per week, and so on. Show that average productivity

P(x1,x2,,xn)x1

increases as x1 increases if and only if

x1P(x1,x2,,xn)>P(x1,x2,,xn)x1.

Exercise 3.2.E.11

Suppose f:RnR is C1 on an open ball about the point c.

(a) Given a unit vector u, what is the relationship between Duf(c) and Duf(c)?

(b) Is it possible that Duf(c)>0 for every unit vector u?

Answer

(a) Duf(c)=Duf(c)

(b) No


This page titled 3.2.E: Directional Derivatives and the Gradient (Exercises) is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform.

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