.lst-kix_9ycbvglffw5i-8 > li:before{content:"" counter(lst-ctn-kix_9ycbvglffw5i-8,decimal) ". "}.lst-kix_9ycbvglffw5i-7 > li:before{content:"" counter(lst-ctn-kix_9ycbvglffw5i-7,lower-roman) ". "}ol.lst-kix_9ycbvglffw5i-0.start{counter-reset:lst-ctn-kix_9ycbvglffw5i-0 0}ol.lst-kix_2ffinenr8m70-2.start{counter-reset:lst-ctn-kix_2ffinenr8m70-2 0}.lst-kix_2ffinenr8m70-6 > li{counter-increment:lst-ctn-kix_2ffinenr8m70-6}.lst-kix_9ycbvglffw5i-0 > li:before{content:"" counter(lst-ctn-kix_9ycbvglffw5i-0,lower-latin) ". "}ol.lst-kix_naw8ljaj8cmi-3.start{counter-reset:lst-ctn-kix_naw8ljaj8cmi-3 0}ol.lst-kix_9ycbvglffw5i-3.start{counter-reset:lst-ctn-kix_9ycbvglffw5i-3 0}.lst-kix_9mitsmlx4h3o-3 > li{counter-increment:lst-ctn-kix_9mitsmlx4h3o-3}ol.lst-kix_naw8ljaj8cmi-0.start{counter-reset:lst-ctn-kix_naw8ljaj8cmi-0 0}.lst-kix_naw8ljaj8cmi-6 > li{counter-increment:lst-ctn-kix_naw8ljaj8cmi-6}.lst-kix_9ycbvglffw5i-1 > li{counter-increment:lst-ctn-kix_9ycbvglffw5i-1}.lst-kix_wvx8hu4gm1uj-0 > li:before{content:"● "}.lst-kix_wvx8hu4gm1uj-1 > li:before{content:"○ "}.lst-kix_wvx8hu4gm1uj-2 > li:before{content:"■ "}.lst-kix_2ffinenr8m70-7 > li{counter-increment:lst-ctn-kix_2ffinenr8m70-7}.lst-kix_2ffinenr8m70-4 > li{counter-increment:lst-ctn-kix_2ffinenr8m70-4}.lst-kix_9mitsmlx4h3o-1 > li{counter-increment:lst-ctn-kix_9mitsmlx4h3o-1}.lst-kix_wvx8hu4gm1uj-7 > li:before{content:"○ "}.lst-kix_9mitsmlx4h3o-4 > li{counter-increment:lst-ctn-kix_9mitsmlx4h3o-4}.lst-kix_naw8ljaj8cmi-8 > li{counter-increment:lst-ctn-kix_naw8ljaj8cmi-8}.lst-kix_wvx8hu4gm1uj-5 > li:before{content:"■ "}.lst-kix_wvx8hu4gm1uj-6 > li:before{content:"● "}ol.lst-kix_9mitsmlx4h3o-6.start{counter-reset:lst-ctn-kix_9mitsmlx4h3o-6 0}.lst-kix_wvx8hu4gm1uj-3 > li:before{content:"● "}.lst-kix_wvx8hu4gm1uj-4 > li:before{content:"○ "}ol.lst-kix_9mitsmlx4h3o-3.start{counter-reset:lst-ctn-kix_9mitsmlx4h3o-3 0}.lst-kix_9mitsmlx4h3o-6 > li{counter-increment:lst-ctn-kix_9mitsmlx4h3o-6}ol.lst-kix_naw8ljaj8cmi-5.start{counter-reset:lst-ctn-kix_naw8ljaj8cmi-5 0}.lst-kix_naw8ljaj8cmi-5 > li{counter-increment:lst-ctn-kix_naw8ljaj8cmi-5}.lst-kix_2ffinenr8m70-5 > li:before{content:"" counter(lst-ctn-kix_2ffinenr8m70-5,decimal) ". "}.lst-kix_2ffinenr8m70-6 > li:before{content:"" counter(lst-ctn-kix_2ffinenr8m70-6,lower-latin) ". "}ol.lst-kix_2ffinenr8m70-7.start{counter-reset:lst-ctn-kix_2ffinenr8m70-7 0}ol.lst-kix_2ffinenr8m70-1.start{counter-reset:lst-ctn-kix_2ffinenr8m70-1 0}.lst-kix_2ffinenr8m70-3 > li{counter-increment:lst-ctn-kix_2ffinenr8m70-3}.lst-kix_2ffinenr8m70-4 > li:before{content:"" counter(lst-ctn-kix_2ffinenr8m70-4,lower-roman) ". "}.lst-kix_2ffinenr8m70-8 > li:before{content:"" counter(lst-ctn-kix_2ffinenr8m70-8,decimal) ". "}.lst-kix_9mitsmlx4h3o-0 > li{counter-increment:lst-ctn-kix_9mitsmlx4h3o-0}.lst-kix_2ffinenr8m70-7 > li:before{content:"" counter(lst-ctn-kix_2ffinenr8m70-7,lower-roman) ". "}.lst-kix_9mitsmlx4h3o-4 > li:before{content:"" counter(lst-ctn-kix_9mitsmlx4h3o-4,lower-roman) ". "}ol.lst-kix_9mitsmlx4h3o-4.start{counter-reset:lst-ctn-kix_9mitsmlx4h3o-4 0}ol.lst-kix_naw8ljaj8cmi-6.start{counter-reset:lst-ctn-kix_naw8ljaj8cmi-6 0}.lst-kix_9mitsmlx4h3o-6 > li:before{content:"" counter(lst-ctn-kix_9mitsmlx4h3o-6,lower-latin) ". "}.lst-kix_2ffinenr8m70-0 > li:before{content:"" counter(lst-ctn-kix_2ffinenr8m70-0,lower-latin) ". "}.lst-kix_9mitsmlx4h3o-2 > li:before{content:"" counter(lst-ctn-kix_9mitsmlx4h3o-2,decimal) ". "}.lst-kix_9mitsmlx4h3o-5 > li:before{content:"" counter(lst-ctn-kix_9mitsmlx4h3o-5,decimal) ". "}ol.lst-kix_9ycbvglffw5i-0{list-style-type:none}.lst-kix_2ffinenr8m70-1 > li:before{content:"" counter(lst-ctn-kix_2ffinenr8m70-1,lower-roman) ". "}.lst-kix_2ffinenr8m70-2 > li:before{content:"" counter(lst-ctn-kix_2ffinenr8m70-2,decimal) ". "}ol.lst-kix_9ycbvglffw5i-1{list-style-type:none}ol.lst-kix_9ycbvglffw5i-2{list-style-type:none}ol.lst-kix_9ycbvglffw5i-3{list-style-type:none}ol.lst-kix_9ycbvglffw5i-4{list-style-type:none}ol.lst-kix_9ycbvglffw5i-8.start{counter-reset:lst-ctn-kix_9ycbvglffw5i-8 0}ol.lst-kix_9mitsmlx4h3o-0{list-style-type:none}.lst-kix_2ffinenr8m70-3 > li:before{content:"" counter(lst-ctn-kix_2ffinenr8m70-3,lower-latin) ". "}ol.lst-kix_9ycbvglffw5i-5{list-style-type:none}ol.lst-kix_9mitsmlx4h3o-1{list-style-type:none}ol.lst-kix_9ycbvglffw5i-6{list-style-type:none}.lst-kix_9mitsmlx4h3o-3 > li:before{content:"" counter(lst-ctn-kix_9mitsmlx4h3o-3,lower-latin) ". "}.lst-kix_9ycbvglffw5i-1 > li:before{content:"" counter(lst-ctn-kix_9ycbvglffw5i-1,lower-roman) ". "}ol.lst-kix_9ycbvglffw5i-7{list-style-type:none}ol.lst-kix_9ycbvglffw5i-8{list-style-type:none}ol.lst-kix_9mitsmlx4h3o-4{list-style-type:none}ol.lst-kix_9mitsmlx4h3o-5{list-style-type:none}ol.lst-kix_9mitsmlx4h3o-2{list-style-type:none}ol.lst-kix_2ffinenr8m70-8.start{counter-reset:lst-ctn-kix_2ffinenr8m70-8 0}.lst-kix_9ycbvglffw5i-2 > li:before{content:"" counter(lst-ctn-kix_9ycbvglffw5i-2,decimal) ". "}ol.lst-kix_9mitsmlx4h3o-3{list-style-type:none}.lst-kix_9ycbvglffw5i-3 > li{counter-increment:lst-ctn-kix_9ycbvglffw5i-3}ol.lst-kix_9mitsmlx4h3o-8{list-style-type:none}.lst-kix_9ycbvglffw5i-6 > li{counter-increment:lst-ctn-kix_9ycbvglffw5i-6}ol.lst-kix_9mitsmlx4h3o-6{list-style-type:none}.lst-kix_9ycbvglffw5i-3 > li:before{content:"" counter(lst-ctn-kix_9ycbvglffw5i-3,lower-latin) ". "}ol.lst-kix_9mitsmlx4h3o-7{list-style-type:none}.lst-kix_9mitsmlx4h3o-8 > li:before{content:"" counter(lst-ctn-kix_9mitsmlx4h3o-8,decimal) ". "}.lst-kix_9ycbvglffw5i-4 > li:before{content:"" counter(lst-ctn-kix_9ycbvglffw5i-4,lower-roman) ". "}.lst-kix_9ycbvglffw5i-6 > li:before{content:"" counter(lst-ctn-kix_9ycbvglffw5i-6,lower-latin) ". "}ol.lst-kix_9ycbvglffw5i-2.start{counter-reset:lst-ctn-kix_9ycbvglffw5i-2 0}.lst-kix_9ycbvglffw5i-0 > li{counter-increment:lst-ctn-kix_9ycbvglffw5i-0}.lst-kix_9mitsmlx4h3o-7 > li:before{content:"" counter(lst-ctn-kix_9mitsmlx4h3o-7,lower-roman) ". "}.lst-kix_9ycbvglffw5i-5 > li:before{content:"" counter(lst-ctn-kix_9ycbvglffw5i-5,decimal) ". "}.lst-kix_2su7wlywvg5v-4 > li:before{content:"- "}ol.lst-kix_9ycbvglffw5i-7.start{counter-reset:lst-ctn-kix_9ycbvglffw5i-7 0}.lst-kix_2su7wlywvg5v-3 > li:before{content:"- "}.lst-kix_2su7wlywvg5v-5 > li:before{content:"- "}ol.lst-kix_naw8ljaj8cmi-0{list-style-type:none}ol.lst-kix_naw8ljaj8cmi-2{list-style-type:none}ol.lst-kix_naw8ljaj8cmi-1{list-style-type:none}ol.lst-kix_naw8ljaj8cmi-4{list-style-type:none}ol.lst-kix_naw8ljaj8cmi-3{list-style-type:none}.lst-kix_2su7wlywvg5v-0 > li:before{content:"- "}.lst-kix_2su7wlywvg5v-8 > li:before{content:"- "}ol.lst-kix_2ffinenr8m70-6.start{counter-reset:lst-ctn-kix_2ffinenr8m70-6 0}.lst-kix_2su7wlywvg5v-1 > li:before{content:"- "}.lst-kix_naw8ljaj8cmi-2 > li{counter-increment:lst-ctn-kix_naw8ljaj8cmi-2}ol.lst-kix_naw8ljaj8cmi-7.start{counter-reset:lst-ctn-kix_naw8ljaj8cmi-7 0}.lst-kix_2su7wlywvg5v-2 > li:before{content:"- "}ol.lst-kix_9mitsmlx4h3o-5.start{counter-reset:lst-ctn-kix_9mitsmlx4h3o-5 0}ol.lst-kix_9ycbvglffw5i-4.start{counter-reset:lst-ctn-kix_9ycbvglffw5i-4 0}ul.lst-kix_wvx8hu4gm1uj-8{list-style-type:none}ul.lst-kix_2su7wlywvg5v-0{list-style-type:none}ul.lst-kix_wvx8hu4gm1uj-7{list-style-type:none}ul.lst-kix_wvx8hu4gm1uj-6{list-style-type:none}ul.lst-kix_wvx8hu4gm1uj-5{list-style-type:none}ul.lst-kix_wvx8hu4gm1uj-4{list-style-type:none}ul.lst-kix_wvx8hu4gm1uj-3{list-style-type:none}ul.lst-kix_wvx8hu4gm1uj-2{list-style-type:none}ul.lst-kix_wvx8hu4gm1uj-1{list-style-type:none}ul.lst-kix_wvx8hu4gm1uj-0{list-style-type:none}ul.lst-kix_2su7wlywvg5v-8{list-style-type:none}.lst-kix_9ycbvglffw5i-5 > li{counter-increment:lst-ctn-kix_9ycbvglffw5i-5}ul.lst-kix_2su7wlywvg5v-7{list-style-type:none}.lst-kix_9mitsmlx4h3o-0 > li:before{content:"" counter(lst-ctn-kix_9mitsmlx4h3o-0,lower-latin) ". "}.lst-kix_9mitsmlx4h3o-1 > li:before{content:"" counter(lst-ctn-kix_9mitsmlx4h3o-1,lower-roman) ". "}ul.lst-kix_2su7wlywvg5v-6{list-style-type:none}ul.lst-kix_2su7wlywvg5v-5{list-style-type:none}ul.lst-kix_2su7wlywvg5v-4{list-style-type:none}.lst-kix_2ffinenr8m70-0 > li{counter-increment:lst-ctn-kix_2ffinenr8m70-0}ol.lst-kix_9mitsmlx4h3o-2.start{counter-reset:lst-ctn-kix_9mitsmlx4h3o-2 0}ul.lst-kix_2su7wlywvg5v-3{list-style-type:none}ul.lst-kix_2su7wlywvg5v-2{list-style-type:none}ul.lst-kix_2su7wlywvg5v-1{list-style-type:none}ol.lst-kix_9ycbvglffw5i-1.start{counter-reset:lst-ctn-kix_9ycbvglffw5i-1 0}.lst-kix_2ffinenr8m70-1 > li{counter-increment:lst-ctn-kix_2ffinenr8m70-1}.lst-kix_naw8ljaj8cmi-0 > li{counter-increment:lst-ctn-kix_naw8ljaj8cmi-0}.lst-kix_9ycbvglffw5i-7 > li{counter-increment:lst-ctn-kix_9ycbvglffw5i-7}ol.lst-kix_naw8ljaj8cmi-1.start{counter-reset:lst-ctn-kix_naw8ljaj8cmi-1 0}.lst-kix_naw8ljaj8cmi-3 > li{counter-increment:lst-ctn-kix_naw8ljaj8cmi-3}.lst-kix_9mitsmlx4h3o-7 > li{counter-increment:lst-ctn-kix_9mitsmlx4h3o-7}.lst-kix_9ycbvglffw5i-4 > li{counter-increment:lst-ctn-kix_9ycbvglffw5i-4}ol.lst-kix_2ffinenr8m70-3.start{counter-reset:lst-ctn-kix_2ffinenr8m70-3 0}ol.lst-kix_naw8ljaj8cmi-6{list-style-type:none}ol.lst-kix_2ffinenr8m70-0.start{counter-reset:lst-ctn-kix_2ffinenr8m70-0 0}ol.lst-kix_naw8ljaj8cmi-4.start{counter-reset:lst-ctn-kix_naw8ljaj8cmi-4 0}ol.lst-kix_naw8ljaj8cmi-5{list-style-type:none}.lst-kix_2su7wlywvg5v-7 > li:before{content:"- "}ol.lst-kix_naw8ljaj8cmi-8{list-style-type:none}ol.lst-kix_naw8ljaj8cmi-7{list-style-type:none}.lst-kix_2su7wlywvg5v-6 > li:before{content:"- "}ol.lst-kix_9mitsmlx4h3o-0.start{counter-reset:lst-ctn-kix_9mitsmlx4h3o-0 0}ol.lst-kix_naw8ljaj8cmi-2.start{counter-reset:lst-ctn-kix_naw8ljaj8cmi-2 0}.lst-kix_wvx8hu4gm1uj-8 > li:before{content:"■ "}.lst-kix_naw8ljaj8cmi-7 > li:before{content:"" counter(lst-ctn-kix_naw8ljaj8cmi-7,lower-latin) ". "}.lst-kix_naw8ljaj8cmi-8 > li:before{content:"" counter(lst-ctn-kix_naw8ljaj8cmi-8,lower-roman) ". "}ol.lst-kix_naw8ljaj8cmi-8.start{counter-reset:lst-ctn-kix_naw8ljaj8cmi-8 0}.lst-kix_naw8ljaj8cmi-5 > li:before{content:"" counter(lst-ctn-kix_naw8ljaj8cmi-5,lower-roman) ". "}.lst-kix_9ycbvglffw5i-2 > li{counter-increment:lst-ctn-kix_9ycbvglffw5i-2}ol.lst-kix_2ffinenr8m70-7{list-style-type:none}ol.lst-kix_2ffinenr8m70-8{list-style-type:none}.lst-kix_9ycbvglffw5i-8 > li{counter-increment:lst-ctn-kix_9ycbvglffw5i-8}.lst-kix_naw8ljaj8cmi-6 > li:before{content:"" counter(lst-ctn-kix_naw8ljaj8cmi-6,decimal) ". "}ol.lst-kix_9ycbvglffw5i-6.start{counter-reset:lst-ctn-kix_9ycbvglffw5i-6 0}ol.lst-kix_9mitsmlx4h3o-7.start{counter-reset:lst-ctn-kix_9mitsmlx4h3o-7 0}ol.lst-kix_2ffinenr8m70-4.start{counter-reset:lst-ctn-kix_2ffinenr8m70-4 0}.lst-kix_naw8ljaj8cmi-0 > li:before{content:"" counter(lst-ctn-kix_naw8ljaj8cmi-0,decimal) ". "}ol.lst-kix_2ffinenr8m70-0{list-style-type:none}ol.lst-kix_2ffinenr8m70-1{list-style-type:none}ol.lst-kix_2ffinenr8m70-2{list-style-type:none}ol.lst-kix_2ffinenr8m70-3{list-style-type:none}ol.lst-kix_2ffinenr8m70-4{list-style-type:none}.lst-kix_naw8ljaj8cmi-1 > li:before{content:"" counter(lst-ctn-kix_naw8ljaj8cmi-1,lower-latin) ". "}ol.lst-kix_2ffinenr8m70-5{list-style-type:none}ol.lst-kix_2ffinenr8m70-5.start{counter-reset:lst-ctn-kix_2ffinenr8m70-5 0}ol.lst-kix_2ffinenr8m70-6{list-style-type:none}.lst-kix_naw8ljaj8cmi-4 > li:before{content:"" counter(lst-ctn-kix_naw8ljaj8cmi-4,lower-latin) ". "}.lst-kix_9mitsmlx4h3o-2 > li{counter-increment:lst-ctn-kix_9mitsmlx4h3o-2}.lst-kix_naw8ljaj8cmi-2 > li:before{content:"" counter(lst-ctn-kix_naw8ljaj8cmi-2,lower-roman) ". "}.lst-kix_naw8ljaj8cmi-3 > li:before{content:"" counter(lst-ctn-kix_naw8ljaj8cmi-3,decimal) ". "}ol.lst-kix_9mitsmlx4h3o-1.start{counter-reset:lst-ctn-kix_9mitsmlx4h3o-1 0}.lst-kix_naw8ljaj8cmi-7 > li{counter-increment:lst-ctn-kix_naw8ljaj8cmi-7}ol.lst-kix_9ycbvglffw5i-5.start{counter-reset:lst-ctn-kix_9ycbvglffw5i-5 0}ol.lst-kix_9mitsmlx4h3o-8.start{counter-reset:lst-ctn-kix_9mitsmlx4h3o-8 0}.lst-kix_2ffinenr8m70-2 > li{counter-increment:lst-ctn-kix_2ffinenr8m70-2}.lst-kix_9mitsmlx4h3o-5 > li{counter-increment:lst-ctn-kix_9mitsmlx4h3o-5}.lst-kix_2ffinenr8m70-5 > li{counter-increment:lst-ctn-kix_2ffinenr8m70-5}.lst-kix_9mitsmlx4h3o-8 > li{counter-increment:lst-ctn-kix_9mitsmlx4h3o-8}.lst-kix_naw8ljaj8cmi-4 > li{counter-increment:lst-ctn-kix_naw8ljaj8cmi-4}.lst-kix_2ffinenr8m70-8 > li{counter-increment:lst-ctn-kix_2ffinenr8m70-8}.lst-kix_naw8ljaj8cmi-1 > li{counter-increment:lst-ctn-kix_naw8ljaj8cmi-1}

Math 264 If you want to learn more check out What are the different types of social media?

Final Exam Study Guide

1. Write down an equation of the plane passing through point P=(1,0,-2) and:

1. Normal to vector n = (3,-1,2)

3(x-1)-1(y-0)+2(z+2)=0We also discuss several other topics like Define bender’s hypothesis.

3x-y+2z=-1

1. Parallel to the plane -3x+2y+5z=2018

                -3(x-1)+2(y-0)+5(z+2)=0

                -3x+2y+5z=-13

1. Consider the function f(x,y)=ln(4-x²+y)

1. Find the domain of the function:

                Since ln is undefined at negative value: u-x²+y>0Don't forget about the age old question of What do you mean by marginal utility?

                Domain: {(x,y)|4-x²+y>0}

1. Write down an equation of the level curve of the function passing through point (0,1) and sketch it:

ln(u-x²+y)=cIf you want to learn more check out What are the 5 essential personality traits?

u-x²+y=c

At (0,1): u-0+1=cDon't forget about the age old question of Why do plant-based products do not meet the legal definition of milk?

        c=sIf you want to learn more check out What are the significant functions of the respiratory tract?

Equation: u-x²+y=5

        y-x²+1

1. Find the limit if it exists or show that it does not exist:

1. lim        e^-xy cos (x+y)

(x,y)➝(1,1)

lim                e^-xy cos(x+y) = e^1 cos (1-1) = e^1 cos(c) = e^1

(x,y)➝(1,-1)

1. lim                 xycosy

(x,y)➝(0,0)        3x²+y²

lim                 xycosy = 0/0

(x,y)➝(0,0)        3x²+y²

Along x-axis (y=0):

                lim                 0

(x,y)➝(0,0)        3x² = 0

Along x=y line:

                lim                 x²cosx        =  x²cosx =  cosx = 1/4

(x,y)➝(0,0)        3x² = 0    3x²+x²         4

1. Consider a function:

1. Determine a region where the function is continuous. Provide an explanation:

- for (x,y) ≠ (0,0) the function f(x,y) = x²y^3 / 2x²+y² is a continuous and differentiable function since the denominator does not vanish

- along x axis (y=0):                along y axis (x=0)

                So the limit:

Thus

Continuous or differentiable at (0,0)

5. Compute the partial derivative of:

1. 

        

b.

c.  

6. Verify that the function M =Ae^x cos y where A is an arbitrary constant, is a solution to the laplace equation:

7. If z=x²siny , x = t and y = t^3. Find dz/dt:

8. Consider the function f(x,y,z) = xe^(x/y) + z sin (xy) + z

1. Find the directional derivative of the function in the direction of the vector v=(1,1,1) at (0,1,1)

Definition:

1. Find the maximal rate of change of the function at point (0,1,1):

1. Find the direction of the maximal rate of change at point (0,1,1):

9. Consider the surface y=x²-z². Find the equation of the tangent plane to this surface at point (4,7,3):

10. Consider the local minimum and maximum values and saddle points of the function

*three critical points: (0,0),(1,1),(-1,-1):

• D (0.0) = -16 < 0➝(0,0) is saddle point
• D (1,1) = 144-16>0, since fxx = 12 > 0 - local min
• D (-1,-1) = 144-16>0, since fxx = 12 > 0 - local min

11. Find the absolute max and absolute min value of the  function f(x,y) = 2x^3 + y^4 on the domain D{(x,y)|x²+y²≤1}

References: