Find equations of the tangent plane and the normal line to the given surface at the specified point. x + y + z = 8exyz, (0, 0, 8) (a) the tangent plane (b) the normal line (x(t), y(t), z(t)) =

1 answer
Question:

Find equations of the tangent plane and the normal line to the given surface at the specified point. x + y + z = 8exyz, (0, 0, 8) (a) the tangent plane (b) the normal line (x(t), y(t), z(t)) =

Answers

Let [tex]f(x,y,z)=x+y+z-8e^{xyz}[/tex]. The tangent plane to the surface at (0, 0, 8) is[tex]\nabla f(0,0,8)\cdot(x,y,z-8)=0[/tex]The gradient is[tex]\nabla f(x,y,z)=\left(1-8yze^{xyz},1-8xze^{xyz},1-8xye^{xyz}\right)[/tex]so the tangent plane's equation is[tex](1,1,1)\cdot(x,y,z-8)=0\implies x+y+(z-8)=0\implies x+y+z=8[/tex]The normal vector to the plane at (0, 0, 8) is the same as the gradient of the surface at this point, (1, 1, 1). We can get all points along the line containing this vector by scaling the vector by [tex]t[/tex], then ensure it passes through (0, 0, 8) by translating the line so that it does. Then the line has parametric equation[tex](1,1,1)t+(0,0,8)=(t,t,t+8)[/tex]or [tex]x(t)=t[/tex], [tex]y(t)=t[/tex], and [tex]z(t)=t+8[/tex].(See the attached plot; the given surface is orange, (0, 0, 8) is the black point, the tangent plane is blue, and the red line is the normal at this point)

Similar Solved Questions

1 answer

Everyone likes to have Lydia working on their committees because she seems to have the unique ability to consider a variety of options and quickly narrow the list of options down to the one or two best alternatives. In other words, Lydia is skilled in ___________.a. divergent thinking. b. transformational logic. c. convergent thinking. d. field dependence.

Everyone likes to have Lydia working on their committees because she seems to have the unique ability to consider a variety of options and quickly narrow the list of options down to the one or two best alternatives. In other words, Lydia is skilled in ___________.a. divergent thinking. b. transforma...
1 answer

Although video poker machines are outlawed in Pennsylvania, Conley placed them in bars and clubs throughout the state. He used profits from the machines to buy more machines. What crime has he committed, other than violating the law prohibiting video poker machines?

Although video poker machines are outlawed in Pennsylvania, Conley placed them in bars and clubs throughout the state. He used profits from the machines to buy more machines. What crime has he committed, other than violating the law prohibiting video poker machines?...
2 answers

Iafp is mainly interested in protecting the _____ food supply. local state national global

Iafp is mainly interested in protecting the _____ food supply. local state national global...
1 answer

Which of the following examples exhibits the reciprocity condition in the context of a quarantine? a) providing stimulus checks, because some people are not able to work during the quarantine. b) neighbors helping each other keep their morale during the quarantine. c) public health officials communicating clearly. d) buying all of the toilet paper.

which of the following examples exhibits the reciprocity condition in the context of a quarantine? a) providing stimulus checks, because some people are not able to work during the quarantine. b) neighbors helping each other keep their morale during the quarantine. c) public health officials comm...
2 answers

The absolute value of -5

the absolute value of -5...
1 answer

Please help! Create a set of five positive numbers (repeats allowed) that have a median of 10 and a mean of 14. What thought process did you use to create your numbers?

Please help! Create a set of five positive numbers (repeats allowed) that have a median of 10 and a mean of 14. What thought process did you use to create your numbers?...
2 answers

Help please, and thank you Francine and Genevieve were arguing about whether a recursive formula or an explicit formula was better to know to find the 376th term in a sequence. Francine believes that it is the recursive formula, while Genevieve believes that it is the explicit formula. Who is correct? 1st correct answer gets a thanks :)

Help please, and thank you Francine and Genevieve were arguing about whether a recursive formula or an explicit formula was better to know to find the 376th term in a sequence. Francine believes that it is the recursive formula, while Genevieve believes that it is the explicit formula. Who is correc...
1 answer

Find the line through (3, 1, −2) that intersects and is perpendicular to the line x = −1 + t, y = −2 + t, z = −1 + t. (HINT: If (x0, y0, z0) is the point of intersection, find its coordinates. Enter your answers as a comma-separated list of equations.)

Find the line through (3, 1, −2) that intersects and is perpendicular to the line x = −1 + t, y = −2 + t, z = −1 + t. (HINT: If (x0, y0, z0) is the point of intersection, find its coordinates. Enter your answers as a comma-separated list of equations.)...
2 answers

15.Which list shows the numbers below inorder from greatest to least? (8.18, 8.17)1 1/ 0.7, 14%, 10100 3.0.7, 14%6 14%, 5.0.7.12​

15.Which list shows the numbers below inorder from greatest to least? (8.18, 8.17)1 1/ 0.7, 14%, 10100 3.0.7, 14%6 14%, 5.0.7.12​...
2 answers

Is 15 a multiple of each of its factors

Is 15 a multiple of each of its factors...
1 answer

What is the absolute value of |4x|- 12 = 4​

what is the absolute value of |4x|- 12 = 4​...
1 answer

What does 'innate' mean in line 21 of “Dog Proves As Smart As Average Toddler”? *A taught by humansB learned over timeC present from birthD taken from memory

What does 'innate' mean in line 21 of “Dog Proves As Smart As Average Toddler”? *A taught by humansB learned over timeC present from birthD taken from memory...
1 answer

Help please !! Thank you

Help please !! Thank you...
2 answers

Complete the following statement: If the sin 90° = 1, then cos_______= 1

Complete the following statement: If the sin 90° = 1, then cos_______= 1...
2 answers

Does anyone on here know anything about the bass clarinet

Does anyone on here know anything about the bass clarinet...
1 answer

The solubility of CO2 in water is 0.161 g/100 mL at 20oC and a partial pressure of CO2 of 760 mmHg. What partial pressure of CO2 is necessary in a soft drink canning process in order to allow the solubility of CO2 to equal 0.886 g/100 mL?

The solubility of CO2 in water is 0.161 g/100 mL at 20oC and a partial pressure of CO2 of 760 mmHg. What partial pressure of CO2 is necessary in a soft drink canning process in order to allow the solubility of CO2 to equal 0.886 g/100 mL?...
1 answer

Find the equation of the line that is perpendicular to the given line and passes through the given point. Enter your answer in slope-intercept form (y = mx + b) with the values of m and b given as decimals, rounded to the nearest hundredth if necessary. y = 4x ; (2, −4) The equation is y = .

Find the equation of the line that is perpendicular to the given line and passes through the given point. Enter your answer in slope-intercept form (y = mx + b) with the values of m and b given as decimals, rounded to the nearest hundredth if necessary. y = 4x ; (2, −4) The equation is y = ....

-- 0.011993--