You are standing above the point (5,5)(5,5) on the surface z=15−(2x2+2y2)z=15−(2x2+2y2). (a) in which direction should you walk
to descend fastest? (give your answer as a unit 2-vector.)
1 answer:
Since the surface is the top part of a sphere centred on the origin, the directions of steepest descent are any vector originating from the origin (0,0).
At (5,5), the direction would be dictated by the line joining (5,5) and the origin (0,0), i.e. along <5,5> or equivalently along <1,1>.
You can also go through vector calculus to arrive at the same result.
At (5,5), the direction would be dictated by the line joining (5,5) and the origin (0,0), i.e. along <5,5> or equivalently along <1,1>.
You can also go through vector calculus to arrive at the same result.
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Answer:
The intersection is .
The Problem:
What is the intersection point of and
?
Step-by-step explanation:
To find the intersection of and
, we will need to find when they have a common point; when their
and
are the same.
Let's start with setting the 's equal to find those
's for which the
's are the same.
By power rule:
Since implies
:
Squaring both sides to get rid of the fraction exponent:
This is a quadratic equation.
Subtract on both sides:
Comparing this to we see the following:
Let's plug them into the quadratic formula:
So we have the solutions to the quadratic equation are:
or
.
The second solution definitely gives at least one of the logarithm equation problems.
Example: has problems when
and so the second solution is a problem.
So the where the equations intersect is at
.
Let's find the -coordinate.
You may use either equation.
I choose .
The intersection is .
Hey there!
3x+6y=12
Divide both sides by 3 and your equation would now look like:
x+2y= 4
Move the x to the other side
2y= -x+4
Seprate the y by dividing both sides by 2
y=
x+2
3x+6y=12
Divide both sides by 3 and your equation would now look like:
x+2y= 4
Move the x to the other side
2y= -x+4
Seprate the y by dividing both sides by 2
y=