# Tangent Line MultiVar Calc Problem

• harrietstowe
In summary, the problem asks for the slope of the tangent line to the curve of intersection of the vertical plane x-y+1= 0 and the surface z=x^2+ y^2 at the point (1, 2, 5). The suggested method is to find the normal vectors of the two surfaces and take the cross product to get a tangent vector. Then, using the definition of slope as dz/du, where du is the position change projected on the xy-plane and dz is the change in vertical position, the slope can be determined by taking the ratio of the z component to the projection on the xy plane. Alternatively, the tangent vector direction can be found by parameterizing the line in terms of t=x and
harrietstowe

## Homework Statement

Find the slope of the tangent line to the curve of intersection of the vertical plane x-y+1= 0 and the surface z=x^2+ y^2 at the point (1, 2, 5) .

## The Attempt at a Solution

Normal Vector 1 is i-j
Normal Vector 2 is 2i+4j-k
Cross them to get i+j+6k
then turn that vector into a unit vector and after that I am unsure what to do

first you need to decide what exactly you mean by slope...

Something like dz/du, where du is position change projected on the xy-plane, and dz is the change in vertical position. This lines up with the usual notion of slope.

If you are confident in you tangent vector (which i haven't checked) call it t:
t = xi+yj+zk

then the slope as defined above will be given by the ratio of z component to the projection on the xy plane
slope = z/(sqrt(x^2 + y^2))

Another easy way to find the tangent vector direction would be to parameterise the line in terms of t=x, giving r(t). Then differentiate in terms of t.

## What is a tangent line in multivariable calculus?

In multivariable calculus, a tangent line is a line that touches a curve at a single point, following the same direction as the curve at that point.

## How is a tangent line calculated in multivariable calculus?

In order to calculate a tangent line in multivariable calculus, one must first find the partial derivatives of the function at the point of interest. These partial derivatives are then used to calculate the slope of the tangent line, which can be plugged into the point-slope formula to find the equation of the tangent line.

## Why is finding the tangent line important in multivariable calculus?

The tangent line is important in multivariable calculus because it allows us to approximate the behavior of a function at a single point. This can be useful in understanding the behavior of a function in a specific area, as well as in optimization problems.

## What are some real-world applications of tangent lines in multivariable calculus?

Tangent lines in multivariable calculus have many real-world applications, such as in physics, engineering, and economics. For example, in physics, the tangent line can represent the instantaneous velocity of an object at a specific point in time. In economics, the tangent line can represent the marginal cost or revenue of a product at a particular level of production.

## What are some common challenges when solving tangent line problems in multivariable calculus?

Some common challenges when solving tangent line problems in multivariable calculus include determining the partial derivatives correctly, understanding the concept of a tangent line in higher dimensions, and applying the point-slope formula accurately. Additionally, setting up the problem and interpreting the results can also be challenging for some students.

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