# Multivariable Calculus - Tangent Line

• Larrytsai
In summary, the parametric equations for the tangent line at the point (cos (-5*pi/6), sin (-5*pi/6), -5*pi/6) on the curve x(t) = cos t, y(t) = sin t, z(t) = t are x(t) = 1 - (1/2)t, y(t) = (-√3/2)t, z(t) = t. These equations are parametrized so that the tangent line passes through the given point at t=0.
Larrytsai

## Homework Statement

Find parametric equations for the tangent line at the point
(cos (-5*pi/6), sin (-5*pi/6), -5*pi/6) on the curve
x(t) = cos t
y(t) = sin t
z(t) = t

(Your line should be parametrized so that it passes through the given point at t=0).

Im not really understanding the question that well, can anyone help me understand and solve this problem?

## The Attempt at a Solution

What I have so far is,

r'(t) = ( - sin t , cos t, 1)
so
x(t) = -sin t + cos(-5pi/6)
thats all I have so far and its wrong.

Last edited:

To find the parametric equations for the tangent line at the given point, we first need to find the slope of the tangent line at that point. This can be done by taking the derivative of each component of the given curve and evaluating it at t=-5pi/6. So, we have:

x'(t) = -sin t
y'(t) = cos t
z'(t) = 1

x'(-5pi/6) = -sin (-5pi/6) = -1/2
y'(-5pi/6) = cos (-5pi/6) = -√3/2
z'(-5pi/6) = 1

So, the slope of the tangent line at t=-5pi/6 is given by m = (-1/2, -√3/2, 1).

Next, we need to find a point on the tangent line. This can be done by simply plugging in t=0 into the given curve, since we want the tangent line to pass through the given point at t=0.

x(0) = cos 0 = 1
y(0) = sin 0 = 0
z(0) = 0

So, a point on the tangent line is (1, 0, 0).

Now, we can use the point-slope form of a line to find the parametric equations for the tangent line:

x(t) = 1 + (-1/2)t
y(t) = 0 + (-√3/2)t
z(t) = 0 + t

These equations can also be written in vector form as:

r(t) = (1, 0, 0) + t(-1/2, -√3/2, 1)

So, the parametric equations for the tangent line at the given point are:

x(t) = 1 - (1/2)t
y(t) = (-√3/2)t
z(t) = t

## 1. What is the definition of a tangent line in multivariable calculus?

A tangent line in multivariable calculus is a line that touches a curve or surface at a single point and has the same slope as the curve or surface at that point. It represents the instantaneous rate of change of the curve or surface at that point.

## 2. How is the equation for a tangent line in multivariable calculus different from the equation in single variable calculus?

The equation for a tangent line in multivariable calculus is different from the equation in single variable calculus because it takes into account the different variables and their rates of change. In single variable calculus, the equation only has one independent variable, while in multivariable calculus, the equation has multiple independent variables and their corresponding coefficients.

## 3. What is the process for finding the equation of a tangent line to a curve or surface in multivariable calculus?

The process for finding the equation of a tangent line in multivariable calculus involves finding the partial derivatives of the curve or surface with respect to each independent variable, evaluating them at the point of tangency, and then using these values to construct the equation of the tangent line. This equation will include the coordinates of the point of tangency and the partial derivatives at that point.

## 4. Can a tangent line intersect a curve or surface in more than one point?

No, a tangent line can only intersect a curve or surface in one point in multivariable calculus. This is because the tangent line represents the instantaneous rate of change at a single point, so it can only touch the curve or surface at that point.

## 5. How is the concept of a tangent line used in real-world applications?

The concept of a tangent line is used in many real-world applications, such as physics, engineering, and economics. For example, in physics, the velocity of an object at a specific point can be represented by a tangent line to the object's position-time graph. In economics, the marginal cost of a product at a certain level of production can be represented by a tangent line to the cost function. Tangent lines are also used in optimization problems to find the maximum or minimum value of a function.

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