Implicit Differentiation Tangent lines

In summary, to find where the graph of 25x^2 + 16y^2 + 200x - 160y + 400 = 0 has a horizontal tangent line, use implicit differentiation and set dy/dx to zero. There should be two points on the ellipse where this is true.
  • #1
htoor9
17
0

Homework Statement



Where does the graph of 25x^2 + 16y^2 + 200x - 160y + 400 = 0 have a horizontal tangent line.

Homework Equations



dy/dx dx/dy or something not sure.

The Attempt at a Solution



Well I know that a horizontal tangent line would mean the slope is zero...but what exactly do I do?
 
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  • #2
htoor9 said:

Homework Statement



Where does the graph of 25x^2 + 16y^2 + 200x - 160y + 400 = 0 have a horizontal tangent line.

Homework Equations



dy/dx dx/dy or something not sure.

The Attempt at a Solution



Well I know that a horizontal tangent line would mean the slope is zero...but what exactly do I do?
The easiest thing to do is to use implicit differentiation to get dy/dx, then set it to zero. Since the graph is an ellipse, there should be two points where dy/dx = 0.
 

Related to Implicit Differentiation Tangent lines

1. What is implicit differentiation?

Implicit differentiation is a method used in calculus to find the derivative of an equation that is not explicitly expressed in terms of a single variable. It is particularly useful when dealing with equations that are difficult to solve for a specific variable, such as equations involving multiple variables or higher order equations.

2. How is implicit differentiation used to find tangent lines?

Implicit differentiation can be used to find the slope of the tangent line to a curve at a given point. By finding the derivative of both sides of an implicit equation and plugging in the coordinates of the given point, we can determine the slope of the tangent line. This slope, along with the given point, can then be used to write the equation of the tangent line using the point-slope form.

3. What is the difference between implicit and explicit differentiation?

Explicit differentiation is used to find the derivative of an equation that is explicitly expressed in terms of a single variable. It is the most common form of differentiation and can be used for simple equations. On the other hand, implicit differentiation is used for equations that are not explicitly expressed in terms of a single variable, making them more complex and difficult to solve.

4. Can implicit differentiation be used for any type of equation?

Implicit differentiation can be used for most types of equations, including polynomial, trigonometric, exponential, and logarithmic equations. However, it may not always be the most efficient method for solving equations, and there may be other techniques that are more appropriate depending on the specific equation.

5. What are some real-world applications of implicit differentiation?

Implicit differentiation has many real-world applications, such as in physics, economics, and engineering. It can be used to find the slope of a curve on a graph, calculate the rate of change in a dynamic system, or optimize a function for maximum or minimum values. It is also commonly used in the study of curves and surfaces in 3-dimensional space.

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