Slope of tangent line to curves cut from surface

In summary, to find the slope of a tangent line to curves cut from the surface z = (3x^2) + (4y^2) - 6 by planes through the point (1,1,1) and parallel to the xz and yz planes, you need to calculate dz/dx and dz/dy for the respective planes and plug in the value of 1 for x and y. The slope of the tangent line parallel to the xz plane is dz/dx, while the slope of the tangent line parallel to the yz plane is dz/dy.
  • #1
chetzread
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Homework Statement


find the slope of tangent line to curves cut from surface z = (3x^2) +(4y^2) - 6 by planes thru the point (1,1,1) and parallel to xz planes and yz planes ...

Homework Equations

The Attempt at a Solution


slope of tnagent that parallel to xz planes is dz/dy , while the slope of tangent that parallel to yz plane is dz /dz . Correct me if i am wrong
 
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  • #2
To be parallel to the xz plane means that the y value is some constant number (y ≡ 1). So there is no dy. y is not involved in that slope except that its constant value (1) might be in the equation. Calculate dz/dx and plug in the 1s.
Similarly, to be parallel to the yz plane means that the x value is some constant number (x ≡ 1). So there is no dx. x is not involved in that slope except that its constant value (1) might be in the equation. Calculate dz/dy and plug in the 1s
 
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  • #3
So
FactChecker said:
To be parallel to the xz plane means that the y value is some constant number (y ≡ 1). So there is no dy. y is not involved in that slope except that its constant value (1) might be in the equation. Calculate dz/dx and plug in the 1s.
Similarly, to be parallel to the yz plane means that the x value is some constant number (x ≡ 1). So there is no dx. x is not involved in that slope except that its constant value (1) might be in the equation. Calculate dz/dy and plug in the 1s
So , slope of tnagent that parallel to xz planes is dz/dx , while the slope of tangent thatparallel to yz plane is dz /dy . ?
 
  • #4
chetzread said:
So

So , slope of tnagent that parallel to xz planes is dz/dx , while the slope of tangent thatparallel to yz plane is dz /dy . ?
yes
 
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FAQ: Slope of tangent line to curves cut from surface

What is the slope of a tangent line?

The slope of a tangent line is a measure of the steepness of a curve at a particular point. It represents the rate of change of the curve at that specific point.

How is the slope of a tangent line calculated?

The slope of a tangent line can be calculated using the derivative of the function at the given point. This is done by taking the limit as the change in x approaches 0, and solving for the slope using the formula f'(x) = (f(x+h) - f(x))/h.

What does the slope of a tangent line tell us about a curve?

The slope of a tangent line tells us the rate of change of the curve at a specific point. It can also indicate the direction of the curve at that point, whether it is increasing or decreasing.

How does the slope of a tangent line relate to the curvature of a curve?

The slope of a tangent line is related to the curvature of a curve in that a larger slope indicates a steeper curve, while a smaller slope indicates a flatter curve. In other words, the slope of the tangent line can be used to determine the curvature of a curve at a specific point.

Can the slope of a tangent line ever be undefined?

Yes, the slope of a tangent line can be undefined at points where the derivative of the function is undefined. This can occur at sharp corners or cusps on a curve, or at points where the function has a vertical tangent.

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