Solving Curve C Tangent P: (-3,-2,2)

In summary, the unique point on the curve with the property that the tangent line at it passes through the point (−3,−2,2) is (x,y,z)=(3−3t,1−t^{2},t+2t^{3}).
  • #1
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Hello, I need help for this problem

Homework Statement


There exist a curve C such that its parametric equation is (x,y,z)=(3−3t,1−t[tex]^{2}[/tex],t+2t[tex]^{3}[/tex]). There is a unique point P on the curve with the property that the tangent line at P passes through the point (−3,−2,2). Find the coordinates of P.

Homework Equations



(C) : (x,y,z)=(3−3t,1−t[tex]^{2}[/tex],t+2t[tex]^{3}[/tex])

The Attempt at a Solution



Attempt to solve it
(x',y'z')= (-3,-2t,1+6t[tex]^{2}[/tex] )
since the above is the direction vector of the tangent T then I tried to express the parametric equation of the tangent in function of t which has given me
x=-3s-3
y=-2ts-2
z=(1+6t[tex]^{2}[/tex])s+2

after that I tried to solve xp=x by replacing x in the line equation by the curve equation but I can't solve that ! I really don't know how to approach this exercise ...

Thank you for your help
 
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  • #2
tifa8 said:
Hello, I need help for this problem

Homework Statement


There exist a curve C such that its parametric equation is (x,y,z)=(3−3t,1−t[tex]^{2}[/tex],t+2t[tex]^{3}[/tex]). There is a unique point P on the curve with the property that the tangent line at P passes through the point (−3,−2,2). Find the coordinates of P.

Homework Equations



(C) : (x,y,z)=(3−3t,1−t[tex]^{2}[/tex],t+2t[tex]^{3}[/tex])

The Attempt at a Solution



Attempt to solve it
(x',y'z')= (-3,-2t,1+6t[tex]^{2}[/tex] )
since the above is the direction vector of the tangent T then I tried to express the parametric equation of the tangent in function of t which has given me
x=-3s-3
y=-2ts-2
z=(1+6t[tex]^{2}[/tex])s+2

after that I tried to solve xp=x by replacing x in the line equation by the curve equation but I can't solve that ! I really don't know how to approach this exercise ...

Thank you for your help

Hey there tifa and welcome to the forums.

Have you ever studied or covered linear interpolation? Or have you covered the equation of a line in n dimensions (or just 3)?
 
  • #3
thank you !

No I didn't cover yet linear interpolation but I think we will see it next week. And no didn't see equations of lines in more than 3 dimensions. What I'm covering now is curves and motion in curves.
 
  • #4
What you know is that the difference (3−3t,1-t^2,t+2t^3)-(−3,−2,2) is parallel to your derivative direction (-3,-2t,1+6t^2). Two vectors A and B are parallel if A=k*B for some k. Can you write down an equation expressing that and solve for t?
 
  • #5
thank you ! I found t=3 which was correct :)
 

1. What is Curve C Tangent P?

Curve C Tangent P refers to the point where a line (tangent) touches a curve (Curve C) at a specific point (P). It is the point where the slope of the tangent line is equal to the slope of the curve at that point.

2. How do you solve for Curve C Tangent P?

To solve for Curve C Tangent P, you need to first find the slope of the curve at the point P. This can be done by taking the derivative of the curve equation with respect to the variable. Next, you need to find the slope of the tangent line at point P. This can be done by using the point-slope formula and plugging in the slope of the curve and the coordinates of point P. Finally, you can set the two slopes equal to each other and solve for the x and y values of Curve C Tangent P.

3. Why is Curve C Tangent P important?

Curve C Tangent P is important because it helps us understand the behavior of a curve at a specific point. It tells us the direction in which the curve is changing at that point and can be used to find critical points or maximum/minimum points on the curve. It is also used in various applications such as optimization problems in economics and physics.

4. What are some real-life examples of Curve C Tangent P?

One real-life example of Curve C Tangent P is in the study of motion. The slope of a velocity vs. time graph at a specific point represents the acceleration at that point, which is the Curve C Tangent P. Another example is in engineering, where the slope of a stress vs. strain curve at a specific point represents the Young's modulus, which is the Curve C Tangent P.

5. Can Curve C Tangent P be negative?

Yes, Curve C Tangent P can be negative. The slope of a curve can be positive, negative, or zero, and the slope of the tangent line can also have these three values. If the slope of the curve is negative and the slope of the tangent line is positive, then Curve C Tangent P will be negative. Similarly, if the slope of the curve is positive and the slope of the tangent line is negative, then Curve C Tangent P will be negative.

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