Calculus III Tangent Planes In Calculus I, we found tangent lines to curves at a

Question

Calculus III Tangent Planes In Calculus I, we found tangent lines to curves at a specific point. In Chapter 11. we found tangent vectors to curves given by vector-valued functions. After introducing surfaces, we are going to find tangent planes to these surfaces at a specific point. Recall that to find the cquation of a plane. we need a point and a normal vector to that plane. Theorem 1 Given a surface F(ry, z) 0 and a point (a,b, c) on this surface, the normal vector uo the tangent plane at that point isVF(a,b,c) (Fr (a,b,c).F(a,b.c).F (a,b.c) Thus the equation of the tangent plane is: Fe(a, b,c)(a Note that if the surface was given by z f(x, y), we can rewrite it as F(r, y,z) y) 0 which makes F n the above equati ion. The equation of the tangent pl ane at (a, b.c. where c f(a,b) can be rewritten as: z f.(a,b,e)(r-a)+ f (a, b,c)(y- b) +c f, (a,b,c)(x a) f (a, b,c)(- b) +f(a,b). Example 1 Let z 1 2 be a sui We want to find the tangent plane at (1.1.4) (Note rface. that 4 f(l.). First we rewrite ir as F(,y,z) 1+2 -z 0. The normal vector is (Fe, F.F) (4,2y, -1) (4,2 -1) after substituting the point there. Thus the equation of the iangen plane is In Calculus I we saw that one of the advantages of tangent lines was using them to approximate functions by linear approximations. Similarly, we can use tangent planes to approximate the value of the function z f(r,y) near a fixed point (a,b). Theorem 2 The linear approximation to the surface z f(,y) near the point (a,b.c) where c f(a,b) is given by: Near the point (a,b.c. we can approximate the function z f(x.y) using L(r, y): Example 2 let z Vi +y be a surface with the point (I 1.2) We can approximate the values of this function near this point using L(r.y) To find the linear approximation, we need 2 and 6 1. Thus near the point (1,1,2) we have For example, f(1.1,1.I) V4.63 L(1.l, 1.1) 32(1.1)+ 1.1 -1 2.3.

Explanation / Answer

Q2. To have a horizontal tangent plane to a surface, what would be Fx and Fy?

Answer: A horizontal tangent plane means a plane parallel to xy-plane. Both Fx and Fy become zero for any plane which is parallel to xy-plane. Also, a horizontal tangent plane occurs at a point where the surface has a critical point (i.e. either a maximum or minimum). Recall that we determine the critical points on a surface by setting both Fx and Fy equal to zero.

Q3. Would the linear approximation in Example 2 give a good approximation for f(5,6)?

Answer: No. This type of linear approximation is only valid in the vicinity of the point where it was calculated. This approximation involves calculation of slopes in the vicinity of the given point and these slopes can only be determined with a high accuracy if we stay very near to that point. If we start deviating from the point, the slopes will start becoming crude and carry high error. The point (5,6) is too far away from the given point (1,1,2) or {(1,1) in xy-plane} near which the approximation was carried out.

I hope this helps :) Feel free to comment if you still have any doubt.

Good Luck.

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.