(a) Let FR - R define a differentiable curve that does not pass → through the origin. If the point Xo F(to) is a point on the curve closest to the origin and not an endpoint of the curve, show that F'(to) ± F(to), that is, the velocity vector is orthogonal to the posi- tion vector. Hint: consider y(t) = ||F(t)||². = (b) Apply this to prove anew the well-known fact that the radius vector to any point on a circle is perpendicular to the tangent vector at that point. (c) What can you say about a point X₁ = F(t₁) farthest from the origin?

Q: Evaluate [F·dr if F(x, y)=(4e³)ī +(4xe³)j and the curve C' is line segments going from (0,0) to…

A: To evaluate ∫C F⇀·dr⇀ if F⇀x, y=4eyi⇀+4xeyj⇀ and the curve C is line segments from 0, 0 to 5, 4 to…

Q: How many horizontal tangent lines does the curve parametrized by the equations z(t) =-4t-1,…

A: on doing derivative we get the 1

Q: Find the equation of the tangent line to the curve defined by the equations, æ (t) = (2t + 1)², y(t)…

A: .

Q: Sketch the curve r(t)=f°i+tj and indicate the orientation. Find r'(t) and sketch r(1) & r°(1).

A: First we will plot the given parameters.

Q: 17. Let foxys = exy. Find the equation for the tangent plane to the graph of f at the point (1.1).

A:

Q: For the curve r-y +2xy-y+1=0: a) Find the derivative dy/dx. b) Where do the horizontal tangent lines…

A: follow the next step

Q: Q4. Simplify the following expression: 3-5 x 39 Click on image to enlarge if needed 12 81 243 324

A: To evaluate :- 3-5 * 39

Q: Find the equation of the tangent plane to z = In(x³ + 1) + yª at the point (0,3, 81).

A: We have to find equation of tangent plane at (0,3,81).