4. Assume that the curvature of a path described by the vector-valued function r(t) can be ||r' (t) x r" (t) || expressed by the formula Show that the curvature of a curve y = ||r' (t)||3 the xy-plane can be found using the formula f(x) in |f" (x)| √√1+|f'(x)|2³

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Q: Use the formula (x) K(X) = = y = 3x4 = |f"(x)| [1 + (F'(x))²] ³ 3/2 to find the curvature.

A: y=3x4

Q: Find the extreme values of the function z= &r+2ry-3r+y+1 and show that only one of the two extrema…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

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Q: Use the formula x(x) x(x) = y = 2x4 * 1 + (f'(x))²]3/2 to find the curvature.

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Q: The curvature of a function y = f(x) is given by the expression y" (1 + (y')²)³/2 Determine a…

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Q: Pind the curvature x of the curve given by 7(()- (V2)i+ (e)j+(e)k, at the point P(0, 1, 1),

A: Curvature K = |r' x r"|/|r'|^3

Q: Use the formula x(x) = = K(x) = y = 3x4 |f"(x)| [1 + (²²(x))²] ¹ to find the curvature.

A: Given information:Function: To find:The curvature of the function .Concept used:The curvature of a…

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Q: ||r'(t) × r"(t)|| ||r'(t)||* 3. Use the formula k(t) to find the curvature of the space curve at t =…

A: To find The curvature of the space curve

Q: Find the curvature and the radius of curvature of the curve y = x + 5x2 – 2x3 at the origin (0,0).

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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: 6. Given f(x, y) = 2x³y=x²y³, a) find the directional derivative of f(x, y) at (1, 1) in the…

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Q: (1 point) Find the curvature (t) of the curve r(t) = (4 sin t)i + (4 sin t)j + (5 cost) k

A: we will find the required solution.

Q: 8. [*] for position vector (t-1, 3-2t, 4t-t') find [*] the unit tangent [**] the unit normal, the…

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Q: Write a formula for the curvature K(t) in terms of ( r' (t) and r" 0) )? And prove it?

A: The curvature Kt is given by Kt=r't×r''tr't3 Now prove this formula in the following.

Q: The curvature of the curve 7 (t)={2t, 2t+1, sin2t) at t= is 4 O 1 O 1 O 2 1/818 1/2

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Q: Consider the curve C defined by (x; y) = r(t), where r(t) = (t; (1+t)/t; (1-t^2)/t) ; t > 0: (a)…

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Q: Find the curvature k of the curve r(t)=ti+ 6t^2j+4tk

A: To find: The curvature k of the curve rt=ti+6t2j+4tk. Formula used: The curvature of curve is…