1

find b. and c.

To obtain N for a particular plane curve, choose the one of n or - n that points toward the concave side of the curve, and make it into a unit vector. (See the figure to the right. ) Apply this method to find N for the following curves. b. r(t) = 7ti + 5 e'j c. r(1) = 49 - 912 i + 3tj. - 2 St= = a. To show that n(t) and - n(t) are both normal to r(t), first find v(t). v(t) = f'(t) i+ g'(1) j To show that n is normal to r, show that n . T = 0. If n is normal to r, then so is - n, because the components of - n are the same as those of n with sign changed. How are v and T related? The vector v is parallel How can v(t) be used to show that vector n is normal to the curve r(t)? O A. Show that n . v= - 1. O B. Show that v . T = 0. C. Show that n . v = 0. O D. Show that v . T = - 1. Why is the equation from the previous step satisfied? O A. The sum of the components of v(t) is the negative of T, so the dot product is 0. O B. The components of n(t) are negative reciprocals of the components of T, so the dot product is - 1. O C. The components of n(t) are negative reciprocals of the components of v(t), so the dot product is - 1. D. The components of n(t) are the components of v(t) with the order swapped and the sign of one changed, so the dot product is 0. b. N = (i+ (DiThe position of a particle in space at time t is r(t) as shown below. Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at t= 1. Write the particle's velocity at that time as the product of its speed and direction. r(t) = (2 In (t + 1))i + t' j +k The velocity vector is v(t) = 2 (t + 1 ) i+ 2t j + t k. (Type exact answers, using radicals as needed.) The acceleration vector is a(t) = - 2 i+ 2j + 1 k. ( 1 + 1 ) 2 (Type exact answers, using radicals as needed.) The velocity vector at t= 1 written as a product of the speed and direction is v(1) = (DI( Di + (Di + (DD) k]. (Type an exact answer, using radicals as needed.)7'3: 5 Find an equation for the circle oi curvature ofthe curve r[t} = Tti + 5in(5t}i at the point [?1.] (The curve parameterizes the graph elf)r = sin [?X] in the xy-plane.) (I) An equation for the circle of curvature is |:. (Type an equation. Type an exact answer: using 31: as needed.) [f"' (x)| The formula k(x) = [1 + (F' ( x )) 275 2 expresses the curvature k(x) of a twice-differentiable plane curve y = f(x) as a function of x. Find the curvature function of the following curve. Then graph f(x) together with k(x) over the given interval. f(x ) = 7x-, - 25xs2 The curvature function is k(x) =].Determine the maximum curvature for the graph of f(x) = 7 In (7x). The maximum curvature is at x =