Find the equation of a plane perpendicular to the line x = 2t + 1, y = 3t + 4, z = t 1 and passing through the point (2, 1, 3).

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1 CME 100 Midterm Solutions - Fall CME Midterm Solutions - Fall 004 Problem 1 Find the equation of a lane erendicular to the line x = t + 1, y = 3t + 4, z = t 1 and assing through the oint (, 1, 3). lane: r = (t + 1)î + ( 3t + 4)ĵ + (t 1)ˆk n = i 3ĵ + ˆk (x x 0 ) 3(y y 0 ) + 1(z z 0 ) = 0 (x ) 3(y + 1) + 1(z 3) = 0 x 3y + z = 10 Find the area of a triangle whose vertices are (1,, 3), (, 0, 1), and (1, 1, ). ab = î + ĵ + 4ˆk and ac = î ĵ + 3ˆk î ĵ ˆk ab ac = = î(6 + 4) ĵ( 3 + 4) + ˆk(1 + ) = 10î ĵ + 3ˆk area of the arallelogram = ab ac = = 110 area of the triangle = 1 (area of the arallelogram) = 110 Using the definition of a dot roduct, find the cosine of the angle between the main diagonal of a cube and one of its adjacent sides. a = î + ĵ + ˆk and b = î. a b = 1 = a b cos θ a = 3 and b = 1. So 1 = 3 cos θ cos θ = 1 3

2 CME 100 Midterm Solutions - Fall 004 Part d Comute the scalar rojection of a = î + ĵ + ˆk in the direction of b = î + 4ĵ ˆk. Part e ( a b) roj b ( a) = b a b = = 3 b = b b = ( 1) = 18 = 3 roj b ( a) = 3 3 = You have been asked to create a MATLAB scrit to comuter an aroximate some of the infinite series 1 n=1 1+n. Write a short MATLAB scrit to comute sum using the first 100 terms and to dislay the result in the command window. clear s = 0; for n = 1:100 s = s + 1 / (1 + n^); end s Problem A article is traveling along an ellitical ath given by: r(t) = 5 cos tî + 3 sin tĵ. Comute the velocity vector v(t) v(t) = r(t) = 5 sin tî + 3 cos tĵ Find all value(s) of t for 0 t < π for which the velocity v(t) is erendicular to the osition vector r(t) t(t) r(t) v(t) r(t) = 5 sin(t) cos(t) + 9 sin(t) cos(t) = 0 sin(t) cos(t) = 9 sin(t) cos(t) 5 sin(t) = 0 or cos(t) = 0 t = 0, π, π, 3π

3 CME 100 Midterm Solutions - Fall Comute the curvature of the ath as a function of time t. Part d κ = v a v 3 a = r = v = 5 cos(t)î 3sin(t)ĵ î ĵ ˆk v a = 5 sin(t) 3 cos(t) 0 5 cos(t) 3 sin(t) 0 = (15 sin (t) + 15 cos (t))ˆk = 15ˆk v 3 = (5 sin (t) + 9 cos (t)) 3 = (16 sin (t) + 9) 3 15 κ = (16 sin (t) + 9) 3 Determine the location along the ellise at which the curvature is maximized and minimized. Evaluate the curvature at those locations. dκ dt = d dt (15(16 sin (t) + 9) 3 ( 3 ) = 15 ) 3 sin(t) cos(t)(16sin (t) + 9) sin(t) cos(t) = (16 sin (t) + 9) 5 dκ dt = 0 sin(t) cos(t) = 0 sin(t) = 0 or sin(t) = 0 t = 0, π, π, 3π ( nπ ) { κ = (16 sin ( ) nπ 3 = 7 t = 0, π ) 15 = 3 5 t = π, 3π Substituting back into r(t) we see that κ is maximized at (0, ±3) and minimized at (±5, 0). Part e Sketch the ellise indicating the locations of the maximum and minimum curvatures and exlain briefly how your results in art d can be interreted geometrically y

4 CME 100 Midterm Solutions - Fall The radius of the circles is 1/κ for the oints given in art d, thus these are the tangent circles to the oints. The grah shows that our results from the revious calculation makes sense as where the curvature is maximized the tangent circles have a smaller radius and vice versa. Problem 3 Find all maxima and minima of f(x, y) = 4x + 10y on a circular region bounded by x + y = 4 using the following stes Find the coordinates of all critical oints in the interior of the region So, x = 0 and y = 0 is the only critical oint. f = 8xî + 0yĵ = 0 Assuming the boundary x + y = 4 reresents a constraint, use the method of Lagrange multiliers to find all maxima and minima of f(x, y) along the boundary f = λ g g = xî + yĵ 8x = xλ 0y = yλ } λ = 10 x = 0, y = ± λ = 4 y = 0, x = ± f(0, ±) = 4(0) + 10(±) = 40 max on curve at (0, ±) f(±, 0) = 4(±) + 10(0) = 16 min on curve at (±, 0) Based on your results in arts a and b clearly list the coordinates of the absolute maximum and the absolute minimum (0, 0) is the absolute min of the bounded region. (0, ±) are the absolute max of the bounded region.

5 CME 100 Midterm Solutions - Fall Problem 4 Let f(x, y, z) = zy xy Find the gradient of f(x, y, z) at oint (0, 1, 1) f = yî + (zy x)ĵ + y ˆk f (0,1, 1) = î ĵ + ˆk Find the directional derivative of f(x, y, z) at (0, 1, 1) in the direction of the gradient f = y + (zy x) + y 4 û = f f Dûf = f û = f f f = f Dûf (0,1, 1) = 6 A article is moving along a trajectory given by the following osition vector: r(t) = (t 3 1)î + tĵ + (t )ˆk Find the rate of change of f(x, y, z) with resect to time df dt There are two ways to solve this. First Way: Chain Rule df dt = f x x df dt + f t + f z z x = t 3 1 y = t z = t r(t) = 3tî + ĵ + tˆk at oint (0, 1, 1). Note that at this oint. t = f (t)r = 3ty + (zy x) + ty t=1 = 3(1)(1) + ((1)( 1) (0)) + (1)(1) = 3

6 CME 100 Midterm Solutions - Fall Second Way: Substitution f(t) = (t )(t ) (t 3 1)(t) = t t df dt = 1 4t df dt = 3 t=1 Part d Assume now that the indeendent variables are constrained such that x + y + z = 1. Comute (0, 1, 1) ( ) f at x ( ) f x ( ) f x = f x x + f z z + f = f x (0) + y ( ) + zy x = y + zy x = + (1)( 1) 0 = 4 (0,1, 1) Part e Using linearization, determine the aroximate value of the function f(x, y, z) at oint (0.1, 1.05, 1.1) f = yî + (zy x)ĵ + y ˆk L(x, y) = f(0, 1, 1)+f x (0, 1, 1)(x 0)+f y (0, 1, 1)(y 1)+f z (0, 1, 1)(z+1) = 1 (x) (y 1)+(z+1) Bonus L(0.1, 1.05, 1.1) = (0.05) + ( 0.1) = 1.3 In class we have derived three Kelers Laws that govern the motion of lanets and satellites. These three laws imly two imortant conservation rinciles. One is the conservation of angular momentum, aka equal areas in equal times - this was discussed in lecture. The second is conservation of energy. Your job is, following the stes described below, to use the first two Kelers Laws to show that the total energy (including both otential and kinetic energy) er unit mass of the satellite: E = 1 v µ r

7 CME 100 Midterm Solutions - Fall is conserved. r = da dt 1 + e cos θ r θ = = h 1st Kelers Law nd Kelers Law by differentiating r with resect to time and using the fact that h = r θ, show that: ṙ = eh sin θ dr dt = 1 + e cos θ ( θesinθ) = r θe sin θ = ehsinθ recall that in olar coordinates the velocity vector is given by: v = ṙˆr + r θˆθ. Using your result in a) and the fact that h = µ find an exression for the square of the seed v and show that: v = µe sin θ + µ(1 + e cos θ) ṙ = eh sin θ v = eh sin θ θ = h r ˆr e cos θ hˆθ v = v v = e h sin θ + (1 + e cos θ) h = µe sin θ µ(1 + e cos θ) + determine the sum of the otential and kinetic energy and using the fact that = a(1 e ) show that the total energy is conserved, i.e: E = µ a = constant

8 CME 100 Midterm Solutions - Fall E = 1 v µ r = e µ sin θ µ(1 + e cos θ) µ(1 + e cos θ) + = µ (e e cos θ (1 + e cos θ)) = µ (e 1) = µ a