MTH 277 Test 4 review sheet Chapter , 14.7, 14.8 Chalmeta

Transcription

1 MTH 77 Test 4 review sheet Chapter , 14.7, 14.8 Chalmeta Multiple Choice 1. Let r(t) = 3 sin t i + 3 cos t j + αt k. What value of α gives an arc length of 5 from t = 0 to t = 1? (a) 6 (b) 5 (c) 4 (d) 3. Consider the device shown. An expression of point P as a function of wheel radius R, base length L, and angle θ is: (a) x = R L(1 cos θ) (b) x = R L(cos θ) (c) x = L R(1 cos θ) (d) x = L R(cos θ) 3. Let r(t) = cos ( ) t i + sin ( t ) j. Which of the following statements is FALSE? (a) A particle traveling this path has constant speed. (b) Velocity is always orthogonal to acceleration. (c) Acceleration is zero. (d) Curvature is If r(t) = cos t i + t j, then the maximum magnitude of acceleration occurs at (a) t = 0, π (b) t = π, 3π (c) t = π 4, 3π 4 (d) t = 0, π, π Short Answer 5. The velocity of a particle is given by v(t) = e t i + sin t j + k. Find the position function if the initial position is r(0) = 3 i + 4 j + 5 k. 1

2 MTH 77 Test 4 review sheet Chapter , 14.7, 14.8 Chalmeta 6. You are supervising the construction of a stretch of highway. Your supervisor tells you that at a certain point in a sharp turn, the road is following a circular arc with the radius 100 ft. (a) What is the curvature at this point (P ) on the road? (b) If a vehicle can be expected to hold the road safely when a N 5 ft/sec, then what is the maximum safe speed that should be posted on the warning sign? (Give your answer in ft./sec. If you dont get an answer in part (1), use κ = 36, which is not the correct answer.) 7. Let r(t) = t i + ln (sec t) j over the interval π t π. Find each of the following. (a) velocity v(t) and speed v(t). (b) acceleration a(t). (c) unit tangent vector T (t). (d) unit normal vector N(t). (e) curvature κ(t). Is there any value of t in this interval where the curve has no bend (κ(t) = 0)? If so, what is it? (f) tangential component of acceleration a T. (g) normal component of acceleration a N. 8. A projectile is launched horizontally off a cliff 100 feet high. What should the initial speed be in order to hit a target on the ground 500 feet away? 9. Find all local maxima, local minima, and saddle point(s) of f(x, y) = xy(1 x y) 10. Find all local maxima, local minima, and saddle point(s) of f(x, y) = x 3 xy + y Given f(x, y) = x 4x + y 4y + 1, make a list of all candidates for absolute max and min on the closed triangle bounded by the lines, x = 0, y =, and y = x in the first quadrant. Find the absolute maximum and minimum of f(x, y) on the region. 1. Given f(x, y) = x 4 + y 4 4xy +. (a) Make a list of all candidates for absolute maximums and minimums on the closed rectangular region defined by D = {(x, y) 0 x 3, 0 y }. (b) Find the absolute maximum and minimum values of f(x, y) in the region.

3 MTH 77 Test 4 review sheet Chapter , 14.7, 14.8 Chalmeta 13. Find the absolute maxima and/or minima of f(x, y) = x + y + 4x 4y on the disk x + y 9. Make a table showing all critical points on interior of the disk and it s boundary. The table should include the function value at each point. Be sure to state your final conclusion about the absolute maximum value and absolute minimum value of f(x, y). 14. Find the absolute minimum of f(x, y, z) = xy + 6yz + 8xz subject to the constraint xyz = 1, 000 3

4 MTH 77 Test 4 review sheet Chapter , 14.7, 14.8 Chalmeta 1. c. c 3. c 4. a Answers 5. r(t) = (e t + 1) i + ( cos t + 5) j + (t + 5) k 6. (a) κ = (b) 50 ft/sec 7. Find each of the following. (a) v(t) = i + tan t j, v(t) = sec t (b) a(t) = sec t j (c) T (t) = cos t i + sin t j (d) N(t) = sin t i + cos t j (e) κ(t) = cos t (f) a T = sec t tan t (g) a N = sec t 8. v 0 = 00 ft/sec 9. Saddle points at (0, 0), (0, 1), (1, 0); Maximum at ( 1, ) Saddle point at (0, 0); Minimum at (, ) Maximum: f(0, 0) = 1; Minimum f(1, ) = 5 1. Given f(x, y) = x 4 + y 4 4xy +. (a) (0, 0), (1, 1), (3,0), (3, 3 3), ( 3, ), (3, ), (0, ) (b) Max 83, Min 0 ( ) Maximum f, 3 = 9 + 1, Minimum f(, ) = f(30, 40, 10) = 700 4

5 MTH 77 Test 4 review sheet Chapter , 14.7, 14.8 Chalmeta Given a curve r(t) =< f(t), g(t), h(t) > Arc Length: L = s(t) = b a v(t) dt = t b a r (t) dt Arc Length from t = a to t = b. t 0 v(τ) dτ Arc Length from t 0 to t. Note: ds dt = r (t) = v = speed Important Unit Vectors: T (t) = v(t) v(t) = r (t) r (t) Unit Tangent Vector: Gives the direction in which the object is traveling. This is also the direction of the tangential component of acceleration. N(t) = T (t) T (t) Unit Normal Vector: Gives the direction in which the object is turning. This is also the direction of the normal component of acceleration. Curvature and Components of Acceleration: (Note: all are functions of time but the t s have been left off to be concise) d κ(t) = T /dt v a = ds/dt v 3 Curvature: The rate of change of direction per unit f (x) length along the curve. Expresses how sharply the object κ(x) = [1 + (f (x)) ] 3/ is turning. (For a circle of radius r, κ = 1.) r a = a T T + an N Acceleration: written as a sum of its components. a T = d v dt = d s dt Tangential Component of Acceleration: The acceleration in the direction of motion. (The rate of change of speed.) a N = κ v = κ ( ) ds Normal Component of Acceleration: The part of dt the acceleration that causes the object to turn. 5