(d) If a particle moves at a constant speed, then its velocity and acceleration are perpendicular.

Transcription

1 Math 142 -Review Problems II (Sec ) Work on concept check on pages 734 and 822. More review problems are on pages and nd In-Class Exam, Wednesday, April True - False problems. (a) The curve with vector equation r(t) = 1 + t 2, cos t, t 4 is smooth. (b) If f(t) is a scalar function and u(t) is a vector function, then [f(t)u(t)] = f (t)u (t). (c) If a particle moves at a constant speed, then its acceleration is 0. (d) If a particle moves at a constant speed, then its velocity and acceleration are perpendicular. (e) If s(t) is the arc length function of r(t), then s (t) = r (t). (f) The curvature of the circle of radius a > 0 is κ = 1/a. (g) The curvature of a straight line is. (h) If T(t), N(t), B(t) are the unit tangent vector, unit normal vector, and binormal vector, respectively, then T(t) (B(t) N(t)) = 1. (i) If r(t) is the position vector of a moving particle, then r (t) is its speed. (j) If T(t), N(t) are the unit tangent vector and unit normal vector of the curve r(t), then r (t) = a(t)t(t) + b(t)n(t) for some scalar functions a(t) and b(t). (k) The plane that passes through the point with position vector r 0 and contains nonparallel vectors a and b can be described by the vector function r(s, t) = r 0 + sa + tb. (l) If f(x, y) L as (x, y) approaches (0, 0) along any line through (0, 0), then (m) If lim f(x, y) exists and f(a, b) is defined, then f(a, b) is continuous at (a, b). (x,y) (a,b) (n) If p(x, y) is a polynomial, then ln p(x, y) is continuous on R 2. (o) If f y (x, y) = 0 for all (x, y) R 2, then f(x, y) is a constant function. (p) If f x (a, b) and f y (a, b) exist, then f(x, y) is differentiable at (a, b). (q) If f x (x, y) and f y (x, y) are continuous (a, b), then f(x, y) is continuous at (a, b). lim f(x, y) = L. (r) There exists a function f(x, y) with continuous 2nd-order partial derivatives such that f x = 2x y and f y = x + y. (s) r v (a, b) r u (a, b) is a normal vector of the tangent plane to the surface r(u, v) at the point with the position vector r(a, b). 1

2 (t) If f(x, y) = 0, 0 for all (x, y) R 2, then f(x, y) is a constant function. (u) If u is the unit tangent vector of the level curve f(x, y) = f(a, b) at (a, b), then f(a, b) u = 0. (v) If u = ai + bj is a unit vector then D u f(x, y) = ad i f(x, y) + bd j f(x, y). 2. Find r (t) and r (t). (a) r(t) = sin t 2, sin t, cos(2t) (b) r(t) = te 2t i + t 2 j + t + 1 k 3. Evaluate the integral. ( 3 (a) t i + 4 ) 2 t j dt (b) 2 0 ( ) 4 t + 1 i + et 2 j + te t k dt 4. Find parametric equations for the tangent line to the curve at the given point. (a) x = cos t, y = t, z = sin t, ( 1, π, 0) (b) x = t 2 + 1, y = ln(t + 1), z = e 3t, (1, 0, 1) 5. Let r(t) = t 2, t, t 2 5. Find t at which r(t) and r (t) are perpendicular. 6. Find the length of the curve. (a) r(t) = sin t, cos t, 8t, 0 t 3 (b) r(t) = e t sin(2t), e t, e t cos(2t), 0 t 1 7. Find the curvature. (a) r(t) = ti + (t 2 + t 1)j + tk, (b) y = e 2x 8. Find the curvature of the curve at the given point. (a) r(t) = cos 2t, 2 sin 2t, 4t, ( 1, 0, 2π) (b) y = x, x = 1 9. Find (i) the unit tangent vector T, (ii) the unit normal vector N, (iii) the binormal vector B, (iv) the normal plane, (v) the osculating plane, at the given point. (a) r(t) = t, t 2 1, t, (1, 0, 1) (b) r(t) = 4 cos πti + 4 sin πtj + tk, ( 4, 0, 1) 10. Let r(t) = 4te 2t i+2e 2t j 16t 2 k be the position vector of a moving particle. Find the velocity, acceleration, and speed. 11. Find the position and acceleration vectors of a particle that has the velocity v(t) = (t + 2)i + t 2 j + e t/3 k with the initial position r(0) = 4i 3k. 2

3 12. Find the position and velocity vectors of a particle that has the acceleration a(t) = ti 16k with the initial velocity v(0) = 12, 4, 0 and the initial position r(0) = 5, 0, A force with magnitude 20 N acts directly upward from the xy-plane on an object with mass 4kg, and there is no gravitational force. The object starts at the origin with initial velocity v(0) = 2i j. Find (a) its position function at time t, (b) its speed at time t. 14. A projectile is fired with an initial speed of 100 ft/s and angle of elevation π/6. Find (a) the range of the projectile, (b) the maximum height reached, (c) the speed at impact. 15. A baseball pitcher throws a pitch horizontally from a height of 6 ft with an initial speed of 130 ft/s. Find the position of the ball t seconds after release. If home plate is 60 ft away, how high is the ball when it crosses home plate? 16. A particle moves with the position function r(t) = t, t 2 1, t. Find the tangential and normal components of acceleration. 17. Find the limit, if it exists, or show that the limit does not exist. (a) (c) lim 3x 4 + y x 4 + 2y 2 (b) lim xy x2 + 2y 2 x 3 2x 2 2y 2 (x y) 2 lim (d) lim x 2 + y 2 x 2 + y Determine the set of points at which the function is continuous. (a) F (x, y) = x 2 y 2 1 (b) G(x, y) = ln(3 x 2 + y) (c) f(x, y, z) = z x2 y 2 1 z x 3 y 2 (x, y) (0, 0) (d) f(x, y) = x 2 + 3y 2 0 (x, y) = (0, 0). 19. Find all the first and second partial derivatives. (a) f(x, y) = x 2 e y 4y (b) f(x, y) = x 2 y 4x + 3 sin y (c) f(x, y, z) = x 3 y 2 sin yz 3

4 20. Find equations of the tangent plane and normal line to the given surface at the specified point. (a) z = e x2 y 2, (0, 0, 1) (b) z = x 3 2xy (1, 1, 3) 21. Let C be the intersection of the two surfaces S 1 : x 2 + 4y 2 + z 2 = 6, S 2 : z = x 2 + 2y. Show that the point (1, 1, 1) is on the curve C and find the tangent line to the curve C at the point (1, 1, 1). 22. Find the linear approximation of the function f(x, y) = xe y2 4x at (2, 0) and use it to approximate f(1.85, 0.1). 23. Find the linear approximation of the function f(x, y, z) = sin(yz 2 ) + x 3 z at ( 2, 0, 1). 24. Find the differential of the function (a) z = ye x + sin x (b) w = x + y + z 25. If z = x 2 3y and (x, y) changes from (2, 1) to (2.1, 0.8), compare the values z and dz. 26. The dimension of a closed rectangular box are measured as 80 cm, 60 cm, and 50 cm, respectively, with a possible error of 0.2 cm in each. Use differentials to estimate the maximum error in the calculated volume of the box. 27. Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height 15 cm if the tin is 0.02 cm thick. 28. Find an equation of the tangent plane to the parametric surface r(u, v) = ui + ln(uv)j + vk at (u, v) = (1, 1). 29. Suppose z = f(x, y), where x = g(s, t), y = h(s, t), g(1, 2) = 3, g s (1, 2) = 1, g t (1, 2) = 4, h(1, 2) = 6, h s (1, 2) = 5, h t (1, 2) = 10, f x (3, 6) = 7, and f y (3, 6) = 8. Find z/ s and z/ t when s = 1 and t = Suppose g(r, s) = f(2r s, s 2 4r) and f x (0, 0) = 2, f y (0, 0) = 4. Find g r (1, 2) and g s (1, 2). 31. If z = y + f(x 2 y 2 ), where f is differentiable, show that y z x + x z y = x. 32. The length x of a side of a triangle is increasing at a rate of 3 in/s, the length y of another side is decreasing at a rate of 2 in/s, and the contained angle θ is increasing at a rate of 0.05 radian/s. How fast is the area of the triangle changing when x = 40 in, y = 50 in, and θ = π/6? 33. Find dy/dx if e xy + xy = x 2. 4

5 34. Find z/ x and z/ y if 3yz 2 + e x cos 4z = 4 + 3y Find the directional derivative of the function at the given point in the direction of vector v. (a) f(x, y) = x 2 + y 2, (3, 4), v = 3, 2 (b) f(x, y) = y 2 + 2ye 4x, (0, 2), v given by the angle θ = π/4 (c) f(x, y, z) = e xy+z, (1, 1, 1), v in the direction from (1, 1, 1) to (3, 1, 2) 36. Find (i) the gradient of f, (ii) the gradient of f at the point P. (a) f(x, y) = x 2 + y 2, P (3, 4), (b) f(x, y) = y 2 + 2ye 4x, P (0, 2), (c) f(x, y, z) = e xy+z, P (1, 1, 1) 37. Suppose z(u, v) = f(x, y), where x = uv, y = u v f x (1, 1) = 1, f x (1, 0) = 2, f y (1, 1) = 1, and f y (1, 0) = 4. Find z when u = v = Suppose f(x, y) = xy + y 2. Find (a) the maximum rate of change of f at the point (1, 2) and the directions in which it occurs; (b) the minimum rate of change of f at the point (1, 2) and the direction in which it occurs; (c) the directions in which the directional derivative of f at the point (1, 2) is 0; (d) the directions in which the directional derivative of f at the point (1, 2) is Find equations of (i) the tangent plane and (ii) the normal line to the given surface at the specified point. (a) x 2 + y 2 + z 2 = 6, ( 1, 2, 1), (b) e xy+z = 1, (1, 1, 1). 40. Suppose f(x, y) = x sin(πxy). Find (i) the tangent line and (ii) the normal line to the level curve f(x, y) = 0 at the point (1, 2). 41. Suppose f(x, y, z) = (xy 1)e z. Find (i) the tangent plane and (ii) the normal line to the level surface f(x, y, z) = 1 at the point (1, 2, 0). 42. Find the matrix of partial derivatives for spherical coordinates