1) Let u = (2, −1,3), v = (4,0,-2), and w = (1,1,3). Use vector operations to determine each of the following: (a) 7v+3w (b) 3(u - 7v) (c) 2v (u+w) (d) ||u+v+w|| (e) ||2v - 3u|| (f) ||2v|| - ||3u|| (g) u v (h) vxw 2) Let a = (4,6,9) and b = (-6, -3,2). Find the measure of the acute angle between a and b. Round your answer to four decimal places. 3) Find the standard form of the equation of the plane that passes through the points (5,3,2), (-4, 6, 7), and (0, -3,8). 4) Let r(t) = (2 sin(t), 3 cos(t), t²) be a vector-valued function. Find the following: (a) limr(t) (b) r'(t) (c) ||r'(T)|| (d) fr(t) dt 5) Let f(x, y, z) = xey + 5z (a) Find fx, fy, and fz (b) Find fxx, fyy, and fzz 6) Let f (x, y) = 9 - x² - 7y³. (a) Evaluate fx (3,1) (b) Evaluate fy(3,1) 7) Find the equation of the plane that is tangent to the surface f(x, y) = y² - 4x² at the point (3,-1,-35). 8) Find the equation of the plane that is tangent to the surface f(x,y) = at the point (1,2) x+y 9) Let f(x, y) = x²ey. Find the maximum value of a directional derivative at (-2,0), and find the unit vector in the direction in which the maximum value occurs. 10) Determine the critical points of the function f(x, y) = 3x² - 2xy + y² - 8y. Determine whether each point is a local maxima, local minima, or saddle point. Provide proper mathematical justification to support your conclusions. 11) Evaluate the double integral ff xy² dA over the region bound by the x-axis, the y-axis, y = x², and x = 2. Draw a sketch of the region, then set up and evaluate the integral. Π 1 x² 12) Evaluate the triple integral √ √ √ x cos(y) dz dx dy.
1) Let u = (2, −1,3), v = (4,0,-2), and w = (1,1,3). Use vector operations to determine each of the following: (a) 7v+3w (b) 3(u - 7v) (c) 2v (u+w) (d) ||u+v+w|| (e) ||2v - 3u|| (f) ||2v|| - ||3u|| (g) u v (h) vxw 2) Let a = (4,6,9) and b = (-6, -3,2). Find the measure of the acute angle between a and b. Round your answer to four decimal places. 3) Find the standard form of the equation of the plane that passes through the points (5,3,2), (-4, 6, 7), and (0, -3,8). 4) Let r(t) = (2 sin(t), 3 cos(t), t²) be a vector-valued function. Find the following: (a) limr(t) (b) r'(t) (c) ||r'(T)|| (d) fr(t) dt 5) Let f(x, y, z) = xey + 5z (a) Find fx, fy, and fz (b) Find fxx, fyy, and fzz 6) Let f (x, y) = 9 - x² - 7y³. (a) Evaluate fx (3,1) (b) Evaluate fy(3,1) 7) Find the equation of the plane that is tangent to the surface f(x, y) = y² - 4x² at the point (3,-1,-35). 8) Find the equation of the plane that is tangent to the surface f(x,y) = at the point (1,2) x+y 9) Let f(x, y) = x²ey. Find the maximum value of a directional derivative at (-2,0), and find the unit vector in the direction in which the maximum value occurs. 10) Determine the critical points of the function f(x, y) = 3x² - 2xy + y² - 8y. Determine whether each point is a local maxima, local minima, or saddle point. Provide proper mathematical justification to support your conclusions. 11) Evaluate the double integral ff xy² dA over the region bound by the x-axis, the y-axis, y = x², and x = 2. Draw a sketch of the region, then set up and evaluate the integral. Π 1 x² 12) Evaluate the triple integral √ √ √ x cos(y) dz dx dy.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
Do question two only
Transcribed Image Text:1) Let u = (2, −1,3), v = (4,0,-2), and w = (1,1,3). Use vector operations to determine each of the following:
(a) 7v+3w
(b) 3(u - 7v)
(c) 2v
(u+w)
(d) ||u+v+w||
(e) ||2v - 3u||
(f) ||2v|| - ||3u||
(g) u v
(h) vxw
2) Let a = (4,6,9) and b = (-6, -3,2). Find the measure of the acute angle between a and b. Round your
answer to four decimal places.
3) Find the standard form of the equation of the plane that passes through the points (5,3,2), (-4, 6, 7), and
(0, -3,8).
4) Let r(t) = (2 sin(t), 3 cos(t), t²) be a vector-valued function. Find the following:
(a) limr(t)
(b) r'(t)
(c) ||r'(T)||
(d) fr(t) dt
5) Let f(x, y, z) = xey + 5z
(a) Find fx, fy, and fz
(b) Find fxx, fyy, and fzz
6) Let f (x, y) = 9 - x² - 7y³.
(a) Evaluate fx (3,1)
(b) Evaluate fy(3,1)
7) Find the equation of the plane that is tangent to the surface f(x, y) = y² - 4x² at the point (3,-1,-35).
8) Find the equation of the plane that is tangent to the surface f(x,y) = at the point (1,2)
x+y
9) Let f(x, y) = x²ey. Find the maximum value of a directional derivative at (-2,0), and find the unit vector in
the direction in which the maximum value occurs.
10) Determine the critical points of the function f(x, y) = 3x² - 2xy + y² - 8y. Determine whether each point is
a local maxima, local minima, or saddle point. Provide proper mathematical justification to support your
conclusions.
11) Evaluate the double integral ff xy² dA over the region bound by the x-axis, the y-axis, y = x², and x = 2.
Draw a sketch of the region, then set up and evaluate the integral.
Π
1 x²
12) Evaluate the triple integral √ √ √ x cos(y) dz dx dy.
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