Calc III Final

Find all local maxima, local minima, and saddle points for f(x,y) = x³ - y³ - 2xy + 6.

(0,0) is a max, and (-2/3, 2/3) is a min.

Find the local maximum and minimum values and saddle points of f(x,y) = x⁴ + y⁴ -4xy +1.

(1,1) is a minimum, (0,0) is a saddle point, and (-1,-1) is a local minimum.

Find the point on the plane 2x+3y+z = 6 that is nearest the origin.

(29/10, -3/2)

Find the point of intersection of the lines x = 2t+1, y = 3t+2, z = 4t+3 and x = s+2, y = 2s+4, z = -4s-1 and then find the plane determined by these lines.

-20(x-1) + 12(y-2) + 1(z-3) = 0

Find the derivative of f(x,y) = 2xy-3y² at (5,5) in the direction of u=<4,3>.

-4

Find the equation of the plane through the points (2,-1,4), (1,0,1), and (3,-1,-2).

-6(x-2) -3(y+1) -1(z-4) = 0

Find the equation of the plane through the point (2,1,3) and parallel to z = x+y+3.

1(x-2) + 1(y-1) -1(z-3) = 0

Evaluate ∫∫∫(E) xzdV where E is the solid tetrahedron with vertices (0,0,0), (0,1,0), (1,1,0), and (0,1,1).

1/120

Evaluate the integral ∫(e to 1) ∫(x to 0) lnxdydx.

1/4 e² + 1/4

Evaluate ∫∫∫(E) √x²+z² dV where E is the region bound by the parabaloid y = x²+z² and the plane y=4.

128π/15

Evaluate the integral ∫(3 to 1) ∫(√x to 1-x) x²ydydx.

152/15

Find the directional derivative of f(x,y) = 1+2x√y at the point (3,4) in the direction of v = <4,-3>.

23/10

Using polar coordinates, find the volume of the solid above the cone z=√x²+y² and below the sphere x²+y²+z² = 1.

2π/3

Find the length of the arc of the circular helix with the vector equation r(t) = <cost, sint, t> from the point (1,0,0) to (1,0,2π).

2π√2

Find the equation of the plane through (0,2,-1) normal to <3, -2, -1>.

3(x-0) - 2(y-2) -1(z+1) = 0

Find the equation of the plane through the point (5,0,-3) with normal vector 4i + 5j + 2k.

4(x-5) + 5(y-0) +2(z+3) = 0

Find the volume of the solid that lies under the plane 3x+2y+z = 12 and above the rectangle R = ((x,y)‖0≤x≤1, -2≤y≤3).

45/2

Find a vector that has the same direction as <-2,4,2> but has length 6.

6<-2/√24, 4/√24, 2/√24>

A wagon is pulled a distance of 100m along a horizontal path by a constant force of 70N. The handle of the wagon is held at an angle above the horizontal. Find the work done by the force.

7000 Nm

Evaluate the integral ∫(-1 to 1)∫(x³ to x+1) 3x + 2y dydx

8/3 + 2 - 6/5 - 2/7

Two unit vectors orthogonal to both v = <1, -1, 2> and w = <3, 2, -3>.

<-1/√107, 9/√107, 5/√107> and <1/√107, -9/√107, -5/√107>

Suppose that you are climbing a hill whose shape is given by the equation z = 1000 - .01x² - .02y² and you are standing at a point with coordinates (60, 100, 764). In which direction should you proceed initially in order to reach the top of the hill fastest?

<-6/5, -4>

Find tow unit vectors orthogonal to both <1, -1, 1> and <0, 4, 4>

<-8/√96, -4/√96, 4/√96> and <8/√96, 4/√96, -4/√96>

Find a unit vector that is orthogonal to both i + j and i + k.

<1/√3, -1/√3, -1/√3>

If v lies in the first quadrant and makes an angle of π/3 with the positive x-axis and |v| = 4, find v in component form.

<2,√12>

Find the directions in which the directional derivative of f(x,y) = x²+sinxy at the point (1,0) has the value 1.

<4/5, 3/5> and <0,1>

For f(x,y) = sin(x⁴y²) find f_x, f_y, f_xx, f_yy, and f_xy.

Check Exam 2, problem 9. I don't feel like typing that shit out.

Compute lim (x,y)→(0,0) (x⁴-y²)/(x⁴+y⁴) or show that it doesn't exist.

Evaluate as x approaches 0 (lim = 1) and along y = x² (lim = 0), which aren't equal, and therefore the limit doesn't exist.

Find the directions in which the function f(x,y,z) = lnxy + lnyz + lnxz increase and decrease most rapidly at (1,1,1) and calculate the derivative in these directions.

It increases most rapidly in the direction of <2,2,2> with a derivative 2√3, and decreases most rapidly in the opposite direction with the same derivative.

Sketch the region in the xy-plane bounded by the graphs of x = y³, x+y = 2, and y = 0. Set up the double integral to compute the region, but do not integrate.

Look at Exam 3. ∫(0 to 1) ∫(y³ to 2-y) dxdy

Find the unit normal vector N(t) to r(t) = <4, sin2t, cos2t>.

N(t) = <0, -sin2t, -cos2t>

Compute lim (x,y)→(0,0) xy/x²+y² or show that it doesn't exist.

Show that the limit approaches different points along the x-axis and along the line y=x, so it cannot exist.

Show that the vectors <1,5,-2>, <3,-1,0>, and <5,9,-4> are coplanar.

Since the dot product of b and a is 0, they must be orthogonal. Therefore, all three vectors must lie on the same plane, since b is orthogonal to all three.

Find the maximum volume of a rectangular box in the first octant with three faces in the coordinate plane and one vertex in the plane.

V = xyz = 4/3

For r(t) = <2sint, 5t, 2cost>, find: a) the unit tangent vector T(t) b) the unit normal vector N(t) c) the length of the curve described by r(t) for -5≤t≤5 d) the curvature K(t).

a) 1/√29<2cost, 5, -2sint> b) <-sint, 0, -cost> c) 10√29 d) 2/29.

For the vectors v = <3, -1, 2> and w = <1, 0, -3> find: a) |v| b) v°w c) the angle θ between v and w

a) √14 b) -3 c) arccos(-3/(√14 x√10)

For the vectors v = <1,-2,4> and w = <2,3,-5> find: a) |v| b) a unit vector in the same direction as v c) the angle between v and w d) two vectors orthogonal to both v and w.

a) √21 b) <1/√21, -2/√21, 4/√21> c) arccos(-24/(√21 x√38)) d) <-2, 13, 7> and <2, -13, -7>

For what values of b are the vectors <-6,b,2> and <b,b²,b> orthogonal.

b = -2,0,2

Evaluate the integral ∫(2 to 1) ∫(x to x³) e^(y/x)dydx.

e/2(4-1/2 e⁴ -1 +1/2e)

For f(x,y) = sin²(x-3y) find f_x, f_y, f_xx, f_yy, and f_xy.

f_x = 2sin(x-3y)cos(x-3y) f_y = -6sin(x-3y)cos(x-3y) f_xx = 2cos²(x-3y)-2sin²(x-3y) f_yy = 18cos²(x-3y)-18sin²(x-3y) f_xy = -6cos²(x-3y)+6sin²(x-3y).

Find the parametric equation for the line through (2,3,0) perpendicular to the vectors <1,2,3> and <3,4,5>.

r = <2-2t, 3+4t, -2t>

A particle starts at the origin with an initial velocity <1,2,1>. Its acceleration is a(t) = <t, 1, t²>. Find its position function.

r(t) = <1/6 t³ + t, 1/2 t² + 2t, 1/12 t⁴ + t>

A particle starts at the origin with initial velocity <1,2,1>. Its acceleration is a(t) = <t,1,t²>. Find its position function.

r(t) = <1/6 t³+t, 1/2 t²+2t, 1/12 t⁴+1>.

Find the vector valued function r(t) = <x(t), y(t)> in R² satisfying r'(t) = -r(t) with initial conditions r(0) = <1,2,>.

r(t) = <e^x, 2e^x>

The position of a particle at time t is r(t) = <t², 5t, t²-16t>. When is the speed a minimum?

t=4

Find the values of x so that the two vectors <-2, x, -1> and <3, x, x,> are orthogonal.

x = -2, 3

Find the parametric equation of the line through the point (1,-1,1) parallel to the line with symmetric equation x+2 = y/2 = (z-3)/5

x = 1+t, y = -1+2t, z = 1+5t

Find the parametric equation for the line through (3, -4, -1) parallel to the vector i + j - k.

x = 3+t, y = -4+t, z = -1-t

Find the parametric equation of the line through the points (6,1,-3) and (2,4,5).

x = 6-4t, y = 1+3t, z = -3+8t

Find the derivative of f(x,y,z) = xyz in the direction of the velocity vector of the position vector r(t) = <cos3t, sin3t, 3t> at t = π/3.

π/√2

Evaluate the integral ∫(0 to π/6)∫(0 to π/2) xcosy - ycosx dydx.

π²/72 - π²/16

Find the area of the parallelogram with vertices A(1,2,3), B(1,3,6), C(3,8,6) and D(3,7,3).

√265

Find the volume of the solid bounded by the plane z = 0 and the parabaloid z = 1-x²-y².

∫(0 to 2π)∫(0 to 1) 1-r² rdrdθ