Calc final

15. Let S be the solid bounded by the planes z=0 and y+z=3 and the cylinder x^2 + y^2 = 2y. 3. Set up integral

(0 to pi)(0 to 2sin(theta))(0 to 3-rsin(theta)) r dzdrdtheta

13b. Using any appropriate coordinates evaluate the integral for f(x,y) = e^(-x^2 -y^2) 2. Set up

(0 to pi/4)(0 to 2) e^-r^2 rdrdtheta = (0 to pi/4) dtheta (0 to 2) e^-r^2 r dr

4. Evaluate lim (x,y) -> (1,0) ((y^3 + 2xy^2)/(sqrt(x^2 + y^2) -x) *(e^(x^2 + y^2)) 1. Try plugging in point

(0/0)(1)= no go

sin^2 is equivalent to

(1- cos(2theta))/2

2b. Find the plane through P, Q, and R, expressed in the form ax + by + cz = d 2. Find (xo, yo, zo)

(xo, yo, zo) is a point in the plane. In this case choose P, (1, 1, 1)

14b. Using appropriate coordinates set up a single triple integral that represents the volume of S 1. Convert to appropriate system first octant under the cone z = sqrt( x^2 + y^2) and inside the sphere x^2 + y^2 + z^2 =1

0 <= p <= 1 0 <= phi <= pi/2 0 <= theta <= pi/2

11a. del f(-2,2) = -4i + 3j 1. Find vector at the point f(-2,2) on the graph

<-4-(-2), 3-2> = <-2, 1> which doesn't equal <-4, 3> False

2b. Find the plane through P, Q, and R, expressed in the form ax + by + cz = d 1. Find <a, b, c>

<a, b, c> is perpendicular to the plane and equal to PQ x PR, which is <6, 3, 2>, from part a

2c. Is the line through (1,2,3) and (2,2,0) parallel to the plane in part 2b? Explain why or why not. 1. Find line

<x1 - xo, y1 - yo, z1 - zo> = <2-1, 2-2, 0-3> = <1,0,-3> r(t) = <1,2,3> + t<1,0,-3>

12a. (2 to 3) (1 to 2) 1/(x + y)^2 dx dy 2. Solve (y+1 to y+2) 1/ u^2 du

= (-1/u)| (y+1 to y+2) = -1/(y+2) + 1/(y+1) Outer (2 to 3) -1/(y+2) + 1/(y+1) dy Seperate = - (2 to 3) 1/(y+2) dy + (2 to 3) 1/(y+1) dy = -(ln(y+2))| (2 to 3) + ln(y+1)| (2 to 3) = -ln(5) + ln(4) + ln(4) - ln(3) =-ln(5) + 2ln(4) - ln(3)

2. Let P= (1,1,1), Q= (0,3,1), and R= (0,1,4) a) Find area of the triangle PQR

A = .5 || v x w|| 1. Find the 2 vectors 2. Cross the vectors 3. | | 4. Plug in

if D>0 and fxx(a,b) < 0, then f(a,b) is

A local maximum

11d. The tangent line to the level (contour) curve at the point (2, 1) has i - j as it's direction vector. 1. Go to that point on the graph and go draw a line in direction of that direction vector then see if it is perpendicular or 1. Draw a line tangent and see what possible direction vectors it has

At (2,1) on the graph is a vector of <1,1> *draw* so <1, -1> *draw* is tangent True

15. Let S be the solid bounded by the planes z=0 and y+z=3 and the cylinder x^2 + y^2 = 2y. 2. Convert to appropriate system

Convert to r, theta, and keep z x^2 + y^2 = 2y —> r^2 = 2rsin(theta) —> r= 2sin(theta) y+z=3 —> z= 3- rsin(theta) 0<= theta <= pi 0 <= r <= 2sin(theta) 0 <= z <= 3 - rsin(theta)

15. Let S be the solid bounded by the planes z=0 and y+z=3 and the cylinder x^2 + y^2 = 2y. 1. Sketch

Cylinder is shifted so it is only in positive y with only an edge hitting 0 y + z = 3 cuts through cylinder z=0 bounds the bottom

7c. Find the directional derivative at the point (0,1) in the direction of pi/3 2. del f * unit vector in that direction

Duf(0,1)= <sin(1), 0> * <1/2, sqrt(3)/2> = sin(1)*.5 + 0*sqrt(3)/2 = sin(1)/2

11. The figure below shows various gradient vectors of a function f. Determine whether each of the following statements is TRUE or FALSE. In each case explain or justify your answer. c) f(2.02, 2) > f(2, 2)

False

12b. (1 to 4) (sqrt(y) to 2) sqrt(y)e^(x^4 - 4x) dx dy 2. Solve (1 to 2)(1 to x^2) sqrt(y)e^(x^4 - 4x) dy dx

Inner Take out constant =e^(x^4 - 4x) (1 to x^2) sqrt(y) dy = e^(x^4 - 4x)(2/3 y^(3/2))| (1 to x^2) Outer (1 to 2) e^(x^4 - 4x) (2/3 x^3 - 2/3) U sub

12a. (2 to 3) (1 to 2) 1/(x + y)^2 dx dy 1. U substitution

Inner u = x+y (y+1 to y+2) 1/ u^2 du

5. Show that lim (x,y) -> (0,1) (e^(xy) -1)/(x + y - 1) does not exist 1. Find what it approaches along x axis

Keep x a constant and set y as 1 (e^x -1)/(x) Plug in x (1-1)/0 => LHop => e^x/1= e^0 =1 f(x,y) —> 1 as (x,y) —> (0,1) along the x axis

14. Let S be a solid that lies in the first octant under the cone z = sqrt( x^2 + y^2) and inside the sphere x^2 + y^2 + z^2 =1 a) Sketch S

Only first octant, then draw the one part of the sphere and the cone going up from the origin

4. Evaluate lim (x,y) -> (1,0) ((y^3 + 2xy^2)/(sqrt(x^2 + y^2) -x) *(e^(x^2 + y^2)) 3. Plug in points

Plug in 1 and 0 e^1 (2)(2) = 4e

2c. Is the line through (1,2,3) and (2,2,0) parallel to the plane in part 2b? Explain why or why not. 2. See if parallel r(t) = <1,2,3> + t<1,0,-3> previous(t) = <1,1,1> + s<1,0,-3>

Put into parametric and divide respective coefficients, if they are not equal then they are not parallel x = 1+t y= 2 z = 3-3t x = 1+6s y=1-3s z= 1+2s 1/6 = 0/-3 = -3/2 not true, so not parallel

1b. Find the cosine of the angle PQR 1. Find a*b

QP * QR = (1* 0) + (2* 3) + (0* 3) = 6

1. Let P, Q, and R be the points at 1 on the x axis, 2 on the y axis, and 3 on the z axis, respectively a) Express QP and QR in terms of i, j, and k

QP = 1i + 2j QR = 2j + 3k

9. Let f(x,y)= x^2 - yx^2 + 2y^2. Find and classify the critical points of f. 1. Find critical points

Set first partial derivatives equal to 0 and solve fx= 2x -2yx 0=2x - 2yx 2x(1-y)=0 x=0 —> y=0 —> (0,0) fy= -x^2 +4y -x^2 +4y=0 ^—> y=1 —> x=2 —> (1,2)

If D = 0, then

The test gives no information, and f could have any of the three options at that point

2a. Find area of the triangle PQR 1. Find the 2 vectors P= (1,1,1), Q= (0,3,1), and R= (0,1,4)

To find vectors, second letter minus first PQ = <0-1, 3-1, 1-1> = <-1, 2, 0> PR = <0-1, 1-1, 4-1> = <-1, 0, 3>

6a. Let f(x,y) = (x^3 - y^3 +8)^(1/3). Find fy(-2, 0) 1. Find fy

Treat x as a constant fy = (1/3)(x^3 -y^3 +8)^(-2/3) (-3y^2)

9. Let f(x,y)= x^2 - yx^2 + 2y^2. Find and classify the critical points of f. 3. Plug in and categorize D= (2-2y)(4) - (-2x)^2

Use fxx and D to categorize D(0,0) = 8 >0 fxx(0,0) = 2 >0 Local min at (0,0) D(2,1) = -16 < 0 Saddle point at (2,1)

2b. Find the plane through P, Q, and R, expressed in the form ax + by + cz = d 3. Plug in and rearrange

a(x - xo) + b(y - yo) + c(z - zo) = 0 6(x -1) + 3(y -1) + 2(z - 1) = 0 6x + 3y + 2z = 11

2. Let P= (1,1,1), Q= (0,3,1), and R= (0,1,4) b) Find the plane through P, Q, and R, expressed in the form ax + by + cz = d

a(x - xo) + b(y - yo) + c(z - zo) = 0 1. Find <a, b, c> 2. Find (xo, yo, zo) 3. Plug in and rearrange

1. Let P, Q, and R be the points at 1 on the x axis, 2 on the y axis, and 3 on the z axis, respectively b) Find the cosine of the angle PQR

cos(theta) = (a * b) / (|a||b|) 1. Find a*b 2. Find |a| and |b| 3. Plug in

1b. Find the cosine of the angle PQR 3. Plug in

cos(theta) = 6 / (sqrt(5)*sqrt(13)) = 6/(sqrt(65))

7a. Find the direction in which f increases most rapidly at the point (0,1). What is the maximum rate of increase? 1. Find del f f(x,y) = x^2 + xsiny

del f = <fx, fy> del f = <2x + siny, xcosy>

7a. Find the direction in which f increases most rapidly at the point (0,1). What is the maximum rate of increase? 2. Plug in numbers del f = <2x + siny, xcosy>

del f(0,1) = <sin(1), 0> This is the direction

13b. Using any appropriate coordinates evaluate the integral for f(x,y) = e^(-x^2 -y^2) 1. Convert to r and theta (0 to 2)(x to sqrt(4 - x^2) f(x,y) dy dx

e^-r^2 0 <= r <= 2 0 <= theta <= pi/4

8. Find the maximum and minimum values of f(x,y) = x^2 +y subject to the constraint: x^2 + y^2 =1 3. Plug points into f and categorize

f(-sqrt(3)/2, -.5) = .25 f(-sqrt(3)/2, .5) = 1.25 f(sqrt(3)/2, -.5) = .25 f(sqrt(3)/2, .5) = 1.25 Maximum value of f is f(+-sqrt(3)/2, .5)= 1.25 Minimum value of f is f(+-sqrt(3)/2, -.5) = .25

10. Let f(x,6) = e^(-x^2 + y^2). Using the method of Lagrange multiplier find the maximum and the minimum values of f on the circle x^2 + y^2 = 5 3. Plug points into f and categorize

f(0, -sqrt(5))= e^5 f(0, sqrt(5))= e^5 f(-sqrt(5), 0)= 1/e^5 f(sqrt(5), 0)= 1/e^5 Maximum value of f is f(0, +- sqrt(5))= e^5 Minimum value of f is f(+-sqrt(5), 0)= 1/e^5

8. Find the maximum and minimum values of f(x,y) = x^2 +y subject to the constraint: x^2 + y^2 =1 1. Organize into set of 3 equations

f(x, y) = x^2 + y g(x, y) = x^2 + y^2 g(x, y)= 1

10. Let f(x,6) = e^(-x^2 + y^2). Using the method of Lagrange multiplier find the maximum and the minimum values of f on the circle x^2 + y^2 = 5 1. Organize into set of 3 equations

f(x,y) = e^(-x^2 + y^2) g(x,y) = x^2 + y^2 g(x,y) = 5

6b. If f(x,y) = ln(x/y), then find x(fx) - y^2(fyy) in simplified form 1. Find fx

fx = 1/x

6a. Let f(x,y) = (x^3 - y^3 +8)^(1/3). Find fy(-2, 0) 2. Plug in fy = (1/3)(x^3 -y^3 +8)^(-2/3) (-3y^2)

fy(-2,0) = (1/3)((-2)^3 -(0)^3 +8)^(-2/3) (-3(0)^2) = (1/3)(0)(0)= 0

6b. If f(x,y) = ln(x/y), then find x(fx) - y^2(fyy) in simplified form 2. Find fyy

fy= -1/y fyy= y^-2= 1/y^2

7c. Find the directional derivative at the point (0,1) in the direction of pi/3 1. Find the point equivalent of pi/3 (and change into unit vector if not already)

pi/3 = (1/2, sqrt(3)/2) Sqrt(.5^2 + (sqrt(3)/2)^2) = 1

12b. (1 to 4) (sqrt(y) to 2) sqrt(y)e^(x^4 - 4x) dx dy 1. Switch order

sqrt(y) = x y= x^2 x=2 y=1 x=1 (1 to 2)(1 to x^2) sqrt(y)e^(x^4 - 4x) dy dx

13b. Using any appropriate coordinates evaluate the integral for f(x,y) = e^(-x^2 -y^2) 3. U sub (0 to pi/4) dtheta (0 to 2) e^-r^2 r dr

u = -r^2 du= -2r -du/2=r CHANGE BOUNDS (pi/4)* (0 to -4) e^u -du/2 Have to swap bounds so multiply by -1; take out constant -1(pi/4)(-.5)*(0 to -4) e^u du =pi/8(1 - e^-4)

12b. (1 to 4) (sqrt(y) to 2) sqrt(y)e^(x^4 - 4x) dx dy 3. U sub (1 to 2) e^(x^4 - 4x) (2/3 x^3 - 2/3)

u= (x^4 - 4x) du = 4x^3 - 4 —> 1/6du= (2/3 x^3 - 2/3) New bounds =(-3 to 8) e^u du/6 Take our constant = 1/6(-3 to 8) e^u du =1/6(e^u)|(-3 to 8) =1/6(e^8 - 1/e^3] =(e^11 -1)/6e^3

7b. Find the directional derivative of f at the point (0,1) in the direction of (1,0) 1. del f * unit vector in that direction del f(0,1) = <sin(1), 0>

u= <1,0> Duf(0,1)= del f * u = <sin(1), 0>*<1,0> = sin(1)*1 + 0*0= sin(1)

6b. If f(x,y) = ln(x/y), then find x(fx) - y^2(fyy) in simplified form 3. Plug in

x(1/x) - y^2(1/y^2) = 1-1 = 0

13. Consider the integral (0 to 2)(x to sqrt(4 - x^2) f(x,y) dy dx a) Sketch the region R in the xy plane over which the integration is being performed

y= sqrt(4- x^2) is a half circle The rest are lines Pick the section bounded by all of them

2a. Find area of the triangle PQR 3. | | PQ x PR = <6, 3, 2>

|PQ x PR| = sqrt(6^2 + 3^2 + 2^2) = sqrt(49) = 7

1b. Find the cosine of the angle PQR 2. Find |a| and |b|

|QP| = sqrt(1^2 + 2^2 + 0^2) = sqrt(5) |QR| = sqrt(0^2 + 2^2 + 3^2) = sqrt(13)

7a. Find the direction in which f increases most rapidly at the point (0,1). What is the maximum rate of increase? 3. | | del f(0,1) = <sin(1), 0>

|del f(0,1)| = sqrt(sin(1)^2 + 0^2) = sin(1) This is the max rate

10. Let f(x,6) = e^(-x^2 + y^2). Using the method of Lagrange multiplier find the maximum and the minimum values of f on the circle x^2 + y^2 = 5

1. Organize into set of 3 equations 2. Set del f(x,y) = lambda del g(x,y) and solve for points 3. Plug points into f and categorize

2a. Find area of the triangle PQR 4.Plug in |PQ x PR| = 7

A = .5 || v x w|| A= .5(7) = 7/2

if D>0 and fxx(a,b) > 0, then f(a,b) is

A local minimum

If D < 0, then f(a,b) is

A saddle point

9. Let f(x,y)= x^2 - yx^2 + 2y^2. Find and classify the critical points of f. 2. Find values for D fx= 2x -2yx, fy= -x^2 +4y

D= fxx*fyy - fxy^2 fxx= 2 -2y fyy= 4 fxy= -2x D= (2-2y)(4) - (-2x)^2

5. Show that lim (x,y) -> (0,1) (e^(xy) -1)/(x + y - 1) does not exist 2. Find what it approaches along y axis

Keep y a constant and set x as 0 ((e^(0)*y)-1)/(y-1) Plug in y (1-1)/(1-1) => LHop => e^y/1= e f(x,y) —> e as (x,y) —> (0,1) along the y axis

4. Evaluate lim (x,y) -> (1,0) ((y^3 + 2xy^2)/(sqrt(x^2 + y^2) -x) *(e^(x^2 + y^2)) 2. Rationalize

Multiply by the denominator but second term with opposite sign (forgot the term) Multiply by (sqrt(x^2 + y^2) + x) / (sqrt(x^2 + y^2) + x) Simplify Ends up as (e^(x^2 + y^2))(y+2x)(x + sqrt(x^2 + y^2))

5. Show that lim (x,y) -> (0,1) (e^(xy) -1)/(x + y - 1) does not exist 3. Compare

Since f has 2 different limits along 2 different lines, the given limit does not exist

14b. Using appropriate coordinates set up a single triple integral that represents the volume of S 3. Solve (0 to pi/2)(0 to pi/2)(0 to 1) p^2 sin(phi) dp dphi dtheta

Split up bc bounds are all numbers (0 to pi/2) dtheta * (0 to pi/2) sin(phi) dphi * (0 to 1) p^2 dp =(pi/2)*(-cos(phi))|(0 to pi/2) * (p^3 /3)| (0 to 1) = (pi/2) (0) (1/3) =0

2a. Find area of the triangle PQR 2. Cross the vectors PQ= <-1, 2, 0>, PR= <-1, 0, 3>

To cross, cover up the column of the letter ur finding and multiply j by -1 |-1 2 0| PQ x PR = |-1 0 3|= (2*3 - 0*0)i - (-1*3 - -1*0)j + (-1*0 - -1*2)k = 6i + 3j + 2k = <6, 3, 2>

11b. fx(2, -1) = -i and fy(2, -1) = j 1. Find the vector at the point on the graph and see what x and y change by

x changing by -1, y changing by 1 True

15. Let S be the solid bounded by the planes z=0 and y+z=3 and the cylinder x^2 + y^2 = 2y. 4. Solve (0 to pi)(0 to 2sin(theta))(0 to 3-rsin(theta)) r dzdrdtheta

(0 to pi)(0 to 2sin(theta))(rz)|(0 to 3-rsin(theta)) drdtheta =(0 to pi)(0 to 2sin(theta)) 3r - r^2 sin(theta) drdtheta Split and take out constants =(0 to pi) 3 (0 to 2sin(theta)) r dr dtheta + (0 to pi)-sin(theta) (0 to 2sin(theta)) r^2 drdtheta = (0 to pi) 3 (r^2/2)| (0 to 2sin(theta)) dtheta + (0 to pi)-sin(theta)(r^3/3)|(0 to 2sin(theta)) dtheta = (0 to pi) 6 sin^2(theta) dtheta - (0 to pi) (8sin^4(theta))/3 dtheta Solve left part using sin^2 equivalency Solve right part by taking out constant then using calculator = 2pi

14b. Using appropriate coordinates set up a single triple integral that represents the volume of S 2. Set up

(0 to pi/2)(0 to pi/2)(0 to 1) p^2 sin(phi) dp dphi dtheta

10. Let f(x,6) = e^(-x^2 + y^2). Using the method of Lagrange multiplier find the maximum and the minimum values of f on the circle x^2 + y^2 = 5 2. Set del f(x,y) = lambda del g(x,y) and solve for points

-2x e^(-x^2 + y^2)= lambda 2x 2y e^(-x^2 + y^2) = lambda 2y x^2 + y^2 = 5 (0, +- sqrt(5)), (+-sqrt(5), 0)

14. Let S be a solid that lies in the first octant under the cone z = sqrt( x^2 + y^2) and inside the sphere x^2 + y^2 + z^2 =1 b) Using appropriate coordinates set up a single triple integral that represents the volume of S

1. Convert to appropriate system 2. Set up 3. Solve

13. Consider the integral (0 to 2)(x to sqrt(4 - x^2) f(x,y) dy dx b) Using any appropriate coordinates evaluate the integral for f(x,y) = e^(-x^2 -y^2)

1. Convert to r and theta 2. Set up 3. U sub

9. Let f(x,y)= x^2 - yx^2 + 2y^2. Find and classify the critical points of f.

1. Find critical points 2.Find values for D 3. Plug in

7. Let f(x,y) = x^2 + xsiny a) Find the direction in which f increases most rapidly at the point (0,1). What is the maximum rate of increase?

1. Find del f 2. Plug in numbers 3. | |

6b. If f(x,y) = ln(x/y), then find x(fx) - y^2(fyy) in simplified form

1. Find fx 2. Find fyy 3. Plug in

6a. Let f(x,y) = (x^3 - y^3 +8)^(1/3). Find fy(-2, 0)

1. Find fy 2. Plug in

2. Let P= (1,1,1), Q= (0,3,1), and R= (0,1,4) c) Is the line through (1,2,3) and (2,2,0) parallel to the plane in part 2b? Explain why or why not.

1. Find line 2. See if parallel 3. Explain why or why not

7. Let f(x,y) = x^2 + xsiny c) Find the directional derivative at the point (0,1) in the direction of pi/3

1. Find the point equivalent of pi/3 2. del f * unit vector in that direction

11. The figure below shows various gradient vectors of a function f. Determine whether each of the following statements is TRUE or FALSE. In each case explain or justify your answer. b) fx(2, -1) = -i and fy(2, -1) = j

1. Find the vector at the point on the graph and see what x and y change by

11. The figure below shows various gradient vectors of a function f. Determine whether each of the following statements is TRUE or FALSE. In each case explain or justify your answer. a) del f(-2,2) = -4i + 3j

1. Find vector at the point f(-2,2) on the graph

5. Show that lim (x,y) -> (0,1) (e^(xy) -1)/(x + y - 1) does not exist

1. Find what it approaches along x axis 2. Find what it approaches along y axis 3. Compare

11. The figure below shows various gradient vectors of a function f. Determine whether each of the following statements is TRUE or FALSE. In each case explain or justify your answer. d) The tangent line to the level (contour) curve at the point (2, 1) has i - j as it's direction vector.

1. Go to that point on the graph and go draw a line in direction of that direction vector then see if it is perpendicular or 1. Draw a line tangent and see what possible direction vectors it has

8. Find the maximum and minimum values of f(x,y) = x^2 +y subject to the constraint: x^2 + y^2 =1

1. Organize into set of 3 equations 2. Set del f(x,y) = lambda del g(x,y) and solve for points 3. Plug points into f and categorize

15. Let S be the solid bounded by the planes z=0 and y+z=3 and the cylinder x^2 + y^2 = 2y. a) sketch b) set up triple integral c) solve

1. Sketch 2. Convert to appropriate system 3. Set up integral 4. Solve

12. Evaluate the following double integrals b) (1 to 4) (sqrt(y) to 2) sqrt(y)e^(x^4 - 4x) dx dy

1. Switch order 2. Solve

4. Evaluate lim (x,y) -> (1,0) ((y^3 + 2xy^2)/(sqrt(x^2 + y^2) -x) *(e^(x^2 + y^2))

1. Try plugging in point 2. Rationalize 3. Plug in points

12. Evaluate the following double integrals a) (2 to 3) (1 to 2) 1/(x + y)^2 dx dy

1. U substitution 2. Solve

7. Let f(x,y) = x^2 + xsiny b) Find the directional derivative of f at the point (0,1) in the direction of (1,0)

1. del f * unit vector in that direction

8. Find the maximum and minimum values of f(x,y) = x^2 +y subject to the constraint: x^2 + y^2 =1 2. Set del f(x,y) = lambda del g(x,y) and solve for points

2x = lambda 2x 1= lambda 2y x^2 + y^2 =1 1= lambda Lambda = lambda 2y 1 = 2y y= +- .5 x^2 + .25 =1 x^2 = 3/4 x= +- sqrt(3)/2