calc 3

Question 1

projection

Answer 1

a = a*b/(magb)^2 *b

Question 2

comp a

Answer 2

a*b/(b magnitude)

Question 3

a*(bxc)

Answer 3

a1 a2 a3
b1 b2 b3
c1 c2 c3

Question 4

midpoint

Answer 4

x1+x2 / 2

Question 5

sphere

Answer 5

(x-a)^2 +(y-b).. =r^2

Question 6

cylinder

Answer 6

x^2+y^2=r^2

Question 7

paraboloid

Answer 7

z=x^2+y^2

Question 8

cone

Answer 8

z^2=x^2+y^2

Question 9

unit vector

Answer 9

v/mag(v)

Question 10

parallelogram area

Answer 10

Area = ||a x b||

Question 11

angle between vectors

Answer 11

cos(theta) = a*b / ||a|| ||b||

Question 12

Triangle area

Answer 12

Area = (1/2)*||a x b||

Question 12

Scalar triple product

Answer 12

a·(b x c) = det[[a1,a2,a3],[b1,b2,b3],[c1,c2,c3]]

Question 13

Tetrahedron volume

Answer 13

V = (1/6)*|a·(b x c)|

Question 14

Parallelepiped volume

Answer 14

V = |a·(b x c)|

Question 15

Line (vector form)

Answer 15

r(t) = r0 + t*v

Question 16

Line (parametric)

Answer 16

x = x0 + at, y = y0 + bt, z = z0 + c*t

Question 17

Plane (point-normal)

Answer 17

A(x-x0) + B(y-y0) + C(z-z0) = 0

Question 18

Plane (standard)

Answer 18

Ax + By + C*z = D

Question 19

Distance point to plane Ax+By+Cz+D=0

Answer 19

d = |Ax0 + By0 + C*z0 + D| / sqrt(A^2 + B^2 + C^2)

Question 20

Angle between planes

Answer 20

cos(theta) = |n1·n2| / (||n1||*||n2||)

Question 20

Angle between line and plane

Answer 20

sin(theta) = |v·n| / (||v||*||n||)

Question 21

Vector derivative

Answer 21

r’(t) = <x’(t), y’(t), z’(t)>

Question 22

Vector integral

Answer 22

∫r(t)dt = <∫x(t)dt, ∫y(t)dt, ∫z(t)dt>

Question 23

Velocity

Answer 23

v(t) = r'(t)

Question 24

Speed

Answer 24

speed = ||v(t)||

Question 25

Acceleration

Answer 25

a(t) = r''(t)

Question 26

Unit tangent

Answer 26

T = r'(t) / ||r'(t)||

Question 27

Tangent line to curve at t0

Answer 27

r = r(t0) + s*r'(t0)

Question 28

Polar conversion

Answer 28

x = r*cos(theta), y = r*sin(theta), r = sqrt(x^2+y^2)

Question 29

Area element (Cartesian)

Answer 29

dA = dx*dy

Question 30

Area element (Polar)

Answer 30

dA = r*dr*dtheta

Question 31

Partial derivative definition (x)

Answer 31

fx(a,b) = lim_{h->0} (f(a+h,b)-f(a,b))/h

Question 32

Partial derivative definition (y)

Answer 32

fy(a,b) = lim_{h->0} (f(a,b+h)-f(a,b))/h

Question 33

Tangent plane to z=f(x,y) at (a,b)

Answer 33

z = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b)

Question 34

Linearization

Answer 34

L(x,y) = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b)

Question 35

Differential approximation

Answer 35

dz ≈ fx*dx + fy*dy

Question 36

Gradient (2D / 3D)

Answer 36

∇f = or ∇f =

Question 37

Directional derivative

Answer 37

D_u f = ∇f · u

Question 38

Maximum directional derivative

Answer 38

max D_u f = ||∇f||

Question 39

Tangent plane to level surface F(x,y,z)=c at (a,b,c)

Answer 39

Fx(a,b,c)(x-a) + Fy(a,b,c)(y-b) + Fz(a,b,c)(z-c) = 0

Question 40

Chain rule (one parameter)

Answer 40

dz/dt = fx*dx/dt + fy*dy/dt

Question 41

Chain rule (two parameters) - with s

Answer 41

∂z/∂s = fx*∂x/∂s + fy*∂y/∂s

Question 42

Implicit partial derivatives (zx)

Answer 42

zx = -Fx/Fz

Question 43

Critical points (2 variables)

Answer 43

fx = 0, fy = 0

Question 44

Second derivative test (2 variables)

Answer 44

D = fxx*fyy - (fxy)^2

Question 45

Double integral (rectangle)

Answer 45

∬_R f dA = ∫_a^b ∫_c^d f(x,y) dy dx

Question 46

Double integral (Type I)

Answer 46

∬_R f dA = ∫_a^b ∫_{g1(x)}^{g2(x)} f(x,y) dy dx

Question 47

Double integral (Type II)

Answer 47

∬_R f dA = ∫_c^d ∫_{h1(y)}^{h2(y)} f(x,y) dx dy

Question 47

Double integral (polar)

Answer 47

∬_R f dA = ∫_α^β ∫_{r=a(θ)}^{b(θ)} f(r cos θ, r sin θ) * r dr dθ

Question 48

Area (double integral)

Answer 48

A = ∬_R 1 dA

Question 49

Volume under surface

Answer 49

V = ∬_R f(x,y) dA

Question 50

Average value over region

Answer 50

f_avg = (1/A) * ∬_R f dA

Question 51

Mass (lamina)

Answer 51

m = ∬_R ρ dA

Question 52

Center of mass (x-bar)

Answer 52

x̄ = (1/m) * ∬_R x ρ dA

Question 53

Moment of inertia about x-axis

Answer 53

Ix = ∬_R y^2 ρ dA

Question 54

Radius of gyration (about origin)

Answer 54

k = sqrt(I0/m)