Calc 3 Test 2

Front 1

Sin^2(x)

Back 1

1-cos(2x)/2

Front 2

cos^2(x)

Back 2

1+cos(2x)/2

Front 3

sec^2(x)

Back 3

1+tan^2(x)

Front 4

csc^2(x)

Back 4

1+cot^2(x)

Front 5

intergral of sin^2

Back 5

1/2(x-sin(x)cos(x))+c

Front 6

Back 6

Tangent plane 𝑎(𝑥−𝑥₀)+b(y-y₀)+c(z-z₀) where (a,b,c) from normal vector of plane.

Front 7

Direction derivative and gradients in 3 space

Back 7

start with z=f(x,y)
rewrite 0=f(xy)-z
define function g(x,y,z)=f(x,y)-z
surface is a level surface: g(x,y,z)=0

Front 8

3d graduate

Back 8

Front 9

What is a
directional derivative?

Back 9

Find partial derivatives to calculate slope in any direction result is directional derivative.

Front 10

Directional derivative formula

Back 10

(𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑥)𝑖+(𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑦)𝑗

Front 11

What is the gradient vector

Back 11

direction of the greatest rate of growth on the surface

Front 12

Tangent plane formula

Back 12

a(x-x°)+b(y-y°)+c(z-z°)=0

Front 13

In tangent plane formula where does (a,b,c) come from

Back 13

normal vector of plane

Front 14

A, B, and C
Z =x^2+y^2
At point (-1,1,2)

Back 14

a=-2,
b=2,
c=-1

Front 15

Equation of Tangent plane
Z =x^2+y^2
At point (-1,1,2)

Back 15

• f=x^2+y^2-z
f=partial derivatives of x,yz place in vector form
(2x)I+(2y)j-k
plug in you points (-1,1,2)
−2(𝑥+1)+2(𝑦−1)−(𝑧−2)=0