Midterm 2

Question 1

∫sin^2(x)

Answer 1

(1/2)x - (1/4)sin(2x)

Question 2

∫cos^2(x)

Answer 2

(1/2)x + (1/4)sin(2x)

Question 3

∫sin

Answer 3

-cos

Question 4

∫cos

Answer 4

sin

Question 5

∫tan

Answer 5

ln|sec|

Question 6

Jacobian of spherical coordinates

Answer 6

p^2 * sin(phi)

Question 7

Jacobian of cylindrical coordinates

Answer 7

r

Question 8

Implicit differentiation

Answer 8

Set the original function equal to zero (if the function is of the form f(x,y,z) = d
dz/dx = -Fx/Fz
The derivative with respect to x divided by the derivative with respect to z
dz/dy = -Fy/Fz
The derivative with respect to x divided by the derivative with respect to z

Question 9

Implicit differentiation practice problem
x^3 + y^3 - 6xy = 0
dy/dx = ?

Answer 9

-Fx/Fy
-(3x^2 - 6y)/(3y^2 - 6x)

Question 10

Chain rule
If you have to find the derivative of a function with respect to multiple variables (something like du/ds when you have equations for x, y, z, relative to r, s, t) what do you do?

Answer 10

Follow this tree

……u
../ | \
.x y z
/|\ /|\ /|\
rst rst rst

aka follow the path first to x y and z then to the bottom part (ds)
so if I followed to find the x component, I would find the u function derivative with respect to x, then the u function derivative with respect to s, then I would multiply those together. I would then do the same for y and z, then sum them together for du/ds

AFTER THE SUM YOU NEED TO SUBSTITUTE THE VARIABLES SO THAT DU/DS IS NOT RELATIVE TO X Y Z BUT RST

Question 11

How to read du/dx

Answer 11

Derivative of the u function with respect to x

Question 12

Directional Derivatives given a differentiable function with respect to x and y and u = ⟨a,b⟩

Answer 12

Fx(x,y)a + Fy(x,y)b
(u IS THE UNIT VECTOR)

Question 13

Directional derivatives with respect to an angle (from the direction of the gradient vector) instead of u = ⟨a,b⟩

Answer 13

Fx(x,y)cos(theta) + Fy(x,y)sin(theta)

Question 14

What is a gradient vector?

Answer 14

⟨Fx,Fy⟩

Question 15

How can we rewrite the directional derivative with respect to the gradient vector? (f(x,y) and u = ⟨a,b⟩)

Answer 15

▽F * u
⟨Fx,Fy⟩ * u
(u IS THE UNIT VECTOR)

Question 16

The maximum value of a directional derivative is…

Answer 16

|▽F|
and it occurs when u is in the same direction at the gradient vector ▽F

Question 17

Equations for normal lines and tangent planes

Answer 17

to find the equation for the tangent plane, you simply find Fx Fy and Fz
Then you plug in the point you are given to give you a, b, and c. The tangent plane is defined by a(x - x1) + b(y - y1) + c(z - z1) = 0

The symmetric equations of the normal line are
(x - x1)/a = (y - y1)/b = (z - z1)/c

Question 18

Second Derivatives test for mins and maxes (formula)

Answer 18

D = Fxx(a,b)Fyy(a,b) - [Fxy(a,b)]^2

Question 19

What do the values for the D test mean

Answer 19

D > 0 and Fxx > 0: Local min
D > 0 and Fxx < 0: Local max
D < 0: Saddle point
D = 0: We know nothing

Question 20

How to set up LaGrange multipliers

Answer 20

▽F(x,y) = λ▽G(x,y)
So Fx = λGx, Fy = λGy and then we have the constraint function G(x,y) = C

Question 21

How to use LaGrange multipliers to solve for the closest and/or furthest points from another object

Answer 21

Find the distance equation then square it to find F(x,y,z)
G(x,y,z) is the equation of the plane
The constraint is G(x)
Fx = λGx
Fy = λGy
Fz = λGz
G(x,y,z) = c
Use LaGrange Multipliers

Question 22

Implicit Function Theorem requirements

Answer 22

F is defined on a disc containing a, b
Fy =/= 0
Fx and Fy are continuous on the disc

Question 23

What maximizes the gradient vector?

Answer 23

When u is in the same direction as the gradient vector, therefore theta is zero making cos(theta) = 1

I guess just plug in the point at which you want to maximize the rate of change and that gives you the vector, then the magnitude of that vector is the rate of change

Question 24

How to find local maxes and minimums of a 2d function

Answer 24

Find in the range where Fx(a,b)=0 and Fy(a,b) = 0

Question 25

How to find the absolute maximum and minimum values on a domain

Answer 25

First check local maxes and mins by using Fx and Fy = 0
Then you have to check the edges, this is done by fixing x or y, then you can check those lines by checking the derivative = 0 again
Then you check all the endpoints/corners

Question 26

How to solve LaGrange multipliers

Answer 26

Solve for zero with the different equations then get points for the different circumstances that are possible (if (x-1)(lambda +1) = 0, then you need to check for the points such that x = 1 or lambda = -1)

Question 27

Fubini’s theorem

Answer 27

You can switch bounds around
if x and y are bounded by constants then
bd
∫..∫f(x,y)dydx is the area!!!111!1
ac

Question 28

What is a Type 1 integral?

Answer 28

Where y depends on x, but x has hard boundries

Question 29

What is a Type 2 integral?

Answer 29

Where x depends on y, but y has hard boundries

Question 30

Polar Coordinates and cylindrical cordinates

Answer 30

r^2 = x^2 + y^2
x = rcos(theta)
y = rsin(theta)
Jacobian: r
tan(theta): y/x
(cylindrical)
z = z

Question 31

density and centers of mass

Answer 31

To find mass
∫∫p(x,y)dA
To find x-bar
1/m * ∫∫xp(x,y)dA
To find y-bar
1/m * ∫∫yp(x,y)dA

Question 32

What must be true in an integral that has bounds that are dependent on other variables

Answer 32

The first integral must have bounds in terms of numbers

The second integral can only be in terms of ONE variable, and the dx/dy/dz cannot have the same variable

The third integral can have up to TWO variables and the dx/dy/dz variable must be the one NOT included in the bounds

Question 33

Spherical
Coordinates

Answer 33

x = psin(φ)cos(θ)
y = psin(φ)sin(θ)
z = pcos(φ)
Jacobian: p^2 sin(φ)
p^2 = x^2 + y^2 + z^2
0 <= φ <= pi
0 <= θ <= 2pi
p >= 0

Question 34

How to set up LaGrange multipliers with three functions (f, g, and h)

Answer 34

▽F(x,y,z) = λ▽G(x,y,z)+μ▽H(x,y,z)
G(x,y,z) = c
H(x,y,z) = k

Question 35

In spherical coordinates, r = ?

Answer 35

r = psin(φ)

Question 36

When before solving an equation do you need to move z to the other side of f(x,y)?

Answer 36

In solving for the tangent plane when the initial equation is given in f(x,y)
You also need to solve for z using the x and y given to you

Question 37

How is the Jacobian matrix set up

Answer 37

With xyz variables consistent horizontally and uvw variables consistent vertically

x x x
y y y
z z z

u v w
u v w
u v w

Absolute value of that determinant

Question 38

What is a polar rectangle

Answer 38

a rectangle where r stays constant, like the shape of a rainbow
not dependent on θ

Question 39

U substitution in an iterated integral steps

Answer 39

Find a new u = a formula containing x and y so that the integral is much simpler
differentiate the u formula in terms of the dx or dy that is on the outside of the integral
rewrite the inside in terms of u with the other parts that come from the above steps

Now you have to rewrite the bounds

rewrite the bounds by plugging in the original bounds into the u equation
Example: u = x^2 + y^2 and original bound were y = x to y = 2x
You get u = x^2 + x^2 and u = x^2 + (2x)^2
Then you plug these bounds back in for the full substitution

then SOLVE!!!!!

Question 40

integral of 1/x

Answer 40

ln(abs(x))