Calculus II Chapter 12 Know-Hows

(12.2) How do you set the conditions of the domain?

{(x,y):condition} ex. x =/= 4 OR y < 16 range is {z:condition}

BONUS ROUND: Derivative of b^x when b is a constant

(b ^u)(ln b)(du/dy)

(12.5) What would be the equation to find something like dz/dt, when x and y are functions of t?

(dz/dx)(dx/dt) + (dz/dy)(dy/dt)

What are the two equations for the following: (12.1) The PLANE passing through the POINT with a nonzero normal VECTOR

1) Fx(a,b,c)(x-a) + Fy(a,b,c)(y-b) + Fz(a,b,c)(z-c) = 0 2) ax + by + cz = d

(12.4) Second Order Derivatives: 1) d^2f / dx^2 = ? 2) d^2f / dy^2 = ? 3) d^2f / dxdy = ? 4) d^2f / dydx = ? Which part is found first?

1). d/dx * df/dx 2) d/dy * df/dy 3) d/dx *df/dy 4) d/dx * df/dx The second part

(12.1) To find the plane parallel to (J and K) passing through point Q?

1. Cross Product of Planes 2. Vector in to PLANE equation 3. Plug in point to solve for d 4. write all together

(12.8) What are the 11 steps to finding the absolute extrema?

1. Find the Partials 2. Set Them Both Equal to Zero 3. Solve X,Y for Critical Points 4. Plug the Points in to OG Eq. 5. Parameterize X = Rcos(O) Y = Rsin(O) 6. Replace the x,y of the OG eq with the paramertizations 7. Find the Derivative of that function 8. Set the Derivative equal to 0 and solve for the angles 9. Once you have the angles, put them into the x,y equations to find the critical points 10. Once you have the crit. points put them into the eq at the beginning 11. Find which value is biggest and which value is smallet.

(12.6) How do you find the unit vector when it is not given?

1. Find the partials 2. Plug in (a,b) 3. Find the magnitude of the vector 4.Divide the partials vector @ a,b by the magnitude

(12.8) How do you find a Critical Point usually? b) What's another way?

1. Find the partials 2. Set them both to zero 3. Solve for points b) 1. Find what x/y equals in Fy/Fx when equaled to zero 2. Replace it in the other partial and solve for y/x 3. Once you have y/x solved place it back into the equation from step 1

(12.8) What are the conditions of a Critical Point?

1. It is an interior point of the domain f 2. Fx(a,b) = Fy(a,b) = 0 3. Fx or Fy (or both) do not exist at (a,b)

(12.1) Find the points @ which the plane and curve intersect?

1. Place the curve in to the Plane EQ and solve for t by setting Plane EQ. to 0 2. Plug the points in to the point curve line

(12.7) how do you get the points on the surface where the tangent plane is horizontal?

1. Set Partials Equal to Zero 2. Solve for points 3. Plug Point into OG EQ in order to get the z in the (x,y,z)

(12.1) How do you find the Eq. of a plane parallel to plane Q passing though P?

1. Take the normal vector 2. Take the point 3. Plug them in to the EQ to find d 4. Put it all together (sans point)

(12.1) How do you find the EQ. of the line where Q and R intersect?

1. Take the normal vectors 2. Find the cross product 3. Solve for x and y in the OG. EQ. by setting z to 0 4. Add the points to the vector

(12.3) What are the three conditions for a function to be continuous?

1. f is defined at (a,b) 2. the limit of the function at the point exists 3. the limit of the function as it goes towards the point must EQUAL the function at the point

(12.7) What are dx and dy similar to?

Change in X and Change in Y

(12.8) What is the Discriminant and what do you do with it?

D(x,y) = fxxfyy - (fxy)^2 you find the local extrema

(12.2) What are domain and range?

Domain is every point the equation EXISTS Range is every answer that could come as a result of the equation.

(12.6) No change is found where?

Dot product is 0

(12.6) What is the formula for the direction derivative of f in the direction of the unit vector?

Duf(a,b) = <fx(a,b),fy(a,b)> (dot) <u1,u2>

(12.1) To find the the EQ of the LINE passing through Point P and NORMAL to plane P , what do you do?

Equation of the line is: r(t) = <x,y,z>(point) + t<a,b,c> (vector) x = x0 + at y = y0 + bt z = z0 + ct

(12.4) How do you find Fx and Fy?

Find the derivatives with respect to the variable

(12.7) How do you find the linear approximation for the following function at the given point, with an estimate?

Find the formula with the first by plugging into this formula: L(x,y) = fx(a,b)(x-a) + fy(a,b)(y-b) + f(a,b) and then find the estimate value given the second.

(12.7) What are the equations of the plane tanged to the explicitly defined surface F(x,y,z) at the point P(a,b,c)

Fx(x-a)+Fy(y-b) + F(a,b) = z

(12.7) What are the equations of the plane tanged to the implicitly defined surface F(x,y,z) at the point P(a,b,c)

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

(12.6) To get a directional derivative from two partial derivatives

Get the dot product of the unit vector and the vector formed from the partials: <fx,fy>

(12.1) What do you do when you are given three points and asked to find the equation of the plane?

Get two vectors by doing the distance things between two points, and then find the cross product. Plug that vector into the eqn. along with point.

(12.3) What does it mean when a limit approaches two different limits?

IT DOES NOT EXIST

(12.7) The difference between implicit and explicit use is?

Implicit is equal to zero therefore + fz(z-c) Explicit is equal to z therefore + f(a,b)

(12.3) What do the graphs look like for ln and e?

Ln: wave (ln 1 = 0) e: down slope (never equals zero)

(12.6) What does the magnitude mean and how do you find the deepest descent?

Magnitude of the GRADIENT VECTOR is the rate of increase in the direction of steepest ASCENT. Descent, you just take the vector in the other direction (make it negative)

(12.2) R^n, what is n?

N is the number of independent variables

(12.3) How do you know if a set is open or closed?

Open: < or > closed: >= or =<

(12.5) Whats the difference between a partial d and a regular d?

Partial is when two variables make up a function Regular is when there is a single variable function

(12.2) How do you find the Family EQ?

Set EQ. Equal to a Constant

(12.2) How do you find level curves?

Set Z to a constant Z0

Where does the plane intersect with coordinate axes/planes?

Set the other two variables to 0 and solve

(12.6) Note: When it concerns the directional derivative, what should you be aware of?

That the unit vector is in fact a unit vector

(12.8) What are the two equations for the volume Q?

V =LHW and S = LW + 2LH + 2WL

(12.6) What is the formula for the gradient and what does it mean?

Vf(a,b) = <fx(a,b),fy(a,b)> (so same as if you took the partials and put them into a vector) The direction of steepest ASCENT

(12.3) What are the four ways of potentially getting the limit to work if the limit at the point DNE?

Way #1: Factor Out! Way #2: Multiply the TOP Function with the same function (with reverse operator) Way #3: Along y = x ( so replace x with y Way #4: Along y=mx

(12.1) When are planes identical?

When the EQUATIONS are scalar multiples.

(12.1) When are planes parallel?

When their normal vectors (coefficients) are parallel (meaning the are scalar multiples)

(12.1) When are planes orthogonal?

When their normal vectors are orthogonal (meaning their dot product = 0)

BONUS ROUND: What's the difference between: a) e^x and e^-x b) -e^x and e^x c)-ln x and ln x d) ln -x and ln x

a&d) horizontal mirror b&c) vertical mirror

(12.3) Note: Be Careful with Formatting Limits! a) lim x^2 ? b) lim 10x? c) lim x/y ?

a) (lim x)^2 b) 10 lim x c) lim x / lim y **you can find the limit of above and below!

BONUS ROUND: Derivatives - a) arcsin b) arccos c) arctan d) arccsc e) arcsec f) arccot

a) 1 / (1-x^2)^(1/2) b) -(1 / (1-x^2)^(1/2)) c) 1 / 1+x^2 d) -(1 / |x|(x^2-1)^(1/2)) e)1 / |x|(x^2-1)(^1/2) f)-1 / 1+x^2

(12.8) What does it mean when: a) D > 0 and Fxx < 0 b) D > 0 and Fxx > 0 c) D < 0 d) D = 0

a) Local Maxima b) Local Minima c) Saddle Point d) Inconclusive

BONUS ROUND: What are their identities? a) cos(A + B) b) sin(A + B) c) tan(A + B) d) cos(2x) e)sin(2x) f) tan(2x)

a) cos(a)cos(b) + sin(a)sin(b) b) sin(a)cos(b) + cos(a)sin(b) c) (tan(a) + tan(b)) / (1-tan(a)tan(b)) d) cos^2(x) - sin^2(x) e) 2 sin(x) cos(x) f) 2tan(x)/ (1-tanx)

BONUS ROUND: Derivatives - a)sinx = ? b)cosx =? c)tanx = ? d)cscx = ? e)secx = ? f)cotx =?

a) cosx b) -sinx c) sec^2x d)-cscxcotx e)secxtanx f)-csc^2x

(12.4) a) fx(x,y) = ? b) fy(x,y) = ?

a) df/dx (curly) @ point x,y b) df/dy (curly) @ point x,y

BONUS ROUND: a)e^(x+y) b)e^x(y)

a) e^x * e^y b)e^9x*y)

(12.8) a) Absolute Minimum? b) Absolute Maximum?

a) f(x,y) > f(a,b) b) f(x,y) < f(a,b) for all (x,y)

(12.5) Whats the difference between a) EXPLICIT? b) IMPLICIT? c) BYPASSING IMPLICIT? When finding dy/dx

a) find the derivative of the thing like normal b) find the derivative by taking the derivative of both sides (remember dy/dx) and solve for dy/dx c) - Fx/ Fy (using partials)

(12.4) Treat what variable as a constant for: a) fx b) fy

a) y b) x

BONUS ROUND: a) ln x^2 is the same as what? b) ln xy is the same as what?

a)2 ln x b)ln x + ln y

(12.6) slope of the line tangent to the path of steepest descent

dy/dx = Fx/Fy Find integrals: dy/y = dx/x

(12.7) What are differentials?

dz = fxdx + fydy

(12.4) Derivative of the _____ with respect to ______

the function on top the variable on bottom