Methods 2 Notes 1

Question 1

(1D-N3-1.1) Definition of a function

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Question 2

(1D-N4-1.3) How to form a graph from a function 2D

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Question 3

(1D-N4-1.4) How to form a graph from a function 3D

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Question 4

(1D-N5-1.6) A coordinate plane

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Question 5

(1D-N6-1.7) A cross section

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Question 6

(1E-N6-1.8) Find the cross-section of h by the plane z = 2

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Question 7

(1D-N6-1.9) Contour (or level curve) of a function f

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Question 8

(1D-N8-1.11) Level surface

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Question 9

(1E-N8-1.12) (Level surface of x^2 + y^2 + z^2 = c) explain why this is wrong

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Question 10

(1R-N8–) Explain relationship between level surface and graph in different dimensions. How to represent x2 + y2 + z2 = 1 as a graph of a function

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Question 11

(1R-N9-) Polar coordinates. Explain

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Question 12

(1R-N9-) Cylindrical coordinates

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Question 13

(1R-N9-) Spherical coordinates. (i know you don’t know this)

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Question 14

(1D-N11-1.16) Let f : R2 to R. The partial derivative of f with respect to x, y at (x0, y0)

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Question 15

(1D-N12-1.17) Let f : R2 to R. The partial derivative of f with respect to x, y, z at (x0, y0, z0)

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Question 16

(1T-N14-1.19) (Second partial derivates are same no matter what order done in) nope

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Question 17

(1D-N14–) If h : R ! R, then Taylor’s theorem tells us…

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Question 18

(1D-N15–) Equation for the tangent plane at the point (x0,y0,f(x0,y0) is …

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Question 19

(1D-N17-1.20) Let f : R3 to R. The gradient of f

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Question 20

(1D-N17-1.21) Let f : R3 to R. The directional derivative of f, at a point, in the direction of u (vector) is ..

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Question 21

(1E-N18-1.22) Directional derivative of f(x,y,z) = x^2 + y^2 + z^2. At point (1,2,3) in the direction of u (vector)

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Question 22

(1T-N20-1.24) Chain rule for Let f : R^2 to R and y : R to R. For the function w : R to R defined by w(t) := f(x(t),y(t)), we have the following formula for its derivative (with respect to t):

Answer 22

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Question 23

(1E-N20-1.25) Let f(x,y) = xsin(y) and suppose that x and y in turn depend upon t as follows: x(t) = t2 and y(t) = 2t+1.

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Question 24

(1D-N21-) Chain rule more general. Function of three variables f:R^3 to R. With u,v, r. : R^2 to R

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Question 25

(1E-N21-1.26) See if you can compute the derivative of the composite function w(x, y) := f(u(x,y),v(x,y),r(x,y)) using the more general version of the chain rule if f(x,y,z) = cos(x2 +y2)+z, u(x,y)=xcos(y), v(x,y)=xsin(y), and r(x,y)=1.

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Question 26

(1D-N22-1.27) A vector field

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Question 27

(1R-N22-) We define the vector function G : R3 to R3 by G(x, y, z) := yi + xj. Still valid

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Question 28

(1D-N23-) Gradient Vector Function

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Question 29

(1D-N26-) Dot product of differential vector operator with F force. Cross product of same two (curl)

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Question 30

(1R-N27-) Know how to construct the unit normal. Know what its formula is for Z = f(x,y) surface

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Question 31

(1E-N27-) We have f : R2 ! R defined by f(x,y) := root (4- x2 - y2 ) fin nd a formula for the unit normal vector n(x,y).

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Question 32

(1R-N28-) Important link between gradient of f and ….. With an equation ..(29)

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Question 33

(1R-N30-) Important thing to understand with directional derivative of G and tangent to Level surface

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Question 34

(1R-N31-) Geometric properties of the gradient vector field

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Question 35

(1D-N31-) Mean value theorem and Taylors formula scalar case we recall that for f : R to R

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Question 36

(1D-N31-) Mean value theorem and Taylors formula function of several variables. Also be able to derive

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Question 37

(1D-N31-) Taylor’s formula, el regular

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Question 38

(1T-N32-1.35) Taylor’s formula, f:R^3 to R. Also second order approximation in a neighbourhood

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Question 39

(1T-N34-1.36) Minimum point in R^n

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Question 40

(1E-N34-1.37) i) f : (0,1) to R with f(x) = x. ii) f : [1,1) to R with f(x) = 1/x iii) Even if domain is unbounded f may…

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Question 41

(1R-N34-) When solving the minimisation problem two situations can arise:

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Question 42

(1D-N35-1.39) Br(a):={x element Rn : |x-a|

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Question 43

(1D-N35-1.40) A point x element of (D) is called a critical or stationary point if …

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Question 44

(1D-N35-1.41) Suppose that f : (D) to R has a local extremum (maximum or minimum) at an interior point x ̄ element of (D). Then upside down triangle f(x ̄) = 0.

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Question 45

(1R-N36-) We see that if the Hessian satisfies…for all y in some (suciently small) neighbourhood of x then…and

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Question 46

(1R-N36-) To find out if critical point is… we study the …of the …. These are the solutions to the equation… We get the following cases

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Question 47

(1R-N36-) To find local min/ max in (D) we…

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Question 48

(1R-N36-) Find min/ max of x=(x,y), f(x)=x2+(1/3) y3 y2 and (D) :={(x,y) to R2 :x2+y2 less than or equal to 4

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Question 49

(1R-N39-) Putting it all together

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