MATH1052 Final

what is the formula for an ellipse?

(x²/a²) + (y²/b²) = 1

what is the formula for a x hyperbola?

x²/a² - y²/b² = 1

what is the formula for a y hyperbola?

-x²/a² + y²/b² = 1

what are the asymptotes for a hyperbola?

y = +- bx/a

what are contour diagrams?

when you define the equation as z = f(x,y) and let z = constants

when is f continuous at f = a for one-variable functions?

when lim _{x→ a-} f(x) = lim _{x → a+} f(x) = f(a)

when does a limit exist for multivariable functions?

if f(x,y) → L1 as (x,y) → (a,b) along path C1 and f(x,y) → L2 as (x,y) → (a,b) along path C1, such that L1 =/ L2, limit does not exist

when is f(x,y) continuous at (a,b)?

lim_{(x,y) → (a,b)} f(x,y) = f(a,b)

what is the derivative formula?

f_x(a,b) = lim_h→0 (f(a+h,b) - f(a,b))/h

what is the tangent plane formula for 1 variable?

y - f(a) = f’(a)(x-a)

what is the tangent plane formula for multivariable?

z = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)

what is the formula for linear approximation?

f(x,y) ~- f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)

what is the directional derivative?

a rate of change: f_u(a,b) = ∇f ^. u/||u||

= || ∇f || x || u || cos theta

what is the direction of maximum increase?

∇f

when is a vector in the same direction?

a scalar multiple

when are vectors perpendicular?

dot product = 0

what is the direction of maximum decrease?

-∇f

what is the chain rule for f(x(t), y(t))?

df/dt = f_x dx/dt + f_y dy/dt

what is dy/dx using partial derivatives?

-F_x/F_y

what does the second derivative test tell us?

whether f(x) is a local minimum (f’’x > 0) or local maximum (f’’x < 0)

where do global maximum and minimums of a continuous function occur?

either at local minimum/maximum or boundary of domain

when is there a critical point for f(x,y)?

∇f = (0,0) or ∇f is undefined

when does f have a saddle point at P₀?

f(P₁) > f(P₀) and f(P₂) < f(P₀)

→ if Q(x,y) = +- (ax² - by²) then C.P is a saddle point

what is D?

D = | fxx(a,b) fxy(a,b) fyx(a,b) fyy(a,b)|

what happens when f_xx(a,b) > 0 and D > 0?

(a,b) is local minimum

what happens when f_xx(a,b) < 0 and D > 0?

(a,b) is local maximum

what happens when D < 0?

(a,b) is a saddle point

what is the larange multipler formula?

∇f = λ∇g

what is the process of lagrange multipliers?

1. define f(x,y) and g(x,y)

2. set g(x,y) to 0 and set as eq 1

3. use the formula

4. find x and y using the equations 1, 2 and 3 and λ (using ratios e.g. eq 2/ eq 3)

what is the distance formula?

d = sqr (x² + y² + z²)

what are parametric equations?

x = f(t), y = f(t) etc

how do you parametrise a circle?

x - a = r cos θ, y-b = r sin θ

where θ = wt + phi and if w is > 0: anticlockwise and < 0 is clockwise motion

how do you parametrise an ellipse?

x = a cos θ, y = b sin θ where θ is wt + phi

what is the position vector formula?

r(t) = r₀ + tv, where r₀ is point and v is parallel vector (or r₁ - r₀)

= r₀ + (t-t₀)v/Δt

what is the velocity vector?

dr/dt

what is the uniform circular motion formula?

r(t) = A cos(wt)i + A sin (wt)j

what is the process of determining arc length?

1. find r(t)

2. differentiate r(t)

3. magnitude of r’(t)

4. integrate || r’(t) || between bounds

what is the process of finding work done?

1. parametrise and find r(t)

2. differentiate r(t)

3. substitute into F

4. dot product: find F(t) ^. r’(t)

5. integrate W = integral of F(r(t)) ^. r’(t) between bounds a and b

what is another formula for work done?

|| F || || PQ|| cos θ

what is a conservative field?

if the integral of the dot product of F and dr between B and A is independent of path taken

when is a gradient field conservative?

F = ∇f

how to find work when a gradient field is conservative?

W = f(B) - f(A)

how to find work when a gradient field is non-conservative?

use the 5 step method

how to find g(y) after integrating f_x?

differentiate wrt y and compare to f_y to find g’(y), integrate to find g(y) (DONT FORGET CONSTANT C) and thus f = f_x + g(y)

how to check if F is a conservative field when not given f?

F1_y = F2_x

how to find points for work done in conservative field?

1. use r(t)

2. substitute boundaries into r(t) to get x and y values

3. use those values to do f(b) - f(a)