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it has been determined that the cost of producing x units of a certain item is 10x + 375. the demand function is given by p = D(x) = 35 - .45x
find revenue find profit
revenue is demand ( price ) * x (36-.45x)(x) 36x - .45x
profit is all money coming in - all money were spending. revenue - cost. find profit 36x - .45x^2 -(10x+375) 36x - .45x^2 -10x -375) -.45x^2 + 26x -375
lim f(x)
x → 3
limits are saying what y value is x approaching f(3) - what is the y value when x is 3
what y value is the function approaching as x gets closer to a
limit is 8
left sided / left handed limit denoted by: lim x→3^-
right sided limit denoted by: lim x→3^+
-5
-1
4
2
4
0
just means plug in the values. e.g. approaching 3 plug in 2.9 2.999 3.01 3.001
3^2+3(3)-7 9+9-7 18-7 11
2^3-5(2) 8-10 -2
4(0)^2-8(0)+3 3
+inf
3
3
3
( -3.1 + 5 ) / ( -3.1 + 3 ) = +/ - = -inf
posinf
get x→3^- and x→3^+
if they are the same, then the limit exists
3
does not exist
if we can plug in, plug in if we plug in and get a zero in our denominator, we must do more then manipulate until we can plug in (factor, rationalize w sqrt simplify, etc ) if we have vertical asymptote, + or - inf
-6
4(7) - 3 28 - 3 = 25
(2(2)^2+3(2)-1)^2
2^2-4
2-2
0
0
(x-2)(x+2)
(x-2)
(x+2) 2+2 4
factor x^2-4
2x^2+5x-3
x+3
2(-3)^2+5(-3)-3
6
2(9)-15-3
18-15-3 0
2x^2+5x-3
a
b
c
-6
5
6, -1
5
2x^2+6x-x-3 2x(x+3) -(x+3) (x+3)(2x-1)
((-3+3)(2(-3)-1))/(-3+3)
-7
1/sqrt(3) must simplify just multiply by sqrt(3)/sqrt(3)
(sqrt(x)+3)/(sqrt(x)+3)
((sqrt(x)^2+3sqrt(x)-3sqrt(x)-9)
(x-9)(sqrt(x)+3)
x-9
(x-9)(sqrt(x)+3)
1
sqrt(x+3)
1/(sqrt(9)+3) 1/6
6x+6 { x< -6 } -7x^2-8 { x>=-6 }
lim f(x) x→-6
6(-6)+6 -36+6 -30
-7(-6)^2-8 -7(36)-8 -252-8 -260 does not exist
lim(x^2+4x-1) x→2 2^2+4(2)-1 4+8-1
-6x^2-22x-20 a b c
-22x
-2x, 3x
-2x, 11
-6x^2
`
-2x(3x+11)-20
find 2 numbers that multiply to 6 * 20 = 120
and add up to 22
factors of 120 are: 10 12
10 + 12 is 22
-(6x^2+22x+20)
-(6x^2+10x+12x+20) -(2x(3x+5)+4(3x+5) -((2x+4)(3x+5)) -2(x+2)(3x+5)
-2(x+2)(3x+5) / x+2
-2(3x+5) -2(3(-2)+5) -2(-6+5) -2(-1) 2
(x-5)^2 (x-5)(x+5)
1/(x+5) 1/10
f(-2) = -2(-2)^2-6(-2)-3 -2(4)+12-3 -8+12-3 4-3 1
f(-2+h) = -2(-2+h)^2-6(-2+h)-3 -2(-2+h)(-2+h)+12-6h-3 (4-2h)(-2+h)+9-6h -8+4h+4h+2h^2+9-6h -8+8h+2h^2-6h+9 2h^2+2h+1
2h^2+2h+1-1 2h^2+2h
(2h^2+2h)/h
2h+2
(x-5)/(x-5)^2 (x-5)/(x-5)(x+5) 1/(x+5)
DNE
-3x^2+23x+70
-3*70 -210
2 numbers that add to 23 but multiply to -210
10 and 21 -7 and 30 30*-7 = -210
-3x^2+30x-7x+70 -3x(x-10) -7(x-10) (-3x-7)(x-10)
-3x-7
-3(10)-7 -30-7 -37
x^2+2x-35 -35 but add to +2… -5*7
x^2-5x+7x-35 x^2-5x+7(x-5) x(x^2-5)(x-5)
dont have to deal w all that shit. can just say its (x-5)(x+7).
10x^2+11x-6 10*-6 = -60 11 multiply to -60
factors of 60: 15, 4 15, -4 !
Sooooo
(x+15)(x-4)? or am i tripping
NOPE im tripping only do that if it is leading coeff of 1, so like
z^2+2z−35
instead factor by grouping
10x^2+15x-4x-6
5x(2x+3)-2(2x+3)
(5x-2)(2x+3)
revenue = price * quantity profit = revenue - cost
revenue = -0.3x^2+41x cost = 8x+670
-0.3x^2+41x-8x-670 -0.3x^2+33x-670
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