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slope of tangent = slope of curve = instantaneous rate of change
89
lim h→ 0 (f(x+h)-f(x))/h
-1^2+1 1+1 2
(-1+0.01)^2+1 .9081+1 1.9081
(1.9081-2)/.01 -9.19
(((-1+0.001)^2+1)-2)/.001 -1.9999
((-1+0.01)^2+1)-2)/.01
lim h→ 0
(f(x+h) -f(x))/h ([(x+h)^2+1)]-[x^2+1])/h
(x+h)(x+h) x^2+xh+xh+h^2 x^2+2xh+h^2
(x^2+2xh+h^2+1-x^2-1)/h (2xh+h^2)/h (2xh+h^2)/h 2x+h 2x+0 2
f’(2)=4 the slope of the tangent line is 4 when x is 2
all of these just mean the derivative
x^2-x+6
(f(x+h)-f(x))/h
([(x+h)^2-(x+h)+6]-[x^2-x+6])/h
((x+h)(x-h)-(x+h)+6-x^2+x-6)/h (x^2+2xh+h^2-x-h+6-x^2+x-6 x^2+2xh+h^2-h-x^2 (2xh+h^2-h)/h 2x+h-1 2x+0-1 2x-1
(f(x+h)-f(x))/h
([3(x+h)^2+4]-[3x^2+4])/h 3(x^2+2xh+h^2)+4 (3x^2+6xh+3h^2+4-3x^2-4)/h (6xh+3h^2)/h (6x+3h) 6x+3(0) 6x
f(x) = x^n f’(x)=(n)x^(n-1)
f(x) = x^2-x+6 2x-1
for derivative of x, power is 1 so f(x) = x derivative 1
f(x) = x^n f’(x)=(n)x^(n-1)
f(x) = x^2-x+6 2x-1
for derivative of x, power is 1 so f(x) = x derivative 1
using pattern, derivative is -5
(f(x+h)-f(x))/h (-5(x+h)+5(x))/h (-5x-5h+5x)/h (-5h)/h (-5)
yay i know how to use patterns !!
also, slope of -5x is just -5 of course.
-6 of course
i want to say -3 of course… but am i right? yes i am
-1 is a sharp point no can do
f(x) = x^n f’(x)=(n)x^(n-1)
f(x) = x^2-x+6 2x-1
for derivative of x, power is 1 so f(x) = x derivative 1
2x+48 according to pattern
NO wrong stupid
(f(x+h)-f(x))/h (48(x+h)^2-48(x)^2)/h (48(x^2+2xh+h^2)-48(x)^2)/h (48x^2+96xh+48h^2-48x^2)/h (96xh+48h^2)/h 96x+48h
96x+48h
96(1)+48(.01) 96.48
961+48(.001) 96.048
96
25x
(f(x+h)-f(x))/h (25(x+h)^2-25x^2) (25(x^2+2xh+h^2)-25x^2 (25x^2+50xh+25h^2-25x^2)/h (50xh+25h^2)/h 50x+25h
50x
5(-8)x^4 -40x^4
(9/2)x^2+5x-4
(f(x+h)-f(x))/h (((9/2)(x+h)^2+5(x+h)-4)-((9/2)x^2+5x-4)) (((9/2)(x^2+2xh+h^2)+5x+5h-4)-((9/2)x^2+5x-4)) (4.5x^2+9xh+4.5h^2+5x+5h-4-4.5x^2-5x+4) (9xh+4.5h^2+5h)/h 9x+4.5h+5
9(1)+4.5(0)+5 9+0+5 14
9x+5
you literally just take the h term and throw it away but keep the 9x+5 type shi
(f(x+h)-f(x))/h (17(x+h)^2-17x^2)/h (17x^2+34xh+17h^2-17x^2) (34xh+17h^2)/h 34x+17h
34x
26x+13h
1
2x
3x^2
10x^9
15x^2
3x^4 3(4x) 12x^3
x^4+3 4x^3+0
x^(1/3) (1/3)x^-(2/3) (1/3x^(2/3)) 1/(3cbrt(x^2))
-3/2
y’=(-3/2)(1x^0)
x^-3 -3x^(-4)
2x^2+10x-3
x^(1/2) - 6(x^(1/3))
(6x^2-3x+7)/x 6x-3+7/x 6x-3+7x^-1 6-7x^-2 6-7/(x^2)
(-1/2)x^4 +3x^3 - 2x at the point -1 -3/2 -2x^3+9x^2-2 -2(-1)^3+9(-1)^2-2 -2(-1)+9(1)-2 2+9-2 9
9x+b -3/2 = 9(-1) + b -3/2 = -9 + b 18/2-3/2 = b 15/2 = b
9x+15/2 9x+7.5
position: t^3+60t 3t^2+60
3(0)^2+60 60 3(3)^2+60 3(9)+60 27+60= 87
derivative
marginal cost marginal revenue marginal whatever just means derivative of cost derivative of revenue derivative whatever
revenue = demand * x r = (117-(1/4x)*x r = 117x - (1/4)x^2
117x - (1/4)x^2 117 - 2/4x 117 - 1/2(16) 117-8 109
profit = R(X) - C(X) 117-(x^2)/4 - ( 225 + 2x^2 ) 117-(1/4)x^2 -225 -2x^2 -108-(1/4)x^2-(8/4)x^2 -9/4x^2+117x-225 -9/4(2x) +117 (9/2)x+117
-9/2(x)+117 -9/2(10)+117 72
C(x)/x (0.02x^2+100x+2000)/x 0.02x+100+(2000/x) .02x+100+2000x^-1 .02+2000x^-2 .02+2000(1000)^-2 .02+2000/(1000)^2 .018
(4x^5-5x^4)/x
4x^5/x^3 - 5x^4/x^3 4x
8x-5
-7x^(3/4)+5-8x^(7/4) (3/4)(-7)x^(-1/4)+5+(7/4)(-8)x^(3/4) -(21/4)x^(-1/4)-14x^(3/4)
(6x^2+1)(5x^2+6) 6x^25x^2 + 6x^2 * 6 + 15x^2 + 6*1 30x^4 + 36x^2 + 5x^2 + 6 30x^4 + 41x^2 + 6 4(30x^3)+2(41x) 120x^3+82x
2x-8x^2-9x^3 2-16x-27x^2 -27x^2-16x+2
-27(-3)^2-16(-3)+2 -27(9)+48+2 -243+48+2 -193
2(-3)-8(-3)^2-9(-3)^3 -6-8(9)-9(-27) -6-72+243 165
165=-193(-3)+b 165=579+b -414=b
-193x-414
0.4(6)^3-0.1(6)^.5+24 0.4(216)-.1(6)^.5+24 86.4-.1(6)^.5+24 110.4-.1(6)^.5 110.16
0.4x^3-0.1x^.5+24 3(0.4)x^2-.05x^-.5 1.2x^2-.05x^-.5
1.2(7)^2-.05(7)^-.5 1.2(49)-.05(.37796) 58.8-.018898 58.781102 58.78
0.4x^3-0.1x^.5+24
f(0) = 24
f(4)
0.4(4)^3-0.1(4)^.5+24 0.4(64)-0.1(2)+24 25.6-.2+24 49.4
y2-y1
x2-x1
49.4-24
4-0
25.4/4 6.35
1978-1955 23
.009(23)^2-.5(23)+6.84 .009(529)-11.5+6.84 .009(529)-4.66 4.761-4.66 .101
.009x^2-.5x+6.84 2(.009)x-.5 .018x-.5
.018*15-.5 -.23
4(7)x^3-16x 28x^3-16x
24x^2+18x
8x^3+12x^2
8x^7/(x^3) + 2x^4/(x^3) 8x^4+2x
32x^3+2
1.7x^2+20x 2(1.7)x+20 3.4x+20
3.4(8)+20 27.2+20 47.2
-5x^(3/2)-5x+2 (3/2)(-5)x^(1/2)-5 -(15/2)x^(1/2)-5
(4x+4)(5x+5) 4x5x + 4x5 + 45x + 45 20x^2 + 20x + 20x + 20 20x^2 + 40x + 20
40x+40
-4x^3-3x+6x^2 3(-4)x^2-3+12x -12x^2+12x-3
-12(-2)^2+12(-2)-3 -12(4)-24-3 -48-24-3 -75
y=-75x+b 62=-75(-2)+b 62=150+b -88=b
-75x-88
-4(-2)^3-3(-2)+6(-2)^2 -4(-8)+6+24 -4(-8)+30 32+30 62
.006x^2-.35x+6.54
1999-1958 41
.006(41)^2-.35(41)+6.54 .006(1681)-14.35+6.54 10.086-14.35+6.54 2.276
200+17x-.5x^2+.4x^3 17-x+1.2x
1.2x^2-x+17
130+6x-.8x^2 6-1.6x -1.6x+6
(39.7-2.6x)x 39.7x-2.6x^2 -2.6x^2+39.7x
(670+1.6x)/x
670/x+1.6
(2000+28x+.4x^2)/x 2000/x+28+.4x 2000x^-1+28+.4x -2000x^-2+.4
3.3x+30
profit = revenue - cost
(140x-.1x^2) - (2210 +30x + .1x^2) 140x-.1x^2 - 2210 -30x - .1x^2 110x -.2x^2 -2210 -.2x^2 + 110x -2210
-.4x + 110
-.4(60)+110 -24+110 86
profit = revenue - cost revenue = price per unit * number of units sold
( 3.00 - .006x ) * x 3x - .006x
3x - .006x^2 - .007x^2 - .62x - 300 3x -.013x^2 -.62x -300 2.38x -.013x^2 -300
-.026x + 2.38
(6.2-.0007x)x 6.2x-.0007x^2-.0006x^2-3x-4850 3.2x-.0013x
-.0026x+3.2
700+16x-.8x^2+.9x^3 16-1.6x+2.7x^2 2.7x^2-1.6x+16
100+7x-.7x^2 -1.4x+7
2600/x +36 + .1x
2600x^-1 + 36 + .1x
-2600x^-2 + .1
46+2.25x
6.2x - .0007x^2 - .0006x^2 - 3x - 4850 3.2x -.0013x^2 -4850
3.2-.0026x
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