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Excalidraw Data
Text Elements
slope
y2-y1
x2-x1
2(0)+1 = 1
2(3)+1 = 7
7-1 ----
-7(.6)^2-6(.6)+10= 3.88
-7(8)^2-6(8)+10= -486
-485 - 3.88
8-.6
-488.88
7.4 66.06
2(9.1)^2-6(9.1)+9 = 120.02 2(5.3)^2-6(5.3)+9 = 33.8 (120.02-33.38)/3.8= 22.8
)
secant line
through two points
x1
msec = deltay/deltax = (f(x2)-f(x1))/(x2-x1) average rate of change
tangent
through one point
x+h
deltay/deltax
instead of using x2 and x1 (f(x+h)-f(x)/(x+h-x) (f(x+h)-f(x))/h ^ difference quotient lim h→0 (f(x+h)-f(x))/h ^ slope of tangent line. instantaneous rate of change.
x,f(x)
(x+h,f(x+h)
f’(x) = mtan = lim h→ 0: (f(x+h)-f(x))/h f’(x) is read as f prime of x
slope of tangent instantaneous rate of change
the y value at x=8 is 10 the slope of f at x=8 is -2 f prime is not average, it is instantaneous
how much quicker it is getting each hour at the current hour
(f(x+h)-f(x))/h f(x) = 5-3x 5-3(6) 5-18 -13
5-3(6+.001) 5-3(6.001) 5-18.003 -13.003 -.003/.001 -3 slope is -3
(f(6+.1)-f(6))/-1 ((5-3(6.1)-(5-3(6))/.1) (-13.3-(-13))/.1 (-13.3 +13)/.1 slope is -3
(f(4)-f(-3))/(4+3) ((-4(4)+4) - ( -12+4) / (7) (-16+4) - (12+4) /7 (-12)-(16)/7 (-28)/7 -4
x1 1 x2 3.6
(f(x2)-f(x1))/(x2-x1) (f(3.6)-f(1))/(3.6-1) ((5(3.6)^2+3(3.6)-9)-(5(1^2)+3(1)-9))/2.6 ((5(12.96)+10.8-9)-(5+3-9))/2.6 ((64.8+10.8-9)-(-1))/2.6 (66.6+1)/2.6 67.6/2.6 26
x1 0 x2 8
.8(0)^2 -.3sqrt(0)+5 5
.8(8)^2-.3sqrt(8)+5 51.2-.848528+5 55.351472 55.35
55.35-5 = 50.35 50.35/8
-9(.2)+6 -1.8+6 4.2
-9(2)+6 -18+6 -12
-12-4.2 -16.2/1.8
4/(7x+8) 4/(7(-.1)+8) 4/(-.7+8) 4/7.3 .55
4/(7x+8) 4/(7(5.5)+8) 4/(38.5+8
.09
.09-.55 -.46/5.6 -.082
340+1.8(16)-.002(16)^2 340+28.8-.002(256) 368.8-.512 368.288
(368.288-340)/16 1.768
4(-10)+2 -40+2 -38
4(1)+2 4+2 6
6+38 44/11 4
-4(-2.1)+9 8.4+9 17.4
-4(6.4)+9 -25.6+9 -16.6
6.4+2.1 8.5
-16.6-17.4/8.5 -34/8.5 4
6(-9.6)^2+3x-9 6(92.16)-28.8-9 552.96-28.8-9 515.16
6(8.7)^2+3(8.7)-9 6(75.69)+26.1-9 454.14+26.1-9 471.24 471.24-515.16
-43.92/18.3 -2.4
(0,40) (4,35) 35-40=5
-5/4
(5,6) (8,12) 12-6=6 6/3
0.9(0)^2-.3sqrt(0)+14 14
0.9(7)^2-.3sqrt(7)+14 .9(49)-.3sqrt(7)+14 .9(49)-.3(2.645751)+14 44.1-0.7937253+14 57.306275
43.306275/7 6.19
(f(x+h)-f(x))/h (f(2+.1)-f(2))/.1 (f(2+.1)+4))/.1 ((2-3(2.1)+4)/.1 ((2-6.3)+4)/.1 (-0.3)/.1 -3
-4(8.1)-1 -32.4-1 -33.4
(-33.4+33)/.1 (-.4)/.1 -4
-16(3)^2+144(3) -16(9)+432 -144+432 288
-16(3.1)^2+144(3.1) -16(9.61)+446.4 -153.76+446.4 292.64
(292.64-288)/.1 4.64/.1 46.4
-.068x+12 -.068(17)+12 -1.156+12 10.844
-.068(17.1)+12 10.8732
(10.8732-10.844)/.1 .292
-.068(17.01)+12 (10.84332-10.844)/.01 -.068
(f(x+h)-10.844)/h
-16(2)^2+48(2) -16(4)+96 -64+96 32
-16(2.1)^2+48(2.1) 30.24 (30.24-32)/.1 -17.6
-16(2.01)^2+48(2.01) 31.8384
(31.8384-32)/.01 -16.16
-16(2.001)^2+48(2.001) 31.983984
(31.983984-32)/.001) -16.016
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