⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’
Excalidraw Data
Text Elements
f’ > 0 ( increasing)
f’ < 0 ( decreasing )
f” positive ( concave up )
f” negative ( concave down )
2x^3 + 12x^2 + 18x + 6 6x^2 + 24x + 18 12x + 24
6x^2 + 24x + 18 = 0 6(x^2 + 4x + 3) = 0 x^2 + 4x + 3 = 0 (x-1)(x-3) = 0 1 and 3 are critical points
-4 6(-4) + 24(-4) + 18 pos ( i did this part wrong) -2 6(-2) + 24(-2) + 18 negative 0 6(0) + 24(0) + 18 = 18 positive
f increasing (-inf, -3) u (-1, inf) f decreasing (-3, -1)
0 = 12x + 24 -24 = 12x -24/12 = x -2 = x
less than -2 gives - more than -2 gives + concave down less than -2 concave up more than -2
point of inflection -2 max at -3 min at -1
nx^2 +8x + 4
has a min at -4
at minimum, first derivative = 0
g’(-4) = 0
n(2x) + 8 = 0 n(2(-4) + 8 = 0 n(-8) + 8 = 0 n(-8) = -8 n(-8) = -8 n = 1
decreasing
increasing
sharp point at 3
f is concave down.
critical point at -3
decreasing less than -3
increasing more than -3
concave up
f(x) = 3 f’(x) = 0 f’(x) < 0 if x < 1? f(0) = 3, so no.
f(1) = 3 f(x) = x+2 f’(1) = 1, not 0
f(1) = 3 f(x) =
f’(2) = 0 nx^2-2x+4 2nx - 2
2n(-2) -2 = 0 -4n -2 = 0 -4n = 2 n = -2/4 n = -1/2
3x^3 + sx^2 -4x -2
f”(1) = 0
3x^3 + sx^2-4x-2 9x^2 + 2sx - 4 18x + 2s
18(1) + 2s = 0 18 + 2s = 0 2s = -18 s = -9
2x^3-12x^2+18x+9 6x^2 -24x + 18 12x - 24
2x^3-12x^2+18x+9 6x^2 -24x + 18 12x - 24
6x^2-24x+18 = 0 x^2 - 4x + 3 = 0 (x-1)(x-3) = 0 x = 1 and x = 3 critical points.
6(0)^2-24(0)+18 18
1
6(2)^2-24(2)+18 -6
3
6(4)^2-24(4)+18 18
pos → neg ( max at 1 ) then neg → pos ( min at 3 )
increasing -inf, 1 decreasing 1,3 increasing 3, inf
12x - 24 = 0 12x = 24 x = 24/12 x = 2
12(0)-24 -24 concave down -inf, 2
12(4)-24 48-24 24 concave up 2, inf
1/2x^4 - x^2 + 1 2x^3 - 2x 6x^2 - 2
2x^3 - 2x = 0 2x(x^2-1) = 0 critical point at 0 and 1 and -1
-2 2(-2)^3-2(-2) -12
-1
-.5 2(-.5)^3 -2(-.5) .75
0
.5 2(.5)^3-2(.5) -.75
1
2 2(2)^3-2(2) 12
at -1, neg → pos at 0, pos → neg at 1, neg → pos
min at -1 max at 0 min at 1
decreasing -inf, -1 increasing -1, 0 decreasing 0,1 increasing 1, inf
6x^2 - 2 = 0 6x^2 = 2 x^2 = 2/6 x = +-sqrt(1/3)
-1 6(-1)^2 -2 = 4
0 6(0)^2 -2 = -2
1 6(1)^2 -2 = 4
concave up -inf, -sqrt(1/3) concave down -sqrt(1/3), sqrt(1/3) concave up sqrt(1/3),inf
x^4 - 108x + 11 4x^3 - 108 12x^2
4x^3 - 108 4x^3 - 108 = 0 x^3 - 27 = 0 x^3 = 27 x = cbrt(27) x = 3
4(2)^3 - 108 -76
4(4)^3 - 108 148
decreasing -inf, 3 increasing 3, inf
12x^2 = 0 x^2 = 0 x = 0
12(-1)^2 12
12(1)^2 12
concave up everywhere.
.02t(90-t) = 18 t(90-t) = 900 -t^2 + 90t = 900 t^2 - 90t = -900 t^2 - 90t + 2025 = -900 + 2025 (t-45)^2 = 1125 t-45 = +-sqrt(1125) t-45 = +-15sqrt(5) t = 45 +- 15sqrt(5) t = 45 + 15sqrt(5) t = 45 - 15sqrt(5) t = 45 + 15sqrt(5)
.02x(90-x)
f = .02x g = 90-x
f’ = .02 g’ = -1
f’g + fg’
.02(90-x) + .02x(-1) 1.8 -.02x - .02x
1.8 -.04x
1.8 - .04x = 0 -.04x = -1.8 x = 45
.02(45)(90-45) 40.5
(x-2)
f = x^2 g = x-2
f’ = 2x g’ = 1
2(x-2) * 1 2(x-2)
product rule f x-7 g (x-2)^2 f’ 1 g’ 2(x-2)
f’g + fg’ 1(x-2)^2 + (x-7)(2)(x-2) (x-2)^2 + (x-7)(2)(x-2) (x-2)^2 + 2x-14(x-2) (x-2) (x-2 + 2x-14) (x-2)(3x-16) 3x^2 -16x -6x -12 3x^2 -22x + 32
6x-22
(x-2)(3x-16)
x-2 - critical point at 2 3x-16 = 0 3x = 16 x = 16/3 critical point at 16/3
0 3(0)^2-22(0)+32 = 32
2
3 3(3)^2-22(3)+32= -7
16/3
6 3(6)^2-22(6)+32 = 8
increasing -inf, 2 decreasing 2, 16/3 increasing 16/3, inf
Embedded Files
8a98a27e7a3df71d1a2fe28df4c8513899776b20: page=1
d7bd022089bbe76e8395f596f9dbef08282f9095: page=2
047b75b9c355ee84aee8bed128d2e19f123f8265: page=3
61df2d0717be3b4d7f82f1e791bb75662037ca40: page=4
7edf5f3bb7e77f378a0621c431b058ce791f4347: page=5
340ee4234a9199ab5b52e10d66f5c03aad2db29f: page=6
a1bc8471177ef9b23dddfdfaf2a98a45cc286322: Pasted Image 20251023101112_039.png
3b54cd9d84dcf26dc9797ab4140a5f061c043932: Pasted Image 20251023101500_115.png
cb3e6594e9642a6a41acac9c1585904f6ef1f016: Pasted Image 20251027044824_417.png
13ef84fa29b6bf3ce5d9d3d31b5c2f9d59c3deff: Pasted Image 20251027204751_824.png
2cd772a8eb55534118c9d4fb5284cc5a8fecef9c: Pasted Image 20251027205241_485.png
c89a7fde8a4f1e586dab96878ff997ec9fed89a8: Pasted Image 20251027205253_445.png
1e9a98988ba49441a6092e68ac1e6de8ae5f8edf: Pasted Image 20251027205345_914.png
3a9f491552ee143bdc0665eb89c4db1660f6eb0d: Pasted Image 20251027210313_382.png
c9a6d3c5973f45c8a786baad8d54522af57b201b: Pasted Image 20251027210837_551.png
cfd395d2a7f9d953aa424d9e2b71fe7a1c3739c4: Pasted Image 20251027211050_686.png
2b890daf43610892477fe436be4d40567037552a: Pasted Image 20251027211627_437.png
41e5e6fb124d23c7db3cf785698781eea39d8800: Pasted Image 20251027212831_806.png
1140b7417a0e3d64482a008cf450976fec504183: Pasted Image 20251027212844_430.png
b984b82acdbc9f559dd794c86b1672c8964b692d: Pasted Image 20251027214057_903.png
c9c79d4a1ebb6074bc3f6db9feca0c62fb09d3dd: Pasted Image 20251027221002_628.png
61559ac6bb25548d5d40c8446c8f5bc0cfcf0a74: Pasted Image 20251027223152_351.png
0345aafcc2e2d413b298088d40d3e956b929af76: Pasted Image 20251027223452_435.png
ebc0c3276419347ecf80769e9d47373fec276251: Pasted Image 20251027224441_089.png
