Yaadannoo boqonnaa fi gaaffii shaakalaa

Boqonnaa 2 Herrega kutaa 12: Seensa Kaalkulasii

Wanti fuula kana irra jiru hundi boqonnaa 2 kitaaba Herrega kutaa 12 tiif: wanta boqonnaan haguugu, yaadannoo gabaabaa, gaaffii shaakalaa deebii isaanii waliin, fi viidiyoo barnootaa bakka jirutti.

4
Kaardii: 10
Gaaffii shaakalaa boqonnaa 2: 61–158
Fuula kitaabaa boqonnaa 2

Boqonnaan 2 maal haguuga

1.Introduction to Derivatives
2.Application of Derivative
3.Introduction to Integration

Yaadannoo gabaabaa boqonnaa 2

Kaardiiwwan fuula Herrega kutaa 12 irratti boqonnaa kana jala galmaa'an. Tokko tokkoon bani.

Reetii Jijjiirama Gidduu GaleessaaFoormulaa

\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}

\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}

$\Delta y$: jijjiirama ba'ii $1$
$\Delta x$: jijjiirama galtee $1$
$b$: gatii galtee isa dhumaa $1$
$a$: gatii galtee isa jalqabaa $1$
$f(b)$: gatii faankishinii b irratti $1$
$f(a)$: gatii faankishinii a irratti $1$

Yoom akka itti fayyadamtu

Reetii jijjiirama waliigalaa yookiin dalga sarara siikaantii qaphxiiwwan gidduu [a, b] shallaguuf itti fayyadami.

Of eeggadhu

f(b) - f(a) kan jiru a - b tiif hin hirin. Qindoominni qabxiiwwan lakkooftuu fi herregduu keessaa wal simuu qaba.

Grade 12 Mathematics, fuula 61-158 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Reetii Siikaantii fi Taanjeentii Wal-makaa GochuuDogoggora baratame

Yoom akka itti fayyadamtu

Reetii jijjiiramaa gidduutti argamu fi reetii jijjiiramaa qabxii sirrii tokko irratti argamu adda baasuuf itti fayyadami.

Of eeggadhu

Gaaffiin reetii battalaa qabxii murtaa'e irratti yoo gaafate, foormulaa reetii jijjiirama gidduu galeessaa hin gargaaramin. Reetiin battalaa dhuma h gara zirootti dhiyaatu shallaguu gaafata.

Grade 12 Mathematics, fuula 61-158 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Hiika Ji'oomeetirii Reetii JijjiiramaaGabatee wabiiPoosatiivii · Nagaatiivii · Ziroo · +1

Gatii Dalgaa	Kallattii Giraafii	Amala Faankishinii
Poosatiivii	Bitaarraa gara mirgaatti ol gara gala	Dabalaa
Nagaatiivii	Bitaarraa gara mirgaatti gadi gara gala	Hir'ataa
Ziroo	Sarara dalgee	Dhaabbataa
Kan hin ibsamne	Sarara sirrii ol dhaabbataa	Kan hin ibsamne

Yoom akka itti fayyadamtu

Mallattoon dalgaa yookiin deriveetiivii waa'ee giraafii fi amala faankishinii maal akka agarsiisu beekuuf itti fayyadami.

Grade 12 Mathematics, fuula 61-158 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Reetii Jijjiirama BattalaaFoormulaa

\lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}

\lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}

$x_0$: qabxii shallaggichi irratti raawwatamu $1$
$h$: daballii galtee $1$
$f(x_0)$: gatii faankishinii x_0 irratti $1$
$f(x_0 + h)$: gatii faankishinii x_0 + h irratti $1$

Yoom akka itti fayyadamtu

Reetii jijjiirama sirrii ta'e yookiin dalga sarara taanjeentii qabxii tokko x_0 irratti argachuuf itti fayyadami.

Of eeggadhu

Hisaabicha osoo hin salphisin dura h = 0 bakka hin buusin, kun ziroodhaan hiruuf nama saaxila.

Grade 12 Mathematics, fuula 61-158 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Gaaffii shaakalaa boqonnaa 2

Gaaffii filannoo 10 boqonnaa kana irratti barreeffaman, tokkoon tokkoon isaanii deebii fi sababa qabu. Deebii banuu dura yaali. Gaaffiin kun Ingiliffaa tiin.

Gaaffii 1

Find the rate of change for the pair of points (0, 2) and (2, 2).

A.2
B.0
C.undefined
D.-1

Deebii fi sababa agarsiisi

Deebii: B 0

Sababa

The rate of change is the change in y divided by the change in x. Here the change in y is 2 - 2 = 0, and the change in x is 2 - 0 = 2. So the rate is 0 / 2 = 0. When y does not change, the rate of change is zero.

Gaaffii 2

Which pair of points has a negative rate of change?

A.(2, 3) and (3, 1)
B.(1, 2) and (3, 4)
C.(0, 2) and (2, 2)
D.(2, 0) and (2, 2)

Deebii fi sababa agarsiisi

Deebii: A (2, 3) and (3, 1)

Sababa

For (2, 3) and (3, 1), the change in y is 1 - 3 = -2 and the change in x is 3 - 2 = 1. The rate is -2 / 1 = -2, which is negative. The y-value goes down as x goes up. The other pairs give positive, zero, or undefined rates.

Gaaffii 3

A candle in Bahir Dar has a starting length of 10 cm. Thirty minutes after it is lit, the length is 7 cm. What is the average rate of change of the length of the candle?

A.-0.1 cm per minute
B.0.1 cm per minute
C.-3 cm per minute
D.3 cm per minute

Deebii fi sababa agarsiisi

Deebii: A -0.1 cm per minute

Sababa

The change in length is 7 - 10 = -3 cm. The change in time is 30 minutes. The average rate of change is -3 / 30 = -0.1 cm per minute. The minus sign shows the candle is getting shorter.

Gaaffii 4

Find the average rate of change of f(x) = 3x - 2 on the interval [1, 5].

A.2
B.3
C.4
D.12

Deebii fi sababa agarsiisi

Deebii: B 3

Sababa

First find the function values. f(5) = 3(5) - 2 = 13 and f(1) = 3(1) - 2 = 1. The average rate of change is (f(5) - f(1)) / (5 - 1) = (13 - 1) / 4 = 12 / 4 = 3.

Gaaffii 5

For the function f(x) = x², the gradient of the tangent line at any point P(a, a²) is 2a. What is the gradient of the tangent line at P(4, 16)?

A.4
B.8
C.16
D.2

Deebii fi sababa agarsiisi

Deebii: B 8

Sababa

The point P(4, 16) has a = 4. Using the formula 2a, the gradient is 2(4) = 8. You can also say the gradient is found by doubling the x-coordinate.

Gaaffii 6

Use the chain rule to find f'(1) when f(x) = (2x + 1)³.

A.18
B.27
C.54
D.6

Deebii fi sababa agarsiisi

Deebii: C 54

Sababa

Use the chain rule: the derivative of the outside power gives 3(2x + 1)², and the derivative of the inside (2x + 1) is 2. So f'(x) = 3(2x + 1)² * 2 = 6(2x + 1)². At x = 1, 2x + 1 = 3, so f'(1) = 6(3)² = 6(9) = 54.

Gaaffii 7

Suppose f'(x) < 0 for all x in an interval I. What can you say about f on I?

A.f is increasing on I
B.f is decreasing on I
C.f is constant on I
D.f has a critical number on I

Deebii fi sababa agarsiisi

Deebii: B f is decreasing on I

Sababa

The increasing and decreasing test says that if f'(x) < 0 for all x in the interior of an interval I, then f is decreasing on I. A negative derivative means the graph falls as x moves to the right.

Gaaffii 8

Evaluate the instantaneous rate of change of f(x) = 6x² - 3 at x = 1.

A.6
B.12
C.-12
D.24

Deebii fi sababa agarsiisi

Deebii: B 12

Sababa

Write f(1 + h) - f(1) = [6(1 + h)² - 3] - [6(1)² - 3] = 12h + 6h². Divide by h to get 12 + 6h. As h gets closer and closer to zero, this approaches 12. So the instantaneous rate of change at x = 1 is 12.

Gaaffii 9

Use the product rule to find f'(0) when f(x) = (2x + 1)(x² + x - 5).

A.-9
B.9
C.-10
D.1

Deebii fi sababa agarsiisi

Deebii: A -9

Sababa

The product rule says (uv)' = u'v + uv'. Let u = 2x + 1, so u' = 2, and let v = x² + x - 5, so v' = 2x + 1. Then f'(x) = 2(x² + x - 5) + (2x + 1)(2x + 1). At x = 0, f'(0) = 2(-5) + (1)(1) = -10 + 1 = -9.

Gaaffii 10

The biomass of a bacterial culture in milligrams, as a function of time t in minutes, is f(t) = 71t³ + 63. What is the instantaneous rate of growth of the culture when t = 2 minutes?

A.213 mg/min
B.852 mg/min
C.426 mg/min
D.71 mg/min

Deebii fi sababa agarsiisi

Deebii: B 852 mg/min

Sababa

The derivative of 71t³ + 63 is f'(t) = 213t². At t = 2, f'(2) = 213(2)² = 213(4) = 852. The rate is positive, so it is a rate of growth. The answer is 852 mg/min.

Appiin boqonnaa kana irratti gaaffii 11 qaba; utuu hojjettuu ni sirreessa. Kan asii jiran 10 jalqabaa ti.