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미적분 Unit 2 Test Review 2.1-3.1

Authored by Rael Clarke

Mathematics

12th Grade

CCSS covered

Used 2+ times

미적분 Unit 2 Test Review 2.1-3.1
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10 questions

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1.

MULTIPLE CHOICE QUESTION

1 min • 1 pt

거듭제곱 법칙을 사용하여 다음 도함수를 결정합니다:

Use the Power Rule to determine the following derivative: f(x)=3x6−x3f\left(x\right)=3x^6-x^3

f′(x)=18x5−3x2f'(x)=18x^5-3x^2

f′(x)=18x6−3x2f'(x)=18x^6-3x^2

f′(x)=18x4−3xf'(x)=18x^4-3x

f′(x)=6x5−3x3f'(x)=6x^5-3x^3

2.

MULTIPLE CHOICE QUESTION

1 min • 1 pt

체인 규칙을 사용하여 다음 함수의 도함수를 찾습니다:

Use the Chain Rule to find the derivative of the following function: f(x)=sin⁡(sin⁡(x))f\left(x\right)=\sin\left(\sin\left(x\right)\right)

f′(x)=cos⁡(x)⋅sin⁡(sin⁡(x))f'(x)=\cos(x)\cdot\sin(\sin(x))

f′(x)=−cos⁡(sin⁡(x))⋅sin⁡(x)f'(x)=-\cos(\sin(x))\cdot\sin(x)

f′(x)=cos⁡(sin⁡(x))⋅cos⁡(x)f'\left(x\right)=\cos\left(\sin\left(x\right)\right)\cdot\cos\left(x\right)

f′(x)=sin⁡(sin⁡(x))⋅sin⁡(x)f'(x)=\sin(\sin(x))\cdot\sin(x)

3.

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

곱셈 규칙을 사용하여 다음 함수의 미분을 구합니다:

Use the Product Rule to find the derivative of the following function: f(x)=ex(2x−5)f\left(x\right)=e^x\left(2x-5\right)

f′(x)=ex(2x−5)f'(x)=e^x(2x-5)

f′(x)=ex(2x−2)f'(x)=e^x(2x-2)

f′(x)=ex(2x−4)f'(x)=e^x(2x-4)

f′(x)=ex(2x−3)f'(x)=e^x(2x-3)

Answer explanation

f(x)=ex(2x−5)f\left(x\right)=e^x\left(2x-5\right) . u =ex=e^x , v =(2x−5)=\left(2x-5\right)

u' =ex=e^x , v' =2=2 . f' = u'v + v'u

ex(2x−5)+2exe^x\left(2x-5\right)+2e^x . Distribute and Re-Factor:

2xex−5ex+2ex=ex(2x−5+2)=ex(2x−3)2xe^x-5e^x+2e^x=e^x\left(2x-5+2\right)=e^x\left(2x-3\right)

4.

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

몫의 법칙을 사용하여 다음 함수의 도함수를 구합니다:

Use the quotient rule to find the derivative of the following function: f(x)=sin⁡(x)3 xf\left(x\right)=\frac{\sin\left(x\right)}{3\sqrt{\ x}}

2xcos⁡(x)−sin⁡(x)6x32\frac{2x\cos\left(x\right)-\sin\left(x\right)}{6x^{\frac{3}{2}}}

2xcos⁡(x)−sin⁡(x)9 x\frac{2x\cos\left(x\right)-\sin\left(x\right)}{9\sqrt{\ x}}

3cos⁡(x) x9x−3sin⁡(x)2 x\frac{3\cos\left(x\right)\sqrt{\ x}}{9x}-\frac{3\sin\left(x\right)}{2\sqrt{\ x}}

3cos⁡(x) x+3sin⁡(x)2 x3\cos\left(x\right)\sqrt{\ x}+\frac{3\sin\left(x\right)}{2\sqrt{\ x}}

Answer explanation

Media Image

f(x)=sin⁡(x)3 xf\left(x\right)=\frac{\sin\left(x\right)}{3\sqrt{\ x}} . u =sin⁡(x)=\sin\left(x\right) . v =3 x=3\sqrt{\ x}

u' =cos⁡(x)=\cos\left(x\right) , v' =32 x=\frac{3}{2\sqrt{\ x}} .

Quotient Rule: u′v−v′uv2\frac{u'v-v'u}{v^2}

3cos⁡(x) x−3sin⁡(x)2 x9x=6xcos⁡(x)−3sin⁡(x)2 x9x=2xcos⁡(x)−sin⁡(x)6x32\frac{3\cos\left(x\right)\sqrt{\ x}-\frac{3\sin\left(x\right)}{2\sqrt{\ x}}}{9x}=\frac{\frac{6x\cos\left(x\right)-3\sin\left(x\right)}{2\sqrt{\ x}}}{9x}=\frac{2x\cos\left(x\right)-\sin\left(x\right)}{6x^{\frac{3}{2}}}

Tags

CCSS.HSA.APR.D.7

5.

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

곱셈 규칙을 사용하여 다음 함수의 미분을 구합니다:

Use the Product Rule to find the derivative of the following function: f(x)=(3 x)sin⁡(x)f\left(x\right)=\left(3\sqrt{\ x}\right)\sin\left(x\right)

2xcos⁡(x)−sin⁡(x)6x32\frac{2x\cos\left(x\right)-\sin\left(x\right)}{6x^{\frac{3}{2}}}

2xcos⁡(x)−sin⁡(x)9 x\frac{2x\cos\left(x\right)-\sin\left(x\right)}{9\sqrt{\ x}}

3cos⁡(x) x9x−3sin⁡(x)2 x\frac{3\cos\left(x\right)\sqrt{\ x}}{9x}-\frac{3\sin\left(x\right)}{2\sqrt{\ x}}

3cos⁡(x) x+3sin⁡(x)2 x3\cos\left(x\right)\sqrt{\ x}+\frac{3\sin\left(x\right)}{2\sqrt{\ x}}

Answer explanation

Media Image

f(x)=(3 x)sin⁡(x)f\left(x\right)=\left(3\sqrt{\ x}\right)\sin\left(x\right) . u =sin⁡(x)=\sin\left(x\right) . v =3 x=3\sqrt{\ x}

u' =cos⁡(x)=\cos\left(x\right) , v' =32 x=\frac{3}{2\sqrt{\ x}} .

Product Rule: u′v+v′uu'v+v'u

3cos⁡(x) x+3sin⁡(x)2 x3\cos\left(x\right)\sqrt{\ x}+\frac{3\sin\left(x\right)}{2\sqrt{\ x}}

6.

MULTIPLE CHOICE QUESTION

1 min • 1 pt

체인 규칙을 사용하여 다음 함수의 도함수를 찾습니다:

Use the Chain Rule to find the derivative of the following function: f(x)=cos⁡(3x2)f\left(x\right)=\cos\left(3x^2\right)

f′(x)=−6xsin⁡(3x2)f'(x)=-6x\sin(3x^2)

f′(x)=6xcos⁡(3x2)f'(x)=6x\cos(3x^2)

f′(x)=−3xsin⁡(3x2)f'(x)=-3x\sin(3x^2)

f′(x)=3xcos⁡(3x2)f'(x)=3x\cos(3x^2)

7.

MULTIPLE CHOICE QUESTION

1 min • 1 pt

곱셈 규칙을 사용하여 다음 함수의 미분을 구합니다:

Use the Product Rule to find the derivative of the following function: f(x)=x2ln⁡(x)f\left(x\right)=x^2\ln(x)

f′(x)=2xln⁡(x)+xf'(x)=2x\ln(x)+x

f′(x)=x2ln⁡(x)+2xf'(x)=x^2\ln(x)+2x

f′(x)=2xln⁡(x)−xf'(x)=2x\ln(x)-x

f′(x)=x2ln⁡(x)−2xf'(x)=x^2\ln(x)-2x

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