3. Which of the following is the quotient rule for derivatives?

h'(x)=((g(x)f'(x) - f(x)g'(x)) / (g(x))2

h'(x)=((g(x)f'(x) + f(x)g'(x)) / (g(x))2

h'(x)=((g'(x)f'(x) - f(x)g(x)) / (g(x))

h'(x)=((g'(x)f'(x) + f(x)g(x)) / (g(x))

Chapter 12 : Limits And Derivatives

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Questions: 10 | Attempts: 97 | Updated: Apr 6, 2023

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Questions and Answers

1.

1. Which of the following is the product rule for derivatives utilizing the original function h(x) = f(x)g(x) ?
- A.
  
  H^'(x)= f^'(x)g^'(x)
- B.
  
  H^'(x)=f^'(x)g(x) - f(x)g^'(x)
- C.
  
  H^'(x)=f^'(x)g^'(x) + f(x)g(x)
- D.
  
  H^'(x)=f^'(x)g(x) + f(x)g^'(x)
Correct Answer
D. H^'(x)=f^'(x)g(x) + f(x)g^'(x)

Explanation
The product rule states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function. Therefore, the correct answer is h'(x) = f'(x)g(x) + f(x)g'(x).

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2.

2. Which of the following is the derivative of f(x)= x²+ x + 2 ?
- A.
  
  2x
- B.
  
  X²+ x
- C.
  
  2x + 1
- D.
  
  0
Correct Answer
C. 2x + 1

Explanation
The derivative of a function represents the rate at which the function is changing at any given point. To find the derivative of f(x)= x^2 + x + 2, we can apply the power rule of differentiation. The power rule states that the derivative of x^n is n*x^(n-1). Applying this rule to each term of the function, we get the derivative of f(x) as 2x + 1. Therefore, the correct answer is 2x + 1.

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3.

3. Which of the following is the quotient rule for derivatives?
- A.
  
  H^'(x)=((g(x)f^'(x) + f(x)g^'(x)) / (g(x))²
- B.
  
  H^'(x)=((g^'(x)f^'(x) - f(x)g(x)) / (g(x))
- C.
  
  H^'(x)=((g(x)f^'(x) - f(x)g^'(x)) / (g(x))²
- D.
  
  H^'(x)=((g^'(x)f^'(x) + f(x)g(x)) / (g(x))
Correct Answer
C. H^'(x)=((g(x)f^'(x) - f(x)g^'(x)) / (g(x))²

Explanation
The quotient rule for derivatives states that the derivative of a quotient of two functions is equal to the numerator's derivative times the denominator minus the denominator's derivative times the numerator, all divided by the square of the denominator. Therefore, the correct answer is h'(x)=((g(x)f'(x) - f(x)g'(x)) / (g(x))^2.

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4.

4. What rule should be used in deriving f(x) = x⁵
- A.
  
  Constant rule
- B.
  
  Power rule
- C.
  
  Difference rule
- D.
  
  Sum rule
Correct Answer
B. Power rule

Explanation
The correct answer is Power rule. This rule states that when differentiating a function of the form f(x) = x^n, where n is a constant, the derivative is given by f'(x) = n*x^(n-1). In this case, the function f(x) = x^5 can be differentiated using the power rule to obtain f'(x) = 5*x^(5-1) = 5*x^4.

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5.

5. Another word for 'integral' is .. .
- A.
  
  Constant
- B.
  
  Derivative
- C.
  
  Theorem
- D.
  
  Antiderivative
Correct Answer
D. Antiderivative

Explanation
An antiderivative is a function that reverses the process of differentiation. It is the opposite of finding a derivative. In calculus, the antiderivative of a function is also known as the integral of the function. Therefore, the given word 'integral' is synonymous with 'antiderivative'.

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6.

6. What does C represent in an antiderivative?
- A.
  
  A variable
- B.
  
  A constant
- C.
  
  None
Correct Answer
B. A constant

Explanation
In the context of an antiderivative, the variable C represents a constant. When finding the antiderivative of a function, we add a constant term (C) to the solution because the derivative of a constant is always zero. This constant accounts for all possible solutions to the antiderivative equation.

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7.

8. What is the limit?
- A.
  
  20
- B.
  
  12
- C.
  
  Infinity
- D.
  
  DNE
Correct Answer
A. 20

Explanation
The limit refers to the value that a function approaches as the input approaches a certain value. In this case, the limit is 20, which means that as the input approaches a certain value, the function also approaches 20.

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8.
- A.
  
  A
- B.
  
  B
- C.
  
  C
- D.
  
  D
Correct Answer
D. D
9.
- A.
  
  0
- B.
  
  4/3
- C.
  
  -3/2
- D.
  
  Infinity
Correct Answer
D. Infinity
10.

10. ∫ 4 dx
- A.
  
  4
- B.
  
  4x + C
- C.
  
  4t + C
- D.
  
  2x² + C
Correct Answer
B. 4x + C

Explanation
The integral of 4 with respect to x is equal to 4x. The "+ C" represents the constant of integration, which is added because when taking the derivative of a constant, it becomes zero. Therefore, the correct answer is 4x + C.

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