Chapter 12 : Limits and Derivatives

1. 8. What is the limit?

20

12

Infinity

DNE

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.

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.

2. 4. What rule should be used in deriving f(x) = x⁵

Constant rule

Power rule

Difference rule

Sum rule

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.

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.

3. 5. Another word for 'integral' is .. .

Constant

Derivative

Theorem

Antiderivative

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'.

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'.

4. 10. ∫ 4 dx

4

4x + C

4t + C

2x2 + C

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.

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.

5. 6. What does C represent in an antiderivative?

A variable

A constant

None

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.

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.

6. 1. Which of the following is the product rule for derivatives utilizing the original function h(x) = f(x)g(x) ?

H'(x)= f'(x)g'(x)

H'(x)=f'(x)g(x) - f(x)g'(x)

H'(x)=f'(x)g'(x) + f(x)g(x)

H'(x)=f'(x)g(x) + f(x)g'(x)

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).

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).

7. 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))

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))

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.

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.

8.

A

B

C

D

not-available-via-ai

Explanation

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9.

0

4/3

-3/2

Infinity

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Explanation

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10. 2. Which of the following is the derivative of f(x)= x²+ x + 2 ?

2x

X2 + x

2x + 1

0

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.

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.

Chapter 12 : Limits And Derivatives