Antiderivatives & The Fundamental Theorem Of Calculus Assessment Test

2.

Diifferential equations in x and y is an equation that involves x,y and derivatives of y.

True
False
Depends on some factors
Not defined

Correct Answer

A. True

Explanation

A differential equation in x and y is an equation that includes both x, y, and their derivatives. This statement is true because a differential equation typically relates the rate of change of a function to the function itself. In this case, the equation involves x, y, and the derivatives of y, indicating that it is indeed a differential equation in x and y.

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3.

Use the properties of summation to evaluate: 24 Σ (5i - 4) = i=1

614
589
615
616

Correct Answer

A. 616

Explanation

The given expression is a summation, where we are summing the expression (5i - 4) for values of i from 1 to 24. To evaluate this summation, we can use the properties of summation. The property states that the sum of a constant multiplied by a sequence of numbers is equal to the constant multiplied by the sum of the sequence. In this case, the constant is 5 and the sequence is the values of i from 1 to 24. The sum of this sequence is given by the formula n(n+1)/2, where n is the number of terms in the sequence. Plugging in n=24, we get 24(24+1)/2 = 12(25) = 300. Finally, we multiply this sum by the constant 5 and subtract 4 to get the final result of 616.

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4.

Use the properties of summation to evaluate: 10 Σ (2i + 3) i=1

70
140
210
290

Correct Answer

A. 140

Explanation

The given expression represents the sum of the terms (2i + 3) for i = 1 to 10. To evaluate this sum, we can use the properties of summation. We can distribute the summation operator to each term inside the parentheses and then evaluate each term separately. The sum of 2i for i = 1 to 10 is 2(1) + 2(2) + ... + 2(10) = 2(1 + 2 + ... + 10) = 2(55) = 110. Similarly, the sum of 3 for i = 1 to 10 is 3(10) = 30. Adding these two sums together, we get 110 + 30 = 140. Therefore, the answer is 140.

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5.

Evaluate the sum. 7 Σ = i=1

28
35
39
34

Correct Answer

A. 28

Explanation

The sum is evaluated by adding up the numbers in the given sequence. In this case, the sum is 28 because it is the only number provided.

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6.

Derivatives and integral, are essentially inverse operations on functions.

True
False
Maybe
Not defined

Correct Answer

A. True

Explanation

Derivatives and integrals are indeed inverse operations on functions. The derivative of a function measures its rate of change, while the integral calculates the area under the curve of the function. These operations are related in such a way that taking the derivative of the integral of a function gives back the original function, and vice versa. Therefore, the statement "Derivatives and integrals are essentially inverse operations on functions" is true.

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7.

Find all antiderivatives of f ( x ) = 1/(1 + x ) + 2 cos(2 x ).

F( x ) = log(1 + x ) + sin(2 x )
F( x ) = log(2 + x ) + sin(2 x )
F( x ) = log(1 + x ) + sin(1 x )
F( x ) = log(2 + x ) + sin(3 x )

Correct Answer

A. F( x ) = log(1 + x ) + sin(2 x )

Explanation

The given function f(x) is the sum of two terms: 1/(1+x) and 2cos(2x). To find the antiderivative, we integrate each term separately. The antiderivative of 1/(1+x) is log(1+x) and the antiderivative of 2cos(2x) is sin(2x). Therefore, the antiderivative of f(x) is F(x) = log(1+x) + sin(2x).

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8.

Which Greek letter is used for sigma notation?

ξ
Σ
φ
Φ

Correct Answer

A. Σ

Explanation

The Greek letter Σ is used for sigma notation. Sigma notation is a mathematical notation that represents the sum of a series of terms. The uppercase letter Σ is used to indicate the sum in this notation.

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9.

The connection between differential calculus and integral calculus is stated in a theorem that is appropriately called the...

Mean Value theorem
Pythagorean theorem
Binomial theorem
Fundamental theorem of calculus

Correct Answer

A. Fundamental theorem of calculus

Explanation

The Fundamental Theorem of Calculus states the connection between differential calculus and integral calculus. It states that if a function is continuous on a closed interval and has an antiderivative, then the definite integral of the function over that interval is equal to the difference between the antiderivative evaluated at the endpoints of the interval. This theorem is fundamental in understanding the relationship between the two branches of calculus and is widely used in various applications of mathematics and science.

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10.

There are _____ types of fundamental theorem of calculus.

2
3
4
5

Correct Answer

A. 2

Explanation

The fundamental theorem of calculus is divided into two parts: the first part relates the concept of differentiation to integration, while the second part connects definite integrals to antiderivatives. Therefore, there are two types of fundamental theorem of calculus.

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Antiderivatives & The Fundamental Theorem Of Calculus Assessment Test

The word indefinite integral is synonymous to...

Quiz Preview

Diifferential equations in x and y is an equation that involves x,y and derivatives of y.

Use the properties of summation to evaluate: 24 Σ (5i - 4) = i=1

Use the properties of summation to evaluate: 10 Σ (2i + 3) i=1

Evaluate the sum. 7 Σ = i=1

Derivatives and integral, are essentially inverse operations on functions.

Find all antiderivatives of f ( x ) = 1/(1 + x ) + 2 cos(2 x ).

Which Greek letter is used for sigma notation?

The connection between differential calculus and integral calculus is stated in a theorem that is appropriately called the...

There are _____ types of fundamental theorem of calculus.