Antiderivatives & The Fundamental Theorem Of Calculus Assessment Test

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Antiderivatives & The Fundamental Theorem Of Calculus Assessment Test - Quiz

We apply the derivative rules in reverse when we are finding antiderivatives, which are functions whose derivatives are ƒ. Take this assessment test to find out more.


Questions and Answers
  • 1. 

    The word indefinite integral is synonymous to...

    • A.

      Solution

    • B.

      Limit

    • C.

      Antiderivative

    • D.

      Integration

    Correct Answer
    C. Antiderivative
    Explanation
    The word "indefinite integral" refers to the process of finding the antiderivative of a function. The antiderivative of a function is a new function whose derivative is equal to the original function. Therefore, "antiderivative" is the correct answer as it accurately describes the relationship between the indefinite integral and finding the antiderivative of a function.

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  • 2. 

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

    • A.

      True

    • B.

      False

    • C.

      Depends on some factors

    • D.

      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

    • A.

      614

    • B.

      589

    • C.

      615

    • D.

      616

    Correct Answer
    D. 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

    • A.

      70

    • B.

      140

    • C.

      210

    • D.

      290

    Correct Answer
    B. 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

    • A.

      28

    • B.

      35

    • C.

      39

    • D.

      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. 

    • A.

      True

    • B.

      False

    • C.

      Maybe

    • D.

      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 ).

    • A.

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

    • B.

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

    • C.

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

    • D.

      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?  

    • A.

      ξ

    • B.

      Σ

    • C.

      φ

    • D.

      Φ

    Correct Answer
    B. 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...

    • A.

      Mean Value theorem

    • B.

      Pythagorean theorem

    • C.

      Binomial theorem

    • D.

      Fundamental theorem of calculus

    Correct Answer
    D. 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.

    • A.

      2

    • B.

      3

    • C.

      4

    • D.

      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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  • Current Version
  • Mar 20, 2023
    Quiz Edited by
    ProProfs Editorial Team
  • Jan 07, 2018
    Quiz Created by
    Cripstwick
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