# Worksheet: The Fundamental Theorem of Calculus: Functions Defined by Integrals

In this worksheet, we will practice applying the fundamental theorem of calculus to find the derivative of a function defined by an integral.

Q1:

Use the fundamental theorem of calculus to find the derivative of the function .

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Q2:

Given that , find .

Q3:

Use the Fundamental Theorem of Calculus to find the derivative of the function .

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Q4:

Find the derivative of the function .

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Q5:

Find the derivative of the function .

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Q6:

Suppose that is a function on the interval and we are able to define by . We find that is NOT differentiable on . What can we conclude?

• A is discontinuous somewhere in the interval .
• B is not differentiable everywhere on .
• CThere is a mistake because whenever we integrate a function, it must be differentiable and .
• D is discontinuous everywhere on .
• E is continuous everywhere on .

Q7:

The figure shows the graph of the function Which of the following is the graph of ?

• A • Bnone of the above
• C • D • E Q8:

Use the Fundamental Theorem of Calculus to find the derivative of the function .

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• E

Q9:

Use the Fundamental Theorem of Calculus to find the derivative of the function .

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• E

Q10:

Use the fundamental theorem of calculus to find the derivative of the function .

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• E

Q11:

Use the Fundamental Theorem of Calculus to find the derivative of the function .

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• E

Q12:

Use the Fundamental Theorem of Calculus to find the derivative of the function .

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• D
• E

Q13:

Use the Fundamental Theorem of Calculus to find the derivative of the function .

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• E

Q14:

Find the derivative of the function .

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• E

Q15:

Find the derivative of the function .

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Q16:

Given that , find .

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Q17:

Let . Use the Fundamental Theorem of Calculus to find .

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Q18:

Use the Fundamental Theorem of Calculus to find the derivative of the function .

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• E

Q19:

Use the Fundamental Theorem of Calculus to find the derivative of the function .

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Q20:

Given that , find .

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• D

Q21:

Use the Fundamental Theorem of Calculus to find the derivative of the function .

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• E

Q22:

Given that , find .

Q23:

Use the Fundamental Theorem of Calculus to find the derivative of the function .

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• B
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• D
• E

Q24:

The graph of a function is shown. Which is the graph of an antiderivative of ? • A
• B
• C

Q25:

The figure shows the graph of the function Which of the following is the graph of ?  • A(c)
• B(a)
• C(b)
• D(d)