Calculus 2 : Fundamental Theorem of Calculus

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #11 : Fundamental Theorem Of Calculus

Given 

\(\displaystyle f(x)=\int_{0}^{x}({3}t^{2}-7t-12})dt\), what is \(\displaystyle f'(7)\)?

Possible Answers:

\(\displaystyle 86\)

\(\displaystyle 77\)

\(\displaystyle 49\)

\(\displaystyle 57\)

\(\displaystyle 67\)

Correct answer:

\(\displaystyle 86\)

Explanation:

By the Fundamental Theorem of Calculus, for all functions \(\displaystyle f\) that are continuously defined on the interval \(\displaystyle [a,b]\) with \(\displaystyle x\) in \(\displaystyle [a,b]\) and for all functions \(\displaystyle F\) defined by by \(\displaystyle F(x)=\int_{a}^{x}f(t)dt\), we know that \(\displaystyle F(x)=f'(x)\).

Thus, since 

\(\displaystyle f(x)=\int_{0}^{x}({3}t^{2}-7t-12})dt\)\(\displaystyle f'(x)=\frac{d}{dx}\int_{0}^{x}(3t^{2}-7t-12)dt=3x^{2}-7x-12\).

Therefore, 

\(\displaystyle f'(7)=3(7)^{2}-7(7)-12=147-49-12=98-12=86\).

Example Question #12 : Fundamental Theorem Of Calculus

Given \(\displaystyle f(x)=\int_{0}^{x}(5t^{2}-8t+13)dt\), what is \(\displaystyle f'(0)\)?

Possible Answers:

\(\displaystyle 13\)

\(\displaystyle 5\)

\(\displaystyle 0\)

\(\displaystyle -13\)

\(\displaystyle -8\)

Correct answer:

\(\displaystyle 13\)

Explanation:

According to the Fundamental Theorem of Calculus, if \(\displaystyle f\) is a continuous function on the interval \(\displaystyle [a,b]\) with \(\displaystyle F\) as the function defined for all \(\displaystyle x\) on \(\displaystyle [a,b]\) as \(\displaystyle F(x)=\int_{a}^{x}f(t)dt\), then \(\displaystyle F'(x)=f(x)\). Therefore, if \(\displaystyle f(x)=\int_{0}^{x}(5t^{2}-8t+13)dt\) , then \(\displaystyle f'(x)=\frac{d}{dx}\int_{0}^{x}(5t^{2}-8t+13)dt=5t^{2}-8t+13\) . Thus, \(\displaystyle f'(0)=5(0)^{2}-8(0)+13=0-0+13=13\) .

Example Question #161 : Introduction To Integrals

Evaluate the definite integral using the Fundamental Theorem of Calculus.

\(\displaystyle \int_{0}^{3}(x^4 - 3x^2)dx\)

Possible Answers:

\(\displaystyle 75.6\)

\(\displaystyle 21.6\)

\(\displaystyle 162\)

\(\displaystyle 0\)

\(\displaystyle -26.6\)

Correct answer:

\(\displaystyle 21.6\)

Explanation:

The antiderivative of \(\displaystyle f(x) =\) \(\displaystyle x^4 - 3x^2\) is  \(\displaystyle F(x) = \frac{1}{5}x^5 - x^3\).

Evaluating \(\displaystyle F(3) - F(0)\) (by the fundamental theorem of calculus) gives us...

\(\displaystyle \\F(3) - F(0)\\ = 21.6 - 0\\ = 21.6\)

Example Question #161 : Introduction To Integrals

Solve

\(\displaystyle \int_{-2}^{2} (x^3 + 1) dx\)

Possible Answers:

\(\displaystyle -1\)

\(\displaystyle -4\)

\(\displaystyle 4\)

\(\displaystyle 1\)

\(\displaystyle 0\)

Correct answer:

\(\displaystyle 4\)

Explanation:

The antiderivative of \(\displaystyle f(x) = x^3 + 1\)  is  \(\displaystyle F(x) = \frac{1}{4}x^4 + x\).

Evaluating \(\displaystyle F(2) - F(-2)\) (by the fundamental theorem of calculus) gives us...

\(\displaystyle \\F(2) - F(-2)\\ = 6-2\\ = 4\)

 

Example Question #161 : Integrals

Evaluate the definite integral using the Fundamental Theorem of Calculus.

\(\displaystyle \int_{1}^{e^2}\left(\frac{1}{x} + 1\right)dx\)

Possible Answers:

\(\displaystyle 1-e^2\)

\(\displaystyle 2+e^2\)

\(\displaystyle 1+\frac{e^3}{3}\)

\(\displaystyle 2-e^2\)

\(\displaystyle 1+e^2\)

Correct answer:

\(\displaystyle 1+e^2\)

Explanation:

The antiderivative of \(\displaystyle f(x) = \frac{1}{x} + 1\) is \(\displaystyle F(x) = ln\left | x\right | + x\).

By evaluating \(\displaystyle F(e^2) - F(1)\) (by the fundamental theorem of calculus) we get...

 

\(\displaystyle F(e^2) - F(1)\)

\(\displaystyle = (2 + e^2) - (0+1)\)

\(\displaystyle = 1+ e^2\)

Example Question #162 : Introduction To Integrals

Evaluate the definite integral using the Fundamental Theorem of Calculus.

\(\displaystyle \int_{1}^{2}\left(2 - \frac{1}{x^2}\right)dx\)

Possible Answers:

\(\displaystyle -2.5\)

\(\displaystyle 1.5\)

\(\displaystyle -1.5\)

\(\displaystyle 0\)

\(\displaystyle 2.5\)

Correct answer:

\(\displaystyle 1.5\)

Explanation:

The antiderivative of \(\displaystyle f(x) = 2 - \frac{1}{x^2}\) is \(\displaystyle F(x) = 2x + \frac{1}{x}\).

By evaluating \(\displaystyle F(2) - F(1)\) (by the fundamental theorem of calculus) we get...

 

\(\displaystyle F(2) - F(1)\)

\(\displaystyle = (4 + \frac{1}{2}) - (2+1)\)

\(\displaystyle = 4.5 - 3\)

\(\displaystyle = 1.5\)

Example Question #11 : Fundamental Theorem Of Calculus

Given \(\displaystyle f(x)=\int_{0}^{x}(t^{2}-2t+5)dt\), what is \(\displaystyle f'(4)\)?

Possible Answers:

\(\displaystyle 15\)

\(\displaystyle 14\)

\(\displaystyle 16\)

\(\displaystyle 13\)

\(\displaystyle 12\)

Correct answer:

\(\displaystyle 13\)

Explanation:

According to the Fundamental Theorem of Calculus, if \(\displaystyle f\) is a continuous function on the interval \(\displaystyle [a,b]\) with \(\displaystyle F\) as the function defined for all \(\displaystyle x\) on \(\displaystyle [a,b]\) as 

\(\displaystyle F(x)=\int_{a}^{x}f(t)dt\), then \(\displaystyle F'(x)=f(x)\).

Therefore, if 

\(\displaystyle f(x)=\int_{0}^{x}(t^{2}-2t+5)dt\), then 

\(\displaystyle f'(x)=\frac{d}{dx}\int_{0}^{x}(t^{2}-2t+5)dt=x^{2}-2x+5\).

Thus, 

\(\displaystyle f'(4)=4^{2}-2(4)+5=16-8+5=13\).

Example Question #11 : Fundamental Theorem Of Calculus

Given \(\displaystyle f(x)=\int_{0}^{x}(3t^{2}+8t-6)dt\), what is \(\displaystyle f'(6)\)?

Possible Answers:

\(\displaystyle 120\)

\(\displaystyle 150\)

\(\displaystyle 130\)

\(\displaystyle 140\)

\(\displaystyle 110\)

Correct answer:

\(\displaystyle 150\)

Explanation:

According to the Fundamental Theorem of Calculus, if \(\displaystyle f\) is a continuous function on the interval \(\displaystyle [a,b]\) with \(\displaystyle F\) as the function defined for all \(\displaystyle x\) on \(\displaystyle [a,b]\) as 

\(\displaystyle F(x)=\int_{a}^{x}f(t)dt\), then \(\displaystyle F'(x)=f(x)\).

Therefore, if 

\(\displaystyle f(x)=\int_{0}^{x}(3t^{2}+8t-6)dt\), then 

\(\displaystyle f'(x)=\frac{d}{dx}\int_{0}^{x}(3t^{2}+8t-6)dt=3x^{2}+8x-6\).

Thus, 

\(\displaystyle f'(6)=3(6)^{2}+8(6)-6=108+48-6=156-6=150\).

Example Question #162 : Introduction To Integrals

Given \(\displaystyle f(x)=\int_{0}^{x}(-2t^{2}+4t+7)dt\), what is \(\displaystyle f'(1)\)?

Possible Answers:

\(\displaystyle -2\)

\(\displaystyle -1\)

\(\displaystyle 2\)

\(\displaystyle 1\)

\(\displaystyle 9\)

Correct answer:

\(\displaystyle 9\)

Explanation:

According to the Fundamental Theorem of Calculus, if \(\displaystyle f\) is a continuous function on the interval \(\displaystyle [a,b]\) with \(\displaystyle F\) as the function defined for all \(\displaystyle x\) on \(\displaystyle [a,b]\) as 

\(\displaystyle F(x)=\int_{a}^{x}f(t)dt\), then \(\displaystyle F'(x)=f(x)\).

Therefore, if 

\(\displaystyle f(x)=\int_{0}^{x}(-2t^{2}+4t+7)dt\)\(\displaystyle f'(x)=\frac{d}{dx}\int_{0}^{x}(-2t^{2}+4t+7)dt=-2t^2+4t+7\).

Thus, 

\(\displaystyle f'(1)=-2(1)^2+4(1)+7=9\).

Example Question #165 : Introduction To Integrals

Write \(\displaystyle P_{final}-P_{initial}\) in integral form, if \(\displaystyle P\) is position and \(\displaystyle P'(t)=v(t)\) where \(\displaystyle v(t)\) is velocity at time \(\displaystyle t\)

Possible Answers:

\(\displaystyle P_{final}-P_{initial}=\int_{initial}^{final}v(t)dt\)

\(\displaystyle P_{final}-P_{initial}=\int_{initial}^{final}P(t)v(t)dt\)

\(\displaystyle P_{final}-P_{initial}=\frac{1}{2}\int_{initial}^{final}v(t)dt\)

\(\displaystyle P_{final}-P_{initial}=\int_{final}^{initial}v(t)dt\)

Correct answer:

\(\displaystyle P_{final}-P_{initial}=\int_{initial}^{final}v(t)dt\)

Explanation:

To write position in integral form, we can take advantage of the fundamental theorem of calculus. Since the bounds are \(\displaystyle initial\) and \(\displaystyle final\), and \(\displaystyle P'(t)=v(t)\)

\(\displaystyle P_{final}-P_{initial}=\int_{initial}^{final}v(t)dt\)

 

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