Fundamental Theorem of Calculus and Accumulation Functions Practice

Sample questions

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1. Sample 1difficulty 2/5

   The Fundamental Theorem (Part 1) says $\dfrac{d}{dx}\int_a^x f(t)\,dt =$

   • A

     $f(x)$

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   • B

     $F(x) - F(a)$

   • C

     $\int f'$

   • D

     $f'(x)$

   Why

   Differentiation undoes integration.

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2. Sample 2difficulty 3/5

   Let $F(x) = \int_0^x f(t)\,dt$ where $f$ is shown. $F$ has a relative maximum at $x =$:

   • A

     $0$

   • B

     $2$

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   • C

     Nowhere

   • D

     $4$

   Why

   $F'(x) = f(x)$. Since $f$ changes from positive to negative at $x=2$, $F$ has a relative maximum there.

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3. Sample 3difficulty 3/5

   Let $g(x) = \int_1^x t^2\,dt$. Then $g(2) =$

   • A

     $3$

   • B

     $\tfrac{1}{3}$

   • C

     $8$

   • D

     $\tfrac{7}{3}$

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   Why

   $g(2) = [t^3/3]_1^2 = 8/3 - 1/3 = 7/3$.

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4. Sample 4difficulty 3/5

   For $g(x) = \int_2^{e^x} \ln t\,dt$, $g'(x) =$

   • A

     $\ln(e^x)$

   • B

     $e^x \ln(e^x) = x e^x$

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   • C

     $x e^x$

   • D

     $x \cdot e^x$

   Why

   Chain rule: $\ln(e^x) \cdot e^x = x e^x$.

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5. Sample 5difficulty 3/5

   The shaded area to the left of $x = c$ represents

   • A

     $F(c) = \int_a^c f(t)\,dt$

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   • B

     $f(c)$

   • C

     $f'(c)$

   • D

     $F(b) - F(c)$

   Why

   Accumulation function: integral from start to $c$.

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