The Fundamental Theorem of Calculus (FTC) tells us that for any polynomial f : $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( \int _ { 0 } ^ { x } \mathrm { f } ( t ) \mathrm { d } t \right) = \mathrm { f } ( x )$$ A student calculates $\frac { \mathrm { d } } { \mathrm { d } x } \left( \int _ { x } ^ { 2 x } t ^ { 2 } \mathrm {~d} t \right)$ as follows: (I) $\quad \int _ { x } ^ { 2 x } t ^ { 2 } \mathrm {~d} t = \int _ { 0 } ^ { 2 x } t ^ { 2 } \mathrm {~d} t - \int _ { 0 } ^ { x } t ^ { 2 } \mathrm {~d} t$ (II) By FTC, $\frac { \mathrm { d } } { \mathrm { d } x } \left( \int _ { 0 } ^ { x } t ^ { 2 } \mathrm {~d} t \right) = x ^ { 2 }$ (III) By FTC, $\frac { \mathrm { d } } { \mathrm { d } x } \left( \int _ { 0 } ^ { 2 x } t ^ { 2 } \mathrm {~d} t \right) = ( 2 x ) ^ { 2 } = 4 x ^ { 2 }$ (IV) So $\frac { \mathrm { d } } { \mathrm { d } x } \left( \int _ { x } ^ { 2 x } t ^ { 2 } \mathrm {~d} t \right) = 4 x ^ { 2 } - x ^ { 2 }$ (V) giving $\frac { \mathrm { d } } { \mathrm { d } x } \left( \int _ { x } ^ { 2 x } t ^ { 2 } \mathrm {~d} t \right) = 3 x ^ { 2 }$ Which of the following best describes the student's calculation? A The calculation is completely correct. B The calculation is incorrect, and the first error occurs on line (I). C The calculation is incorrect, and the first error occurs on line (II). D The calculation is incorrect, and the first error occurs on line (III). E The calculation is incorrect, and the first error occurs on line (IV). F The calculation is incorrect, and the first error occurs on line (V).
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The Fundamental Theorem of Calculus (FTC) tells us that for any polynomial f :
$$\frac { \mathrm { d } } { \mathrm {~d} x } \left( \int _ { 0 } ^ { x } \mathrm { f } ( t ) \mathrm { d } t \right) = \mathrm { f } ( x )$$
A student calculates $\frac { \mathrm { d } } { \mathrm { d } x } \left( \int _ { x } ^ { 2 x } t ^ { 2 } \mathrm {~d} t \right)$ as follows:\\
(I) $\quad \int _ { x } ^ { 2 x } t ^ { 2 } \mathrm {~d} t = \int _ { 0 } ^ { 2 x } t ^ { 2 } \mathrm {~d} t - \int _ { 0 } ^ { x } t ^ { 2 } \mathrm {~d} t$\\
(II) By FTC, $\frac { \mathrm { d } } { \mathrm { d } x } \left( \int _ { 0 } ^ { x } t ^ { 2 } \mathrm {~d} t \right) = x ^ { 2 }$\\
(III) By FTC, $\frac { \mathrm { d } } { \mathrm { d } x } \left( \int _ { 0 } ^ { 2 x } t ^ { 2 } \mathrm {~d} t \right) = ( 2 x ) ^ { 2 } = 4 x ^ { 2 }$\\
(IV) So $\frac { \mathrm { d } } { \mathrm { d } x } \left( \int _ { x } ^ { 2 x } t ^ { 2 } \mathrm {~d} t \right) = 4 x ^ { 2 } - x ^ { 2 }$\\
(V) giving $\frac { \mathrm { d } } { \mathrm { d } x } \left( \int _ { x } ^ { 2 x } t ^ { 2 } \mathrm {~d} t \right) = 3 x ^ { 2 }$
Which of the following best describes the student's calculation?
A The calculation is completely correct.\\
B The calculation is incorrect, and the first error occurs on line (I).\\
C The calculation is incorrect, and the first error occurs on line (II).\\
D The calculation is incorrect, and the first error occurs on line (III).\\
E The calculation is incorrect, and the first error occurs on line (IV).\\
F The calculation is incorrect, and the first error occurs on line (V).