examday

/TMUA

The Fundamental Theorem of Calculus (FTC) tells us that for any polynomial $f$:

A student calculates $\displaystyle\frac\dd{\dd x}\int_x^{2x}t^2\,\dd t$ as follows:

• (I) $\displaystyle\int_x^{2x}t^2\,\dd t=\int_0^{2x}t^2\,\dd t-\int_0^x t^2\,\dd t$
• (II) By FTC, $\displaystyle\frac\dd{\dd x}\int_0^x t^2\,\dd t=x^2$
• (III) By FTC, $\displaystyle\frac\dd{\dd x}\int_0^{2x}t^2\,\dd t=(2x)^2=4x^2$
• (IV) So $\displaystyle\frac\dd{\dd x}\int_x^{2x}t^2\,\dd t=4x^2-x^2$
• (V) giving $\displaystyle\frac\dd{\dd x}\int_x^{2x}t^2\,\dd t=3x^2$

Which of the following best describes the student's calculation?