This quiz tests the work covered in the lecture on graphical interpretation of the
anti-derivative and corresponds to Section 6.1 of the textbook Calculus: Single and
Multivariable (Hughes-Hallett, Gleason, McCallum et al.).

There are more web quizzes at Wiley, select Section 1.

There are not many relevant web resources but it may be useful to review the
derivative function.

Visual Calculus, as usual, has links to an applet at http://archives.math.utk.edu/visual.calculus/4/ftc.9/ and some useful explanations.

Which of the following statements are correct? (Zero or more options can be correct)

*There is at least one mistake.*

For example, choice (a) should be False.

*There is at least one mistake.*

For example, choice (b) should be True.

*There is at least one mistake.*

For example, choice (c) should be True.

*There is at least one mistake.*

For example, choice (d) should be False.

*Correct!*

*False*Note that ${\int}_{2}^{3}f\left(x\right)=F\left(3\right)-F\left(2\right)=8-4=4$*True*${\int}_{2}^{4}f\left(x\right)=F\left(4\right)-F\left(2\right)=11-4=7$*True*${\int}_{3}^{4}f\left(x\right)=F\left(4\right)-F\left(3\right)=11-8=3$*False*Note that ${\int}_{2}^{3}f\left(x\right)=F\left(3\right)-F\left(2\right)=8-4=4$

Let ${F}^{\prime}\left(x\right)=f\left(x\right)\phantom{\rule{0.3em}{0ex}}.$

Which one of the following statements is true. Exactly one option must be correct)

*Choice (a) is correct!*

*Choice (b) is incorrect*

*Choice (c) is incorrect*

*Choice (d) is incorrect*

Suppose ${F}^{\prime}\left(x\right)=f\left(x\right)$ and $F\left(5\right)=4$. Determine the nature of the critical point of $F$ at $x=9$ and its coordinates.

Which one of the following statements is correct? Exactly one option must be correct)

*Choice (a) is incorrect*

*Choice (b) is correct!*

${\int}_{5}^{9}f\left(x\right)\phantom{\rule{0.3em}{0ex}}dx=6\phantom{\rule{0.3em}{0ex}}.$ Hence $F\left(9\right)=6+4=10$ and the critical point is $\left(9\phantom{\rule{0.3em}{0ex}},10\right)\phantom{\rule{0.3em}{0ex}}.$

The derivative changes from positive to negative at $x=9$ so the critical point is a maximum.

*Choice (c) is incorrect*

*Choice (d) is incorrect*

Recall ${\int}_{a}^{b}f\left(x\right)\phantom{\rule{0.3em}{0ex}}dx=F\left(b\right)-F\left(a\right)\phantom{\rule{0.3em}{0ex}}.$

Suppose ${F}^{\prime}\left(x\right)=f\left(x\right)$ and $F\left(2\right)=1$. Determine the nature of the critical point of $F$ at $x=5$ and its coordinates.

Which one of the following statements is correct? Exactly one option must be correct)

*Choice (a) is incorrect*

Recall ${\int}_{a}^{b}f\left(x\right)\phantom{\rule{0.3em}{0ex}}dx=F\left(b\right)-F\left(a\right)\phantom{\rule{0.3em}{0ex}}.$

*Choice (b) is incorrect*

*Choice (c) is correct!*

${\int}_{2}^{5}f\left(x\right)\phantom{\rule{0.3em}{0ex}}dx=-3\phantom{\rule{0.3em}{0ex}}.$ The area is beneath the $x$-axis so the integral is negative.

Hence $F\left(5\right)=-3+1=2$ and the critical point is $\left(5\phantom{\rule{0.3em}{0ex}},-2\right)\phantom{\rule{0.3em}{0ex}}.$

The derivative changes from negative to positive at $x=5$ so the critical point is a minimum.

*Choice (d) is incorrect*