Suppose that $f$
is continuous everywhere and that
$${\int}_{1}^{5}\phantom{\rule{1em}{0ex}}f\left(x\right)\phantom{\rule{1em}{0ex}}dx=-6,\phantom{\rule{2em}{0ex}}{\int}_{2}^{5}\phantom{\rule{1em}{0ex}}3f\left(x\right)\phantom{\rule{1em}{0ex}}dx=6.$$
Find
$${\int}_{2}^{1}\phantom{\rule{1em}{0ex}}f\left(x\right)\phantom{\rule{1em}{0ex}}dx.$$

*Correct!*

Well done. We have ${\int}_{2}^{1}\phantom{\rule{1em}{0ex}}f\left(x\right)\phantom{\rule{1em}{0ex}}dx=-{\int}_{1}^{2}\phantom{\rule{1em}{0ex}}f\left(x\right)\phantom{\rule{1em}{0ex}}dx$
and also ${\int}_{1}^{5}\phantom{\rule{1em}{0ex}}f\left(x\right)\phantom{\rule{1em}{0ex}}dx-{\int}_{2}^{5}\phantom{\rule{1em}{0ex}}f\left(x\right)\phantom{\rule{1em}{0ex}}dx={\int}_{1}^{2}\phantom{\rule{1em}{0ex}}f\left(x\right)\phantom{\rule{1em}{0ex}}dx.$

*Incorrect.*

*Please try again.*

Remember that ${\int}_{a}^{b}\phantom{\rule{1em}{0ex}}f\left(x\right)\phantom{\rule{1em}{0ex}}dx=-{\int}_{b}^{a}\phantom{\rule{1em}{0ex}}f\left(x\right)\phantom{\rule{1em}{0ex}}dx$
and that ${\int}_{a}^{b}\phantom{\rule{1em}{0ex}}f\left(x\right)\phantom{\rule{1em}{0ex}}dx={\int}_{a}^{c}\phantom{\rule{1em}{0ex}}f\left(x\right)\phantom{\rule{1em}{0ex}}dx+{\int}_{c}^{b}\phantom{\rule{1em}{0ex}}f\left(x\right)\phantom{\rule{1em}{0ex}}dx$.

Suppose that $f$
and $g$
are continuous everywhere and that
$${\int}_{2}^{5}\phantom{\rule{1em}{0ex}}\left(f\left(x\right)+2g\left(x\right)\right)\phantom{\rule{1em}{0ex}}dx=-3,\phantom{\rule{2em}{0ex}}{\int}_{5}^{2}\phantom{\rule{1em}{0ex}}\left(f\left(x\right)-g\left(x\right)\right)=1.$$
Find
$${\int}_{2}^{5}\phantom{\rule{1em}{0ex}}\left(2f\left(x\right)-5g\left(x\right)\right)\phantom{\rule{1em}{0ex}}dx.$$

*Correct!*

*Incorrect.*

*Please try again.*

The data gives us ${\int}_{2}^{5}\left(f+2g\right)=-3$
and ${\int}_{2}^{5}\left(f-g\right)=-1.$
Also, $2f-5g=3\left(f-g\right)-\left(f+2g\right)$.
Now try the calculation again!

The function $g$ is a
continuous function and $a<b$.
Tick each correct statement in the list of options. (Zero or more options can be
correct)

For example, choice (a) should be True.

For example, choice (b) should be True.

For example, choice (c) should be False.

For example, choice (d) should be False.

For example, choice (e) should be True.

*There is at least one mistake.*

For example, choice (a) should be True.

*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 False.

*There is at least one mistake.*

For example, choice (d) should be False.

*There is at least one mistake.*

For example, choice (e) should be True.

*Correct!*

*True**True**False**False**True*

A particle accelerates from rest in a straight line over a 6 second period. The graph of its
acceleration, $f\left(t\right)$
at time $t$,
from $t=0$
seconds to $t=6$
seconds, is shown below. At what time after the start does the particle reach
maximum velocity?

*Correct!*

Well done! Whenever the graph of acceleration is positive,
the velocity is increasing. Since the acceleration is zero at
$t=5$, this
marks the time of maximum velocity.

*Incorrect.*

*Please try again.*

Remember that acceleration is the
derivative of velocity, and that an increasing function has positive derivative.

This question uses the same data as the previous question. A particle accelerates
from rest in a straight line over a 6 second period. The graph of its acceleration
$f\left(t\right)$ at time
$t$, from
$t=0$ seconds
to $t=6$
seconds is shown below.
Tick all the correct statements.
(Zero or more options can be correct)

For example, choice (a) should be False.

For example, choice (b) should be True.

For example, choice (c) should be False.

For example, choice (d) should be False.

For example, choice (e) should be True.

For example, choice (f) should be True.

*There is at least one mistake.*

For example, choice (a) should be False.

How is the average
value of a function $g\left(x\right)$
over an interval $\left[a,b\right]$
calculated? How is it related to the definite integral
${\int}_{a}^{b}\phantom{\rule{0.3em}{0ex}}g\left(x\right)\phantom{\rule{1em}{0ex}}dx$ ?

*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 False.

How is the average
value of a function $g\left(x\right)$
over an interval $\left[a,b\right]$
calculated? How is it related to the definite integral
${\int}_{a}^{b}\phantom{\rule{0.3em}{0ex}}g\left(x\right)\phantom{\rule{1em}{0ex}}dx$ ?

*There is at least one mistake.*

For example, choice (d) should be False.

First obtain an expression for the velocity of the particle as a function of time, over
the first two seconds of travel.

*There is at least one mistake.*

For example, choice (e) should be True.

*There is at least one mistake.*

For example, choice (f) should be True.

*Correct!*

*False*How is the average value of a function $g\left(x\right)$ over an interval $\left[a,b\right]$ calculated? How is it related to the definite integral ${\int}_{a}^{b}\phantom{\rule{0.3em}{0ex}}g\left(x\right)\phantom{\rule{1em}{0ex}}dx$ ?*True**False*How is the average value of a function $g\left(x\right)$ over an interval $\left[a,b\right]$ calculated? How is it related to the definite integral ${\int}_{a}^{b}\phantom{\rule{0.3em}{0ex}}g\left(x\right)\phantom{\rule{1em}{0ex}}dx$ ?*False*First obtain an expression for the velocity of the particle as a function of time, over the first two seconds of travel.*True**True*

Find the value of
$${\int}_{0}^{\frac{3\pi}{2}}\phantom{\rule{1em}{0ex}}|cosx|\phantom{\rule{1em}{0ex}}dx.$$
(A graph helps.)
Exactly one option must be correct)

*Choice (a) is incorrect*

Where does the graph of the cosine function cross the horizontal
axis? This will give you a way of sketching the graph of
$|cosx|$. You can also make
use of symmetry.

*Choice (b) is incorrect*

Where does the graph of the cosine function cross the horizontal
axis? This will give you a way of sketching the graph of
$|cosx|$. You can also make
use of symmetry.

*Choice (c) is incorrect*

Where does the graph of the cosine function cross the horizontal
axis? This will give you a way of sketching the graph of
$|cosx|.$
You can also make use of symmetry.

*Choice (d) is correct!*

The graph of
$|cosx|$ shows that the
required answer is $3{\int}_{0}^{\pi \u22152}\phantom{\rule{1em}{0ex}}cosx\phantom{\rule{1em}{0ex}}dx$,
which equals 3.

*Choice (e) is incorrect*

Find the average value of $g\left(t\right)$
over $\left[0,3\right]$,
where
$$g\left(t\right)=\left\{\begin{array}{cc}0\phantom{\rule{1em}{0ex}}\hfill & \text{if}\phantom{\rule{1em}{0ex}}0\le t\le 1\hfill \\ 1\phantom{\rule{1em}{0ex}}\hfill & \text{if}\phantom{\rule{1em}{0ex}}1<t\le 3\hfill \end{array}\right.$$
Exactly one option must be correct)

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is correct!*

*Choice (d) is incorrect*

*Choice (e) is incorrect*

Which of the following expressions may be evaluated using the Fundamental
Theorem of Calculus Part II? Tick all those that can be. (You do not have to
evaluate them.) (Zero or more options can be correct)

For example, choice (a) should be False.

For example, choice (b) should be True.

For example, choice (c) should be False.

For example, choice (d) should be True.

For example, choice (e) should be False.

For example, choice (f) should be False.

*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 False.

*There is at least one mistake.*

For example, choice (d) should be True.

*There is at least one mistake.*

For example, choice (e) should be False.

*There is at least one mistake.*

For example, choice (f) should be False.

*Correct!*

*False**True**False**True**False**False*

Consider the sum
$$\frac{1}{{2}^{3}}+\frac{1}{{3}^{3}}+\frac{1}{{4}^{3}}+\dots \dots \dots +\frac{1}{1{1}^{3}}.$$
Tick each option that equals this sum. (Zero or more options can be correct)

For example, choice (a) should be False.

For example, choice (b) should be False.

For example, choice (c) should be True.

For example, choice (d) should be True.

For example, choice (e) should be False.

*There is at least one mistake.*

For example, choice (a) should be False.

Remember that the Lower Sums use the minimum function
values on each sub-interval.

*There is at least one mistake.*

For example, choice (b) should be False.

Remember that the Upper Sums use the maximum function
values on each sub-interval.

*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 True.

*There is at least one mistake.*

For example, choice (e) should be False.

*Correct!*

*False*Remember that the Lower Sums use the minimum function values on each sub-interval.*False*Remember that the Upper Sums use the maximum function values on each sub-interval.*True**True**False*

Part of the graph of the derivative function
$\frac{df}{dx}={f}^{\prime}\left(x\right)$ is
shown below. Tick each correct option.
(Zero or more options can be correct)

For example, choice (a) should be False.

For example, choice (b) should be True.

For example, choice (c) should be True.

For example, choice (d) should be True.

For example, choice (e) should be False.

*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 True.

*There is at least one mistake.*

For example, choice (e) should be False.

*Correct!*

*False**True**True**True**False*