## MATH1003 Quizzes

Quiz 2: Fundamental Theorem of Calculus and Riemann Sums
Question 1 Questions
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 ${\int }_{1}^{5}\phantom{\rule{1em}{0ex}}f\left(x\right)\phantom{\rule{1em}{0ex}}dx-{\int }_{2}^{5}\phantom{\rule{1em}{0ex}}f\left(x\right)\phantom{\rule{1em}{0ex}}dx={\int }_{1}^{2}\phantom{\rule{1em}{0ex}}f\left(x\right)\phantom{\rule{1em}{0ex}}dx.$
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!
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. Tick each correct statement in the list of options. (Zero or more options can be correct)
 a) $\left|{\int }_{a}^{b}\phantom{\rule{1em}{0ex}}g\left(x\right)\phantom{\rule{1em}{0ex}}dx\right|\le {\int }_{a}^{b}\phantom{\rule{1em}{0ex}}|g\left(x\right)|\phantom{\rule{1em}{0ex}}dx$ b) $\left|{\int }_{a}^{b}\phantom{\rule{1em}{0ex}}g\left(x\right)\phantom{\rule{1em}{0ex}}dx\right|\ge {\int }_{a}^{b}\phantom{\rule{1em}{0ex}}g\left(x\right)\phantom{\rule{1em}{0ex}}dx$ c) $\left|{\int }_{a}^{b}\phantom{\rule{1em}{0ex}}g\left(x\right)\phantom{\rule{1em}{0ex}}dx\right|\le {\int }_{a}^{b}\phantom{\rule{1em}{0ex}}g\left(x\right)\phantom{\rule{1em}{0ex}}dx$ d) ${\int }_{a}^{b}\phantom{\rule{1em}{0ex}}|g\left(x\right)|\phantom{\rule{1em}{0ex}}dx\le {\int }_{a}^{b}\phantom{\rule{1em}{0ex}}g\left(x\right)\phantom{\rule{1em}{0ex}}dx$ e) ${\int }_{a}^{b}\phantom{\rule{1em}{0ex}}|g\left(x\right)|\phantom{\rule{1em}{0ex}}dx\ge {\int }_{a}^{b}\phantom{\rule{1em}{0ex}}g\left(x\right)\phantom{\rule{1em}{0ex}}dx$

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!
1. True
2. True
3. False
4. False
5. 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.
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)
 a) The average acceleration of the particle over these 6 seconds is $3.5m∕{s}^{2}$. b) The velocity at time $t=4$ is $6m∕s.$ c) The average acceleration of the particle over these 6 seconds is $1.4m∕{s}^{2}$. d) The distance travelled between $t=0$ and $t=2$ is 2 metres. e) The average acceleration of the particle over these 6 seconds is $1.08\stackrel{̇}{3}m∕{s}^{2}$. f) The distance travelled between $t=0$ and $t=2$ is $\frac{4}{3}$ metres.

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!
1. 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 ${\int }_{a}^{b}\phantom{\rule{0.3em}{0ex}}g\left(x\right)\phantom{\rule{1em}{0ex}}dx$ ?
2. True
3. 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 ${\int }_{a}^{b}\phantom{\rule{0.3em}{0ex}}g\left(x\right)\phantom{\rule{1em}{0ex}}dx$ ?
4. False First obtain an expression for the velocity of the particle as a function of time, over the first two seconds of travel.
5. True
6. 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)
 a) $\frac{1}{2}$ b) $1$ c) $2\pi$ d) 3 e) None of the above.

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 $3{\int }_{0}^{\pi ∕2}\phantom{\rule{1em}{0ex}}cosx\phantom{\rule{1em}{0ex}}dx$, 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 Exactly one option must be correct)
 a) $3$ b) $\frac{3}{2}$ c) $\frac{2}{3}$ d) $\frac{5}{2}$ e) None of the above

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)
 a) ${\int }_{1}^{5}\phantom{\rule{1em}{0ex}}\frac{x-3}{{x}^{2}-4}\phantom{\rule{1em}{0ex}}dx$ b) ${\int }_{1}^{5}\phantom{\rule{1em}{0ex}}\frac{x-3}{{x}^{2}+1}\phantom{\rule{1em}{0ex}}dx$ c) ${\int }_{-1}^{1}\phantom{\rule{1em}{0ex}}\frac{1}{x}\phantom{\rule{1em}{0ex}}dx$ d) ${\int }_{-1}^{1}\phantom{\rule{1em}{0ex}}\frac{{e}^{x}}{{e}^{x}+1}\phantom{\rule{1em}{0ex}}dx$ e) ${\int }_{-1}^{1}\phantom{\rule{1em}{0ex}}\frac{{e}^{x}+{e}^{-x}}{{e}^{x}-1}\phantom{\rule{1em}{0ex}}dx$ f) None of the above

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!
1. False
2. True
3. False
4. True
5. False
6. 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)
 a) The Riemann Lower Sum for $f\left(x\right)=\frac{1}{{x}^{3}}$ on the interval $\left[2,10\right]$. b) The Riemann Upper Sum for $f\left(x\right)=\frac{1}{{x}^{3}}$ on the interval $\left[1,10\right]$. c) The Riemann Lower Sum for $f\left(x\right)=\frac{1}{{x}^{3}}$ on the interval $\left[1,11\right]$. d) The Riemann Upper Sum for $f\left(x\right)=\frac{1}{{x}^{3}}$ on the interval $\left[2,12\right]$. e) None of the above.

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!
1. False Remember that the Lower Sums use the minimum function values on each sub-interval.
2. False Remember that the Upper Sums use the maximum function values on each sub-interval.
3. True
4. True
5. 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)
 a) The minimum value of $f\left(x\right)$ on $\left[-1,4\right]$ is 0. b) The function $f$ is an increasing function on $\left[-1,1\right]$. c) $f\left(4\right)>f\left(1\right)$. d) The maximum value of $f\left(x\right)$ on $\left[-1,4.5\right]$ occurs when $x=4$. e) The function $f$ is concave down on $\left[-1,4.5\right]$.

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!
1. False
2. True
3. True
4. True
5. False