## MATH1003 Quizzes

Quiz 1: Lower and Upper Sums
Question 1 Questions
The graph of a function $f$ is shown below. Area ${A}_{1}=3.6$, area ${A}_{2}=1.1$ and area ${A}_{3}=0.7$. What is ${\int }_{0.5}^{6}\phantom{\rule{0.3em}{0ex}}f\left(x\right)\phantom{\rule{0.3em}{0ex}}dx$ ?

Correct!
Areas are always positive quantities, unlike definite integrals which can be sums of positive and negative terms. In this case, the integral equals ${A}_{1}-{A}_{2}+{A}_{3}$.
Remember that areas are always positive, but definite integrals can be sums of positive and negative terms. Think about how to combine the three areas to arrive at the correct value of the definite integral.
Find U, the Riemann Upper Sum for $f\left(x\right)={x}^{2}$ on $\left[0,2\right]$, using 4 equal sub-intervals. Exactly one option must be correct)
 a) $U=\frac{15}{4}$ b) $U=\frac{13}{4}$ c) $U=\frac{7}{4}$ d) $U=\frac{7}{2}$ e) None of the above

Choice (a) is correct!
Choice (b) is incorrect
Remember that to find U, you must use the maximum value of $f\left(x\right)$ on each sub-interval.
Choice (c) is incorrect
Remember that to find U, you must use the maximum value of $f\left(x\right)$ on each sub-interval.
Choice (d) is incorrect
Remember that to find U, you must use the maximum value of $f\left(x\right)$ on each sub-interval.
Choice (e) is incorrect
Find L, the Riemann Lower Sum for $f\left(x\right)={x}^{2}$ on $\left[0,2\right]$, using 4 equal sub-intervals. Exactly one option must be correct)
 a) $L=\frac{5}{4}$ b) $L=\frac{3}{4}$ c) $L=\frac{7}{4}$ d) $L=\frac{7}{2}$ e) None of the above

Choice (a) is incorrect
Remember that to find L, you must use the minimum value of $f\left(x\right)$ on each sub-interval.
Choice (b) is incorrect
Remember that to find L, you must use the minimum value of $f\left(x\right)$ on each sub-interval.
Choice (c) is correct!
Choice (d) is incorrect
Remember that to find L, you must use the minimum value of $f\left(x\right)$ on each sub-interval.
Choice (e) is incorrect
When calculating the Riemann Upper and Lower Sums (U and L) for the function $f\left(x\right)={x}^{2}$ on the interval $\left[0,2\right]$, what is the smallest number of (equal) sub-intervals needed to make $U-L\le 0.1$ ? Exactly one option must be correct)
 a) 65 b) 70 c) 75 d) 80 e) None of the above

Choice (a) is incorrect
This function is an increasing function on $\left[0,2\right]$, and so the difference between U and L when $n$ equal sub-intervals are used is just the length of each sub-interval times the difference between $f\left(2\right)$ and $f\left(0\right)$.
Choice (b) is incorrect
This function is an increasing function on $\left[0,2\right]$, and so the difference between U and L when $n$ equal sub-intervals are used is just the length of each sub-interval times the difference between $f\left(2\right)$ and $f\left(0\right)$.
Choice (c) is incorrect
This function is an increasing function on $\left[0,2\right]$, and so the difference between U and L when $n$ equal sub-intervals are used is just the length of each sub-interval times the difference between $f\left(2\right)$ and $f\left(0\right)$.
Choice (d) is correct!
Choice (e) is incorrect
The function $f$ is a continuous function defined on the interval $\left[a,b\right]$, where $a. Read all the statements below and tick all those that are correct. (Zero or more options can be correct)
 a) If $f\left(t\right)$ is positive for all $t$ in $\left[a,b\right]$, then so is ${\int }_{a}^{b}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt.$ b) If $f\left(t\right)$ is negative for all $t$ in $\left[a,b\right]$, then the area between the graph of $f$, the $t-$ axis, and the lines $t=a$ and $t=b$, is negative. c) If $f\left(t\right)$ takes both positive and negative values when $t$ is in $\left[a,b\right]$, then ${\int }_{a}^{b}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt\phantom{\rule{1em}{0ex}}\le \phantom{\rule{1em}{0ex}}{\int }_{a}^{b}\phantom{\rule{0.3em}{0ex}}|f\left(t\right)|\phantom{\rule{0.3em}{0ex}}dt.$ d) If $f\left(t\right)$ is positive for all $t$ in $\left[a,b\right]$ and $a, then ${\int }_{a}^{c}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt\phantom{\rule{1em}{0ex}}\le \phantom{\rule{1em}{0ex}}{\int }_{a}^{b}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt$ only if $f\left(c\right)\le f\left(b\right)$. e) None of the above is correct.

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 False.
Remember that area is always a positive quantity, by definition.
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.
When $f\left(t\right)$ is positive, ${\int }_{a}^{c}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt\phantom{\rule{1em}{0ex}}\le \phantom{\rule{1em}{0ex}}{\int }_{a}^{b}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt$ regardless of the values of $f\left(c\right)$ and $f\left(b\right)$.
There is at least one mistake.
For example, choice (e) should be False.
Correct!
1. True
2. False Remember that area is always a positive quantity, by definition.
3. True
4. False When $f\left(t\right)$ is positive, ${\int }_{a}^{c}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt\phantom{\rule{1em}{0ex}}\le \phantom{\rule{1em}{0ex}}{\int }_{a}^{b}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt$ regardless of the values of $f\left(c\right)$ and $f\left(b\right)$.
5. False
Estimate the value of ${\int }_{1}^{3}lnx\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}dx$ using the Riemann Upper Sum U for $f\left(x\right)=lnx$ on the interval $\left[1,3\right]$, with 4 equal sub-intervals. Exactly one option must be correct)
 a) $\frac{1}{2}\phantom{\rule{1em}{0ex}}ln7.5$ b) $ln3$ c) $\frac{1}{2}\phantom{\rule{1em}{0ex}}ln22.5$ d) $\frac{1}{2}\phantom{\rule{1em}{0ex}}ln15$ e) None of the above.

Choice (a) is incorrect
Have you used the correct function values in your expression for U? For example, on the first sub-interval $\left[1,\frac{3}{2}\right]$, the maximum function value is $ln\frac{3}{2}.$
Choice (b) is incorrect
Have you used the correct function values in your expression for U? For example, on the second sub-interval $\left[\frac{3}{2},2\right]$, the maximum function value is $ln2.$
Choice (c) is correct!
Choice (d) is incorrect
Have you used the correct function values in your expression for U? For example, on the third sub-interval $\left[2,\frac{5}{2}\right]$, the maximum function value is $ln\frac{5}{2}.$
Choice (e) is incorrect
Given that $cos\sqrt{x}$ decreases on the interval $\left[0,9\right]$, estimate the value of ${\int }_{0}^{9}cos\sqrt{x}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}dx$ using the Riemann Lower Sum L on this interval with three unequal sub-intervals $\left[0,1\right],\phantom{\rule{0.3em}{0ex}}\left[1,4\right],\phantom{\rule{0.3em}{0ex}}\left[4,9\right].$ Enter your answer correct to two decimal places.

Correct!
Well done!
Water leaks out of a tank at a decreasing rate. The rate of outflow is monitored every five minutes for twenty minutes, and the results are tabulated below. One estimate for the number of litres of water lost over the 20 minute period is the average of L and U for the function describing this loss. Which option equals the average of L and U? Exactly one option must be correct)
 a) 135 litres b) 160 litres c) 185 litres d) 150 litres

Choice (a) is incorrect
Remember that for a decreasing function, the minimum function value on a sub-interval occurs at the right hand end and the maximum value occurs at the left-hand end.
Choice (b) is incorrect
Remember that for a decreasing function, the minimum function value on a sub-interval occurs at the right hand end and the maximum value occurs at the left-hand end.
Choice (c) is incorrect
Remember that for a decreasing function, the minimum function value on a sub-interval occurs at the right hand end and the maximum value occurs at the left-hand end.
Choice (d) is correct!
Here $U=5\left(10+9+8+6\right)=165$ and $L=5\left(9+8+6+4\right)=135$ and so the average is $150$ litres/minute.
The function $g$ whose graph appears below has these values at the indicated values of $t$: $\begin{array}{ccccccccc}\hfill t\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill & \hfill 7\hfill \\ \hfill g\left(t\right)\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill -2\hfill & \hfill -3\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill \hfill \end{array}$ What is the value of the lower Riemann sum for $g\left(t\right)$ on $\left[0,7\right]$? (Use 7 equal intervals of length 1.)

Correct!
Well done!
Obtain the minimum function values on each sub-interval from the values given in the table and by inspecting the graph. For example, on the sub-interval $\left[4,5\right]$, the minimum value is $f\left(5\right)$.
Suppose that $U$ and $L$ are the upper and lower Riemann sums for $f\left(t\right)$ on the interval $\left[a,b\right]$, using $n$ equal subdivisions. Read the two statements and then select the correct option.
(1)   For all functions $f$, $U$ and $L$ are never equal. In fact, $L$ is always less than $U$.
(2)   For all functions $f$, ${\int }_{a}^{b}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt\phantom{\rule{1em}{0ex}}<\phantom{\rule{1em}{0ex}}U$. Exactly one option must be correct)
 a) (1) and (2) are both true. b) (1) is true and (2) is false. c) (1) is false and (2) is true. d) (1) and (2) are both false.

Choice (a) is incorrect
For statement (1), for example, think about the way $L$ and $U$ are defined, in terms of the minimum and maximum values (respectively) of the function on each sub-interval. For which type of function could they be equal?
Choice (b) is incorrect
For statement (1), for example, think about the way $L$ and $U$ are defined, in terms of the minimum and maximum values (respectively) of the function on each sub-interval. For which type of function could they be equal?
Choice (c) is incorrect
For (2), think about what would happen if $f\left(t\right)=1$ for all $t$.
Choice (d) is correct!
The statements are not true for all functions. For example, when $f\left(t\right)$ is constant on the interval $\left[a,b\right]$, $U=L={\int }_{a}^{b}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt.$ However, for non-constant functions, both statements are true.