Quiz 1: Lower and Upper Sums

Question

Question 1

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 ${\int }_{0.5}^{6}\phantom{\rule{0.3em}{0ex}}f\left(x\right)\phantom{\rule{0.3em}{0ex}}dx$ ?

Your answer is correct
Areas are always positive quantities, unlike definite integrals which can be sums of positive and negative terms. In this case, the integral equals ${A}_{1}-{A}_{2}+{A}_{3}$.
Not correct. You may try again.
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.

Question 2

Find U, the Riemann Upper Sum for $f\left(x\right)={x}^{2}$ on $\left[0,2\right]$, using 4 equal sub-intervals.
 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

Your answer is correct.
Not correct. Choice (b) is false.
Remember that to find U, you must use the maximum value of $f\left(x\right)$ on each sub-interval.
Not correct. Choice (c) is false.
Remember that to find U, you must use the maximum value of $f\left(x\right)$ on each sub-interval.
Not correct. Choice (d) is false.
Remember that to find U, you must use the maximum value of $f\left(x\right)$ on each sub-interval.
Not correct. Choice (e) is false.

Question 3

Find L, the Riemann Lower Sum for $f\left(x\right)={x}^{2}$ on $\left[0,2\right]$, using 4 equal sub-intervals.
 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

Not correct. Choice (a) is false.
Remember that to find L, you must use the minimum value of $f\left(x\right)$ on each sub-interval.
Not correct. Choice (b) is false.
Remember that to find L, you must use the minimum value of $f\left(x\right)$ on each sub-interval.
Your answer is correct.
Not correct. Choice (d) is false.
Remember that to find L, you must use the minimum value of $f\left(x\right)$ on each sub-interval.
Not correct. Choice (e) is false.

Question 4

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 $U-L\le 0.1$ ?
 a) 65 b) 70 c) 75 d) 80 e) None of the above

Not correct. Choice (a) is false.
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 $f\left(0\right)$.
Not correct. Choice (b) is false.
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 $f\left(0\right)$.
Not correct. Choice (c) is false.
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 $f\left(0\right)$.
Your answer is correct.
Not correct. Choice (e) is false.

Question 5

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.
 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.
Your answers are 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.

Question 6

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.
 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.

Not correct. Choice (a) is false.
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 $ln\frac{3}{2}.$
Not correct. Choice (b) is false.
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 $ln2.$
Your answer is correct.
Not correct. Choice (d) is false.
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 $ln\frac{5}{2}.$
Not correct. Choice (e) is false.

Question 7

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.

Your answer is correct
Well done!
Not correct. You may try again.
Is your calculator set to radian mode? If not, change it.

Question 8

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?
 a) 135 litres b) 160 litres c) 185 litres d) 150 litres

Not correct. Choice (a) is false.
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.
Not correct. Choice (b) is false.
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.
Not correct. Choice (c) is false.
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.
Your answer 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.

Question 9

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.)

Your answer is correct
Well done!
Not correct. You may try again.
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 $f\left(5\right)$.

Question 10

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 $f$, ${\int }_{a}^{b}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt\phantom{\rule{1em}{0ex}}<\phantom{\rule{1em}{0ex}}U$.
 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.

Not correct. Choice (a) is false.
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?
Not correct. Choice (b) is false.
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?
Not correct. Choice (c) is false.
For (2), think about what would happen if $f\left(t\right)=1$ for all $t$.
Your answer 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.
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