Finding intervals on the graph of a function
$begingroup$
I'm having a little trouble with a problem that seems suspiciously easy and I wanted to make sure I'm on the right path. The problem reads as follows:
The rate of r (in liters per minute) at which water is entering a tank is given for t > 0 by the graph. A negative rate means that water is leaving the tank. In parts (a) – (d), give the largest interval on which:
(a) The volume of water is increasing.
(b) The volume of water is constant.
(c) The volume of water is increasing fastest.
(d) The volume of water is decreasing.
From my point of view it seems that the largest interval on which the function is increasing is also the largest interval on which it is increasing fastest. It also seems the largest intervals on which the function remains constant are the same length. Am I on the right track or am I misinterpreting the question?
calculus algebra-precalculus
asked Jul 5 '14 at 5:30
Irresponsible NewbIrresponsible Newb
52821225
$endgroup$
add a comment |
$begingroup$
I'm having a little trouble with a problem that seems suspiciously easy and I wanted to make sure I'm on the right path. The problem reads as follows:
The rate of r (in liters per minute) at which water is entering a tank is given for t > 0 by the graph. A negative rate means that water is leaving the tank. In parts (a) – (d), give the largest interval on which:
(a) The volume of water is increasing.
(b) The volume of water is constant.
(c) The volume of water is increasing fastest.
(d) The volume of water is decreasing.
From my point of view it seems that the largest interval on which the function is increasing is also the largest interval on which it is increasing fastest. It also seems the largest intervals on which the function remains constant are the same length. Am I on the right track or am I misinterpreting the question?
calculus algebra-precalculus
asked Jul 5 '14 at 5:30
Irresponsible NewbIrresponsible Newb
52821225
$endgroup$
add a comment |
$begingroup$
I'm having a little trouble with a problem that seems suspiciously easy and I wanted to make sure I'm on the right path. The problem reads as follows:
The rate of r (in liters per minute) at which water is entering a tank is given for t > 0 by the graph. A negative rate means that water is leaving the tank. In parts (a) – (d), give the largest interval on which:
(a) The volume of water is increasing.
(b) The volume of water is constant.
(c) The volume of water is increasing fastest.
(d) The volume of water is decreasing.
From my point of view it seems that the largest interval on which the function is increasing is also the largest interval on which it is increasing fastest. It also seems the largest intervals on which the function remains constant are the same length. Am I on the right track or am I misinterpreting the question?
calculus algebra-precalculus
asked Jul 5 '14 at 5:30
Irresponsible NewbIrresponsible Newb
52821225
$endgroup$
I'm having a little trouble with a problem that seems suspiciously easy and I wanted to make sure I'm on the right path. The problem reads as follows:
The rate of r (in liters per minute) at which water is entering a tank is given for t > 0 by the graph. A negative rate means that water is leaving the tank. In parts (a) – (d), give the largest interval on which:
(a) The volume of water is increasing.
(b) The volume of water is constant.
(c) The volume of water is increasing fastest.
(d) The volume of water is decreasing.
From my point of view it seems that the largest interval on which the function is increasing is also the largest interval on which it is increasing fastest. It also seems the largest intervals on which the function remains constant are the same length. Am I on the right track or am I misinterpreting the question?
calculus algebra-precalculus
calculus algebra-precalculus
asked Jul 5 '14 at 5:30
Irresponsible NewbIrresponsible Newb
52821225
asked Jul 5 '14 at 5:30
Irresponsible NewbIrresponsible Newb
52821225
asked Jul 5 '14 at 5:30
Irresponsible NewbIrresponsible Newb
52821225
asked Jul 5 '14 at 5:30
Irresponsible NewbIrresponsible Newb
52821225
asked Jul 5 '14 at 5:30
Irresponsible NewbIrresponsible Newb
52821225
52821225
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Look at the gradients of the lines. The interval for which the volume of water is increasing fastest is clearly $t=10$ to $t=20$. The gradient is steepest there.
Then, the longest interval for which the volume of water is strictly increasing is just 10 minutes, whichever you pick. Between $t=10$ and $t=20$ we have a $10$-minute interval of strict increase. The same goes for $t=30$ to $t=40$ and $t=80$ to $t=90$. So pick any of the three.
By the way, one really important distinction here is the difference between strictly increasing and non-decreasing. If the interval is strictly increasing, then your $y$-value has to get bigger on every step. If it's nondecreasing, then it has to either get bigger or stay the same. The practical difference between the two is that being nondecreasing allows you to include flat lines (where $r$ is constant) in your interval, and being strictly increasing doesn't allow that.
answered Jul 5 '14 at 5:43
NewbNewb
13.4k93781
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f856786%2ffinding-intervals-on-the-graph-of-a-function%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Look at the gradients of the lines. The interval for which the volume of water is increasing fastest is clearly $t=10$ to $t=20$. The gradient is steepest there.
Then, the longest interval for which the volume of water is strictly increasing is just 10 minutes, whichever you pick. Between $t=10$ and $t=20$ we have a $10$-minute interval of strict increase. The same goes for $t=30$ to $t=40$ and $t=80$ to $t=90$. So pick any of the three.
By the way, one really important distinction here is the difference between strictly increasing and non-decreasing. If the interval is strictly increasing, then your $y$-value has to get bigger on every step. If it's nondecreasing, then it has to either get bigger or stay the same. The practical difference between the two is that being nondecreasing allows you to include flat lines (where $r$ is constant) in your interval, and being strictly increasing doesn't allow that.
answered Jul 5 '14 at 5:43
NewbNewb
13.4k93781
$endgroup$
add a comment |
$begingroup$
Look at the gradients of the lines. The interval for which the volume of water is increasing fastest is clearly $t=10$ to $t=20$. The gradient is steepest there.
Then, the longest interval for which the volume of water is strictly increasing is just 10 minutes, whichever you pick. Between $t=10$ and $t=20$ we have a $10$-minute interval of strict increase. The same goes for $t=30$ to $t=40$ and $t=80$ to $t=90$. So pick any of the three.
By the way, one really important distinction here is the difference between strictly increasing and non-decreasing. If the interval is strictly increasing, then your $y$-value has to get bigger on every step. If it's nondecreasing, then it has to either get bigger or stay the same. The practical difference between the two is that being nondecreasing allows you to include flat lines (where $r$ is constant) in your interval, and being strictly increasing doesn't allow that.
answered Jul 5 '14 at 5:43
NewbNewb
13.4k93781
$endgroup$
add a comment |
$begingroup$
Look at the gradients of the lines. The interval for which the volume of water is increasing fastest is clearly $t=10$ to $t=20$. The gradient is steepest there.
Then, the longest interval for which the volume of water is strictly increasing is just 10 minutes, whichever you pick. Between $t=10$ and $t=20$ we have a $10$-minute interval of strict increase. The same goes for $t=30$ to $t=40$ and $t=80$ to $t=90$. So pick any of the three.
By the way, one really important distinction here is the difference between strictly increasing and non-decreasing. If the interval is strictly increasing, then your $y$-value has to get bigger on every step. If it's nondecreasing, then it has to either get bigger or stay the same. The practical difference between the two is that being nondecreasing allows you to include flat lines (where $r$ is constant) in your interval, and being strictly increasing doesn't allow that.
answered Jul 5 '14 at 5:43
NewbNewb
13.4k93781
$endgroup$
Look at the gradients of the lines. The interval for which the volume of water is increasing fastest is clearly $t=10$ to $t=20$. The gradient is steepest there.
Then, the longest interval for which the volume of water is strictly increasing is just 10 minutes, whichever you pick. Between $t=10$ and $t=20$ we have a $10$-minute interval of strict increase. The same goes for $t=30$ to $t=40$ and $t=80$ to $t=90$. So pick any of the three.
By the way, one really important distinction here is the difference between strictly increasing and non-decreasing. If the interval is strictly increasing, then your $y$-value has to get bigger on every step. If it's nondecreasing, then it has to either get bigger or stay the same. The practical difference between the two is that being nondecreasing allows you to include flat lines (where $r$ is constant) in your interval, and being strictly increasing doesn't allow that.
answered Jul 5 '14 at 5:43
NewbNewb
13.4k93781
answered Jul 5 '14 at 5:43
NewbNewb
13.4k93781
answered Jul 5 '14 at 5:43
NewbNewb
13.4k93781
answered Jul 5 '14 at 5:43
NewbNewb
13.4k93781
13.4k93781
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f856786%2ffinding-intervals-on-the-graph-of-a-function%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
