17Calculus Derivatives - Using Derivatives To Analyze and Graph Functions
We can use derivatives to help when graphing a function. But wait, don't we have calculators and computers for that? Yes, we do. But calculators are very imprecise when graphing and really just give us a feel for what the function looks like. And, although computer graphing is more precise than calculators, they still don't usually give us exact values and are really just used to show generally what is going on. Once you learn how to use derivatives to describe a graph and how to interpret the results, you will have a much better idea of what is going on and how to use the graph for your application.
Recommended Books on Amazon | ||
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First and Second Derivatives
The first derivative and the second derivative give you information about the shape of the graph. Each derivative tells you different things but they do parallel one another, i.e. the pattern with the first derivative is repeated in the second derivative, as shown in this table. These may not make sense initially but, after you have studied each, they will.
1. increasing/decreasing intervals |
2. critical points \(f'(x)=0\) |
3. first derivative test |
4. concavity |
5. inflection points \(f''(x)=0\) |
6. second derivative test |
For the discussion of a topic, select a link above, starting with the the first derivative, increasing and decreasing intervals. The topics need to be discussed in order, since they build on one another. Once you have studied each of these topics, come back to this page and we will help you put all your newly-learned knowledge together.
Asymptotes
We touched on asymptotes on the continuity page. After you are comfortable with that material, come back here and watch this next video. It has lots of good examples of specific types of vertical asymptotes.
Krista King Math - How To Find Vertical Asymptotes [9min-56secs]
video by Krista King Math |
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Putting It All Together
Okay, at this point you should have read all six topics listed in the table above and worked those practice problems. These practice problems put all those concepts together to build a picture of what a graph should look like and how it behaves. In addition, you will need to have covered specifics about domain and range.
Practice
Unless otherwise instructed, use the first and second derivative as well as other concepts linked to on this page to sketch these functions. Show your work by building tables and label your graph carefully and completely.
Basic |
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\(\displaystyle{y=\frac{x-1}{x^2}}\)
Problem Statement |
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Use the first and second derivative to sketch the function \(\displaystyle{y=\frac{x-1}{x^2}}\). Show your work by building tables and label your graph carefully and completely.
Solution |
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This problem is solved in two videos.
1355 video
video by PatrickJMT |
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1355 video
video by PatrickJMT |
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close solution
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\(\displaystyle{f(x)=\frac{x}{x+4}}\)
Problem Statement |
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Use the first and second derivative to sketch the function \(\displaystyle{f(x)=\frac{x}{x+4}}\). Show your work by building tables and label your graph carefully and completely.
Solution |
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1357 video
video by PatrickJMT |
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close solution
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\(f(x)=3x^2-6x+5\)
Problem Statement |
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Use the first and second derivative to sketch the function \(f(x)=3x^2-6x+5\). Show your work by building tables and label your graph carefully and completely.
Solution |
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1926 video
video by Krista King Math |
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close solution
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Intermediate |
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\(\displaystyle{f(x)=\frac{x}{\sqrt{x^2+1}}}\)
Problem Statement |
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Use the first and second derivative to sketch the function \(\displaystyle{f(x)=\frac{x}{\sqrt{x^2+1}}}\). Show your work by building tables and label your graph carefully and completely.
Solution |
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This problem is solved in four videos.
1356 video
video by PatrickJMT |
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1356 video
video by PatrickJMT |
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1356 video
video by PatrickJMT |
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1356 video
video by PatrickJMT |
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close solution
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Log in to rate this practice problem and to see it's current rating. |
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You CAN Ace Calculus
Topics You Need To Understand For This Page
Related Topics and Links
Trig Formulas
The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
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Set 1 - basic identities | |||
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\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) |
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) |
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) |
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) |
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Set 2 - squared identities | ||
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\( \sin^2t + \cos^2t = 1\) |
\( 1 + \tan^2t = \sec^2t\) |
\( 1 + \cot^2t = \csc^2t\) |
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Set 3 - double-angle formulas | |
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\( \sin(2t) = 2\sin(t)\cos(t)\) |
\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\) |
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Set 4 - half-angle formulas | |
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\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\) |
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) |
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\) |
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\) |
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\) | |
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) |
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\) | |
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
Trig Integrals
\(\int{\sin(x)~dx} = -\cos(x)+C\) |
\(\int{\cos(x)~dx} = \sin(x)+C\) | |
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\) |
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\) | |
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) |
\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\) |
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Topics Listed Alphabetically
Single Variable Calculus |
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Multi-Variable Calculus |
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Differential Equations |
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Precalculus |
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Practice Instructions
Unless otherwise instructed, use the first and second derivative as well as other concepts linked to on this page to sketch these functions. Show your work by building tables and label your graph carefully and completely.
