“우리 대부분은 초라한 옷차림과 엉터리 가구들을 부끄럽게 여기지만, 초라한 생각과 엉터리 철학을 부끄럽게 여길 줄 알아야 한다.” – Albert Einstein

00210101
calculus derivatives and limits

https://www.eeweb.com/tools/calculus-derivatives-and-limits-reference-sheet

Definition of Limit

The limit is a method of evaluating an expression as an argument approaches a value. This value can be any point on the number line and often limits are evaluated as an argument approaches infinity or minus infinity. The following expression states that as x approaches the value c the function approaches the value L.

latex!encoded:base64,XGxhcmdlIFxsaW1fe3hcdG8gY31cWyBmKHgpID0gTCBcXQ==

Right Hand Limit

The following expression states that as x approaches the value c and x > c the function approaches the value L.

latex!encoded:base64,XGxhcmdlIFxsaW1fe3hcdG8gY14rfVxbIGYoeCkgPSBMIFxd

Left Hand Limit

The following expression states that as x approaches the value c and x < c the function approaches the value L.

latex!encoded:base64,XGxhcmdlIFxsaW1fe3hcdG8gY14tfVxbIGYoeCkgPSBMIFxd

Limit at Infinity

The following expression states that as x approaches infinity, the value c is a very large and positive number, the function approaches the value L.

latex!encoded:base64,XGxhcmdlIFxsaW1fe3hcdG8gXGluZnR5IH1cWyBmKHgpID0gTCBcXQ==

Also the limit as x approaches negative infinity, the value of c is a very large and negative number, is expressed below.

latex!encoded:base64,XGxhcmdlIFxsaW1fe3hcdG8gLVxpbmZ0eSB9XFsgZih4KSA9IEwgXF0=

Properties of Limits

Given the following conditions:

latex!encoded:base64,XGxhcmdlIFxsaW1fe3hcdG8gY31cWyBmKHgpIFwsXHRleHR1cHthbmR9XCxcbGltX3t4XHRvIGN9XFsgZyh4KVwsXHRleHR1cHtib3RofVwsXHRleHR1cHtleGlzdH1cLFx0ZXh0dXB7YW5kfVwsY1wsXHRleHR1cHtpc31cLFx0ZXh0dXB7YW55fVwsXHRleHR1cHtyZWFsfVwsXHRleHR1cHtudW1iZXJ9

The following properties exist:

latex!encoded:base64,XGxhcmdlIFxsaW1fe3hcdG8gYX1cW1tjZih4KV0gPSBjXGxpbV97eFx0byBhfVxbZih4KQ==
latex!encoded:base64,XGxhcmdlIFxsaW1fe3hcdG8gYX1cW1tmKHgpXCxccG1cLGcoeCldID0gXGxpbV97eFx0byBhfSBmKHgpIFxwbVxsaW1fe3hcdG8gYX1cW2coeCk=
latex!encoded:base64,XGxhcmdlIFxsaW1fe3hcdG8gYX1cW1tmKHgpXCxnKHgpXSA9IFxsaW1fe3hcdG8gYX1mKHgpXFtcbGltX3t4XHRvIGF9XFtnKHgp
latex!encoded:base64,XGxhcmdlIFxsaW1fe3hcdG8gYX1cbGVmdCBbXGZyYWN7Zih4KX17XCxnKHgpfSBccmlnaHRdID1cZnJhY3tcbGltX3t4XHRvIGF9Zih4KX17XGxpbV97eFx0byBhfWcoeCl9IFwgXCBcdGV4dHJte2lmfSBcLCBcbGltX3t4XHRvIGF9Zyh4KSBcbmVxIDAK
latex!encoded:base64,XGxhcmdlIFxsaW1fe3hcdG8gYX1cW1tmKHgpXV57bn09XGxlZnQgW1xsaW1fe3hcdG8gYX1mKHgpXHJpZ2h0XV57bn0=

Limit Evaluation at +-Infinity

latex!encoded:base64,XGxhcmdlIFxsaW1fe3ggXHRvIFxpbmZ0eSB9XFxlXnt4fSA9IFxpbmZ0eSBcIFwgXHRleHR1cHthbmR9IFwgXGxpbV97eCBcdG8gLVxpbmZ0eSB9IGVee3h9PSBcLDA=
latex!encoded:base64,XGxhcmdlIFxsaW1fe3ggXHRvIFxpbmZ0eX1cbG4oeCkgPSBcaW5mdHkgIFwgXHRleHR1cHthbmR9IFxsaW1fe3ggXHRvIDBeK30gXGxuKHgpPSAtXGluZnR5
latex!encoded:base64,XGxhcmdlIFx0ZXh0dXB7SWYgfXI+MCBcIFwgdGhlXHRleHR1cHt9IFxsaW1fe3ggXHRvIFxpbmZ0eX0gXCBcZnJhY3tjfXt4XnJ9ID0gMA==
latex!encoded:base64,XGxhcmdlIFx0ZXh0dXB7SWZ9IFwgciA+IDAgXCBcdGV4dHVwe2FuZH0gXCB4XntyfSBcIFx0ZXh0dXB7aXMgcmVhbCBmb3J9IFwgeCA8MCBcIFx0ZXh0dXB7dGhlbn0gIFxsaW1fe3ggXHRvIC1caW5mdHl9IFwgXGZyYWN7Y317eF57cn19ID0gMA==
latex!encoded:base64,XGxhcmdlIFxsaW1fe3ggXHRvIFxwbSAgXGluZnR5fSB4XntyfSA9ICBcaW5mdHkgXCAgXHRleHR1cHtmb3J9IFwgXHRleHR1cHtldmVufSBcIHI=
latex!encoded:base64,XGxhcmdlIFxsaW1fe3ggXHRvICBcaW5mdHl9ICB4XntyfSA9IFxpbmZ0eSAgXCBcJiBcbGltX3t4IFx0byAtIFxpbmZ0eX0geF57cn0gPSAgLSBcaW5mdHkgXCAgXHRleHR1cHtmb3J9IFwgXHRleHR1cHtvZGR9IFwgcg==

Limit Evaluation Methods

Continuous Functions

If f(x) is continuous at a then:

latex!encoded:base64,XGxhcmdlIFxsaW1fe3hccmlnaHRhcnJvdyBhfSBmKHgpID0gZihhKQ==
Continuous Functions and Compositions

If f(x) is continuous at b:

latex!encoded:base64,XGxhcmdlIFxsaW1fe3ggXHRvIGF9Zyh4KSA9IGIgXCAsIFx0ZXh0dXB7dGhlbn0gXGxpbV97eCBcdG8gYX0gZlxsZWZ0ICggXGxpbV97eCBcdG8gYX0gZyh4KSBccmlnaHQgKSA9IGYoYik=
Factor and Cancel
latex!encoded:base64,XGxhcmdlIFxsaW1fe3ggXHRvIC0zfSAgXGZyYWN7eF4yLXgtMTJ9e3heMiszeH0gPSBcbGltX3t4IFx0byAtM30gXGZyYWN7KHgrMykoeC00KX17eCh4KzMpfSA9IFxsaW1fe3ggXHRvIC0zfSAgXGZyYWN7KHgtNCl9e3h9ID0gXGZyYWN7LTd9ezN9
L’Hospital’s Rule
latex!encoded:base64,XGxhcmdlIFx0ZXh0dXB7SWZ9IFwgXGxpbV97eCBcdG8gYX0gXGZyYWN7Zih4KX17Zyh4KX0gPSBcZnJhY3swfXswfSBcIG9yIFwgXGZyYWN7XHBtIFxpbmZ0eX17XHBtIFxpbmZ0eX0gXCBcdGV4dHVwe3RoZW59IFxsaW1fe3ggXHRvIGF9IFxmcmFje2YoeCl9e2coeCl9ID0gXGxpbV97eCBcdG8gYX0gXGZyYWN7ZicoeCl9e2cnKHgpfQ==

Derivatives Math Help

Definition of a Derivative

The derivative is way to define how an expressions output changes as the inputs change. Using limits the derivative is defined as:

latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShjKSA9IGYnKHgpID0gXGxpbV97aCBcdG8gMH0gXGZyYWN7Zih4K2gpLWYoeCl9e2h9

Mean Value Theorem

This is a method to approximate the derivative. The function must be differentiable over the interval (a,b) and a < c < b.

latex!encoded:base64,XGxhcmdlIGYnKGMpID0gXGZyYWN7ZihiKSAtIGYoYSl9e2ItYX0=

Basic Properties

If there exists a derivative for f(x) and g(x), and c and n are real numbers the following are true:

latex!encoded:base64,XGxhcmdlIFxsZWZ0ICggY2YoeCkgXHJpZ2h0ICknID0gXGxlZnQgYyggZicoeCkgXHJpZ2h0ICk=
latex!encoded:base64,XGxhcmdlIFxsZWZ0ICggZih4KSBccG0gZyh4KSBccmlnaHQgKScgPSBmJyh4KSBccG0gZycoeCk=
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShjKSA9IDA=

Product Rule

The product rule applies when differentiable functions are multiplied.

latex!encoded:base64,XGxhcmdlIFxsZWZ0ICggZih4KWcoeCkgXHJpZ2h0ICknID0gZidnKHgpK2YoeClnJw==

Quotient Rule

Quotient rule applies when differentiable functions are divided.

latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fVxsZWZ0ICggXGZyYWN7Zih4KX17Zyh4KX0gIFxyaWdodCApID0gXGZyYWN7ZicoeClnKHgpLWYoeClnJyh4KX17W2coeCldXjJ9

Power Rule

The power rule applies when a differentiable function is raised to a power.

latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fSh4Xm4pID0gbnhee24tMX0=

Chain Rule

The chain rule applies when a differentiable function is applied to another differentiable function.

latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fVxsZWZ0ICggZihnKHgpKSBccmlnaHQgKSA9IGYnXGxlZnQgKCBnKHgpIFxyaWdodCApZycoeCk=

Common Derivatives

latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fSh4KSA9MQ==
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShcdGV4dHVwe3Npbn0gXCB4KT1cdGV4dHVwe2Nvc30gXCB4
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShcdGV4dHVwe2Nvc30gXCB4KSA9IC1cdGV4dHVwe3Npbn0gXCB4
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShcdGV4dHVwe3Rhbn0gXCB4KSA9IFx0ZXh0dXB7c2VjfV4yIFwgeA==
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShcdGV4dHVwe3NlY30gXCB4KSA9IFx0ZXh0dXB7c2VjfSBcIHggXCBcdGV4dHVwe3Rhbn0gXCB4
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShcdGV4dHVwe2NzY30gXCB4KT0gLVx0ZXh0dXB7Y3NjfSBcIHggXCBcdGV4dHVwe2NvdH0gXCB4
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShcdGV4dHVwe2NvdH0gXCB4KSA9IC1cdGV4dHVwe2NzY31eMiBcIHg=
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShzaW5eey0xfXgpID0gXGZyYWN7MX17XHNxcnR7MSAtIHggXnsyfX19
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShcdGV4dHVwe2Nvc31eey0xfXgpID0gLSBcZnJhY3sxfXtcc3FydHsxLXheMn19
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShcdGV4dHVwe1x0ZXh0dXB7fXRhbn1eey0xfXgpID0gXGZyYWN7MX17MSt4XjJ9
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShhXngpPWFeeFx0ZXh0dXB7bG59KGEp
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShlXngpID0gZV54
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fVxsZWZ0ICggXHRleHR1cHtsbn0oeCkgXHJpZ2h0ICkgPSBcZnJhY3sxfXt4fSBcICwgeCA+IDA=
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fVxsZWZ0IChcdGV4dHVwe2xufVxsZWZ0IHwgeCBccmlnaHQgfCAgXHJpZ2h0ICkgPSBcZnJhY3sxfXt4fSBcICwgXCB4IFxuZXEgIDA=
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fVxsZWZ0ICggbG9nX3thfSh4KSBccmlnaHQgKSA9ICBcZnJhY3sxfXt4IFwsIFx0ZXh0dXB7bG59KGEpfSBcICwgXCB4ID4w

Chain Rule Examples

These are some examples of common derivatives that require the chain rule.

latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShbZih4KV1ebikgPSBuW2YoeCldXntuLTF9ZicoeCk=
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShlXntmKHgpfSkgPSBmJyh4KWVee2YoeCl9
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShcdGV4dHVwe2xufVtmKHgpXSkgPSBcZnJhY3tmJyh4KX17Zih4KX0=
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShcdGV4dHVwe3Npbn1bZih4KV0pID0gZicoeClcdGV4dHVwe2Nvc31bZih4KV0=
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShcdGV4dHVwe2Nvc31bZih4KV0pID0gLWYnKHgpXHRleHR1cHtzaW59W2YoeCld
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShcdGV4dHVwe3Rhbn1bZih4KV0pID0gZicoeClcdGV4dHVwe3NlY31eMltmKHgpXQ==
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShcdGV4dHVwe3NlY31bZih4KV0pID0gZicoeClcdGV4dHVwe3NlY31bZih4KV1cdGV4dHVwe3Rhbn1bZih4KV0=
latex!encoded:base64,XGxhcmdlIFxmcmFje2R9e2R4fShcdGV4dHVwe3Rhbn1eey0xfVtmKHgpXSkgPSBcZnJhY3tmJyh4KX17MStbZih4KV1eMn0=

댓글 남기기 | cat > 사락사락

댓글 남기기

* 표시된 곳은 반드시 입력해주세요