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Flashcards in Derivatives Theory SL Deck (19)
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1
Q

What is the mathematical definition of derivative?

A
2
Q

How are derivatives and tangent lines related?

A

The derivative is the slope of the tangent line for a given x.

3
Q

What is the power rule for finding derivatives?

A
4
Q

What is the derivative of a constant?

A

0.

When you take the derivative of a constant, that term just disappears.

5
Q

How do you find the tangent line of a function at a given point (x1,y1)?

A

Take the derivative of the function to find f’(x), giving you the slope of the tangent line. Since the tangent line is straight, it can be represented as y=mx+c.

Plug in x1 and y1 and solve for c.

y1=f’(x1)*x1+c

6
Q

How do you take the derivative of a function with two or more terms?

For example, f(x)=4x3+2x2-3

A

You apply the sum or difference rule. The derivative of a function that adds or subtracts two or more terms is the derivative of each term added or subtracted together. You just take the derivative of each term.

For example, f’(x)=8x2+4x.

7
Q

What is the derivative of ex?

A

ex

8
Q

What is the derivative of ln(x)?

A
9
Q

What is the product rule and when do you need it?

A

You need the product rule when your function is actually two simpler functions multiplied together.

10
Q

What is the quotient rule and when do you need it?

A

You need the quotient rule when your function is actually two simpler functions divided.

11
Q

When do you need to use the chain rule?

A

When you have functions inside of other functions. For example:

12
Q

What is the chain rule?

A
13
Q

How are derivatives related to the max/min of a function?

A

The tangent at the min or max of a function is a flat line, which means its slope is 0. Therefore, the min/max occurs at f’(x)=0.

You find the derivative and then solve for x when f’(x)=0. (This gives you the x-coordinate. To find the y-coordinate, you have to plug this x into the original function.)

14
Q

How are position, velocity, and acceleration related?

A
  • Position just tells you where a point is or how far something has travelled.
  • Velocity is the derivative of position.
  • Acceleration is the derivative of velocity.
15
Q

How can you use the derivative to graph a sketch of a function, f(x)?

A
  1. Find f’(x)=0 to see where the local min/max are.
  2. Find when f’(x)<0 to see where the function is decreasing.
  3. Find when f’(x)>0 to see where the function is increasing.
  4. Find f’‘(x)=0 to see where the inflection points are.
  5. Find f’‘(x)<0 to see where the function is concave down (frowny face).
  6. Find f’‘(x)>0 to see where the function is concave up (happy face).
16
Q

How can you use the derivative to find out on which intervals the original function is increasing/decreasing?

A
  • When the derivative is negative (f’(x)<0), the function is decreasing.
  • When the derivative is positive (f’(x)>0), the function is increasing.

Practically, you should do the following:

  1. Find the derivative.
  2. Find when f’(x)=0.
  3. Choose points to the left and right of each of the answers to part 2.
  4. Plug those points into f’(x) to see if the derivative is negative or positive. Then you’ll know that the interval containing that point is decreasing or increasing, respectively.
17
Q

What is a “stationary point”?

A

Any point that satisfies the equation f’(x)=0. This is also typically a local min/max.

18
Q

What is an inflexion point?

A

This is the point where concavity changes, which means f”(x)=0.

19
Q

How can you use the second derivative to verify that a point, c, is a maximum or minimum? (Note that you got c by solving f’(x)=0.)

A