Essential Knowledge 2.1A5

Essential Knowledge 2.1A5 Students will know that the derivative can be represented graphically, numerically, analytically, and verbally.

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We have actually done all of these already

Graphically: The value of a derivative at a point equals the slope of the tangent line at the point.

• Pros:

  • This is extremely useful for helping you visualize what you're doing mathematically.

• Cons:

  • But not so useful for getting an accurate answer, as drawing tangent lines free hand is very hard.

Numerically: We were able to get numbers from graph, from the tables, and from the limit of the difference quotient. Usually though, when someone says numerically, they're talking about using a table to estimate a derivative.

• Pros:

  • Don't need an equation!

  • If the curve has no unexpected turns near the derivative, the table can give a nice estimate, sometimes more than sufficient if the table is precise enough.

  • Easy math, just use the slope formula.

• Cons:

  • If you don't have an equation, you have no idea if your answer makes sense, because an unexpected bump could be affecting your slope.

  • Answer isn't perfect

Analytically: A vague word, but it just means the "official" way. In other words solving for a derivative using the limit of a difference quotient (and shortcuts).

• Pros:

  • This gives a mathematically perfect answer.

  • Can give a function or equation as an answer, which is super useful because you never need to take the derivative again.

• Cons:

  • You NEED an equation to do this.

  • Some derivatives are an absolute pain in the butt to solve, even with all the shortcuts

Verbally: I guess this just means saying what a derivative is, like "A derivative is the instantaneous rate of change of a function." or "The derivative of a position function is the rate of change of position, which is known as velocity."

• Pros:

  • In some ways, this is the most important way of expressing a derivative, because you explain the meaning of your answer.

  • If you can't explain what your answer means verbally, you probably don't really understand what you're doing. This means you're probably just trying to memorize rules and remember what you saw the teacher do that one day. You'll mess up something eventually, because you're doing math in the dark.

  • If you can explain it verbally, that means you understand what you're doing, and that makes calculus 10000x easier to memorize. Just think how long it has been for me since I learned all these things, yet I still understand almost everything because I just think about the meaning of the math until I figure out what I should do.

  • Imagine yourself explaining the meaning of all your answers, and you will truly learn calculus

• Cons

  • Doesn't really give you an equation or a number or anything.