Big Idea 2: Derivatives

Big Idea 2: Derivatives

Now that you've finished learning limits, the skeleton of calculus, it's time to put some muscles on those bones so it can do something useful. Derivatives are a mathematically precise method of finding a rate at a desired moment in time. There are many many applications for this, my favorite of which is speed. Here are a list of examples.

You can take the derivative of basically anything to learn its rate, so the possibilities are endless.

Derivatives have some interesting vocabulary which confused me for a long time. Let's talk about it so you don't get confused later.

Let's start with the word we know: derivative. That's the rate of a function.
- Example:
  - "I found the derivative of the function."
So, we need a verb for the act of performing a derivative. Many people think this word would be derive. NO. For some reason, the word is differentiate. Derive means something different.
- Example:
  - "Yo I just derived this function!"
  - "You're either high, or you mean you differentiated it"
Lastly, we need a noun for the act of performing a derivative. Many people think this word would be derivation. NO. For some reason, the word is differentiation. Derivation means something different.
- Example:
  - "Yo did you use derivation on question #31?"
  - "You have smoked yourself brain-dead, sir. Oh wait, are you asking if I used differentiation? Yes, I did."

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Essential Understanding 2.1 Students will understand that the derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

Learning Objective 2.1A Students will be able to identify the derivative of a function as the limit of a difference quotient.

Essential Knowledge 2.1A1 Students will know that the difference quotients

and express the average rate of change of a function over an interval.

Essential Knowledge 2.1A2 Students will know that the instantaneous rate of change of a function at a point can be expressed by

or , provided that the limit exists. These are common forms of the definition of the derivative and are denoted f '(a).

Essential Knowledge 2.1A3 Students will know that the derivative of f is the function whose value at x is provided this limit exists.

Essential Knowledge 2.1A4 Students will know that for y = f(x), notations for the derivative include , f '(x), and y'.

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

Learning Objective 2.1B Students will be able to estimate derivatives.

Essential Knowledge 2.1B1 Students will know that the derivative at a point can be estimated from information given in tables or graphs.

Learning Objective 2.1C Students will be able to calculate derivatives.

Essential Knowledge 2.1C1 Students will know that direct application of the definition of the derivative can be used to find the derivative for selected functions, including polynomial, power, sine, cosine, exponential, and logarithmic functions.

Essential Knowledge 2.1C2 Students will know that specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric.

Essential Knowledge 2.1C3 Students will know that sums, differences, products, and quotients of functions can be differentiated using derivative rules.

Essential Knowledge 2.1C4 Students will know that the chain rule provides a way to differentiate composite functions.

Essential Knowledge 2.1C5 Students will know that the chain rule is the basis for implicit differentiation.

Essential Knowledge 2.1C6 Students will know that the chain rule can be used to find the derivative of an inverse function, provided the derivative of that function exists.

Learning Objective 2.1D Students will be able to determine higher order derivatives.

Essential Knowledge 2.1D1 Students will know that differentiating f 'produces the second derivative f '', provided the derivative of f ' exists; repeating this process produces higher order derivatives of f.

Essential Knowledge 2.1D2 Students will know that higher order derivatives are represented with a variety of notations. For y = f(x), notations for the second derivative include

, f ''(x), and y''. Higher order derivatives can be denoted or f ⁽ⁿ⁾(x).

Essential Understanding 2.2 Students will understand that a function's derivative, which is itself a function, can be used to understand the behavior or the function.

Learning Objective 2.2A Students will be able to use derivatives to analyze properties of a function.

Essential Knowledge 2.2A1 Students will know that first and second derivatives of a function can provide information about the function and its graph including intervals of increase or decrease, local (relative) and global (absolute) extrema, intervals of upward or downward concavity, and points of inflection.

Essential Knowledge 2.2A2 Students will know that key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations.

Essential Knowledge 2.2A3 Students will know that key features of the graphs of f, f ', and f '' are related to one another.

Learning Objective 2.2B Students will be able to recognize the connection between differentiability and continuity.

Essential Knowledge 2.2B1 Students will know that a continuous function may fail to be differentiable at a point in its domain.

Essential Knowledge 2.2B2 Students will know that if a function is differentiable at a point, then it is continuous at that point.

Essential Understanding 2.3 Students will understand that the derivative has multiple interpretations and applications including those that involve instantaneous rates of change.

Learning Objective 2.3A Students will be able to interpret the meaning of a derivative within a problem.

Essential Knowledge 2.3A1 Students will know that the unit for f '(x) is the unit for f divided by the unit for x.

Essential Knowledge 2.3A2 Students will know that the derivative of a function can be interpreted as the instantaneous rate of change with respect to its independent variable.

Learning Objective 2.3B Students will be able to solve problems involving the slope of a tangent line.

Essential Knowledge 2.3B1 Students will know that the derivative at a point is the slope of the line tangent to a graph at that point on the graph.

Essential Knowledge 2.3B2 Students will know that the tangent line is the graph of a locally linear approximation of the function near the point of tangency.

Learning Objective 2.3C Students will be able to solve problems involving related rates, optimization, and rectilinear motion.

Essential Knowledge 2.3C1 Students will know that the derivative can be used to solve rectilinear motion problems involving position, speed, velocity, and acceleration.

Essential Knowledge 2.3C2 Students will know that the derivative can be used to solve related rates problems, that is, finding a rate at which one quantity is changing by relating it to other quantities whose rates of change are known.

Essential Knowledge 2.3C3 Students will know that the derivative can be used to solve optimization problems, that is, finding a maximum or minimum value of a function over a given interval.

Learning Objective 2.3D Students will be able to solve problems involving rates of change in applied contexts.

Essential Knowledge 2.3D1 Students will know that the derivative can be used to express information about rates of change in applied contexts.

Learning Objective 2.3E Students will be able to verify solutions to differential equations.

Essential Knowledge 2.3E1 Students will know that solutions to differential equations are functions or families of functions.

Essential Knowledge 2.3E2 Students will know that derivatives can be used to verify that a function is a solution to a given differential equation.

Learning Objective 2.3F Students will be able to estimate solutions to differential equations.

Essential Knowledge 2.3F1 Students will know that slope fields provide visual clues to the behavior of solutions to first order differential equations.

Essential Understanding 2.4 Students will understand that the Mean Value Theorem connects the behavior of a differentiable function over an interval to the behavior of the derivative of that function at a particular point in the interval.

Learning Objective 2.4A Students will be able to apply the Mean Value Theorem to describe the behavior of a function over an interval.

Essential Knowledge 2.4A1 Students will know that if a function f is continuous over the interval [a, b] and differentiable over the interval (a, b), the Mean Value Theorem guarantees a point within that open interval where the instantaneous rate of change equals the average rate of change over the interval.