Unit 2: Differentiation: Definition and Fundamental Properties
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- 2020
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Topic 2.1
2.1 Defining Average and Instantaneous Rates of Change at a Point
Objectives in this topic
CHA-2.A—Determine average rates of change using difference quotients
- CHA-2.A Determine average rates of change using difference quotients.
- CHA-2.A.1 The difference quotients (f(a + h) − f(a))/h and (f(x) − f(a))/(x − a) express the average rate of change of a function over an interval.
- Enduring understanding CHA-2: Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.
CHA-2.B—Represent the derivative of a function as the limit of a difference quotient
- CHA-2.B Represent the derivative of a function as the limit of a difference quotient.
- CHA-2.B.1 The instantaneous rate of change of a function at x = a can be expressed by lim h→0 (f(a + h) − f(a))/h or lim x→a (f(x) − f(a))/(x − a), provided the limit exists. These are equivalent forms of the definition of the derivative and are denoted f′(a).
- Enduring understanding CHA-2: Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.
Topic 2.2
2.2 Defining the Derivative of a Function and Using Derivative Notation
Objectives in this topic
CHA-2.B—Represent the derivative of a function as the limit of a difference quotient—Topic 2.2
- CHA-2.B Represent the derivative of a function as the limit of a difference quotient.
- CHA-2.B.2 The derivative of f is the function whose value at x is lim h→0 (f(x + h) − f(x))/h, provided this limit exists.
- CHA-2.B.3 For y = f(x), notations for the derivative include dy/dx, f′(x), and y′.
- CHA-2.B.4 The derivative can be represented graphically, numerically, analytically, and verbally.
- Enduring understanding CHA-2: Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.
CHA-2.C—Determine the equation of a line tangent to a curve at a given point
- CHA-2.C Determine the equation of a line tangent to a curve at a given point.
- CHA-2.C.1 The derivative of a function at a point is the slope of the line tangent to a graph of the function at that point.
- Enduring understanding CHA-2: Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.
Topic 2.3
2.3 Estimating Derivatives of a Function at a Point
Objectives in this topic
CHA-2.D—Estimate derivatives
- CHA-2.D Estimate derivatives.
- CHA-2.D.1 The derivative at a point can be estimated from information given in tables or graphs.
- CHA-2.D.2 Technology can be used to calculate or estimate the value of a derivative of a function at a point.
- Enduring understanding CHA-2: Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.
Topic 2.4
2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
Objectives in this topic
FUN-2.A—Explain the relationship between differentiability and continuity
- FUN-2.A Explain the relationship between differentiability and continuity.
- FUN-2.A.1 If a function is differentiable at a point, then it is continuous at that point. In particular, if a point is not in the domain of f, then it is not in the domain of f '.
- FUN-2.A.2 A continuous function may fail to be differentiable at a point in its domain.
- Enduring understanding FUN-2: Recognizing that a function’s derivative may also be a function allows us to develop knowledge about the related behaviors of both.
Topic 2.5
2.5 Applying the Power Rule
Objectives in this topic
FUN-3.A—Calculate derivatives of familiar functions
- FUN-3.A Calculate derivatives of familiar functions.
- FUN-3.A.1 Direct application of the definition of the derivative and specific rules can be used to calculate the derivative for functions of the form f(x) = xʳ.
- Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.
Topic 2.6
2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
Objectives in this topic
FUN-3.A—Calculate derivatives of familiar functions—Topic 2.6
- FUN-3.A Calculate derivatives of familiar functions.
- FUN-3.A.2 Sums, differences, and constant multiples of functions can be differentiated using derivative rules.
- FUN-3.A.3 The power rule combined with sum, difference, and constant multiple properties can be used to find the derivatives for polynomial functions.
- Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.
Topic 2.7
2.7 Derivatives of cos x, sin x, eˣ, and ln x
Objectives in this topic
FUN-3.A—Calculate derivatives of familiar functions—Topic 2.7
- FUN-3.A Calculate derivatives of familiar functions.
- FUN-3.A.4 Specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions.
- Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.
LIM-3.A—Interpret a limit as a definition of a derivative
- LIM-3.A Interpret a limit as a definition of a derivative.
- LIM-3.A.1 In some cases, recognizing an expression for the definition of the derivative of a function whose derivative is known offers a strategy for determining a limit.
- Enduring understanding LIM-3: Reasoning with definitions, theorems, and properties can be used to determine a limit.
Topic 2.8
2.8 The Product Rule
Objectives in this topic
FUN-3.B—Calculate derivatives of products and quotients of differentiable functions
- FUN-3.B Calculate derivatives of products and quotients of differentiable functions.
- FUN-3.B.1 Derivatives of products of differentiable functions can be found using the product rule.
- Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.
Topic 2.9
2.9 The Quotient Rule
Objectives in this topic
FUN-3.B—Calculate derivatives of products and quotients of differentiable functions—Topic 2.9
- FUN-3.B Calculate derivatives of products and quotients of differentiable functions.
- FUN-3.B.2 Derivatives of quotients of differentiable functions can be found using the quotient rule.
- Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.
Topic 2.10
2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
Objectives in this topic
FUN-3.B—Calculate derivatives of products and quotients of differentiable functions—Topic 2.10
- FUN-3.B Calculate derivatives of products and quotients of differentiable functions.
- FUN-3.B.3 Rearranging tangent, cotangent, secant, and cosecant functions using identities allows differentiation using derivative rules.
- Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.