_______________________________________________________________________________________________
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 Name & Learning Objective |
Essential Knowledge |
2.1 Defining Average Value and Instantaneous Rates of Change at a Point
LEARNING OBJECTIVE CHA-2.A Determine average rates of change using difference quotients. |
CHA-2.A.1 The difference quotients and express the average rate of change of a function over an interval. |
CHA-2.B.1 The instantaneous rate of change of a function at x = a can be expressed by or provided the limit exists. These are equivalent forms of the definition of the derivative and are denoted |
Blog Posts:
Local Linearity 1 The graphical manifestation of differentiability with pathological examples.
Local Linearity 2 Using local linearity to approximate the tangent line. A calculator exploration. Discovering the Derivative A graphing calculator exploration
The Derivative 1 Definition of the derivative.
The Derivative 2 Calculators and difference quotients
At Just the Right Time A good problem
What’s a Mean Old Average Anyway? Helping students understand the difference between the average rate of change of a function, the average value of a function, and the Mean Value theorem
_______________________________________________________________________________________________
2.2 Defining the Derivative of a Function and Using Derivative Notation
LEARNING OBJECTIVES CHA-2.B Represent the derivative of a function as the limit of a difference quotient. CHA-2.C Determine the equation of a line tangent to a curve at a given point. |
The derivative of f is the function whose value at x is , provided this limit exists. |
CHA-2.B.3 For , notations for the derivative include , , and . | |
CHA-2.B.4 The derivative can be represented graphically, numerically, analytically, and verballHA | |
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. |
Blog Posts:
Derivative Rules I Using the limit definition
Units The units of derivatives and integrals
Did He or Didn’t He? How Fermat found extreme values without derivatives.
_______________________________________________________________________________________________
2.3 Estimating Derivatives of a Function at a Point
LEARNING OBJECTIVE 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. |
Blog Posts:
Derivative Practice – Numbers Derivative from tables of numbers.
Derivative Practice – Graphs Derivative from graphs
_______________________________________________________________________________________________
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.
_______________________________________________________________________________________________
2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
LEARNING OBJECTIVE 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 f ‘. |
FUN-2.A.2 A continuous function may fail to be differentiable at a point in its domain. |
Blog Posts:
Differentiability Implies Continuity
_______________________________________________________________________________________________
ENDURING UNDERSTANDING
FUN-3 Recognizing opportunities to apply derivative rules can simplify differentiation.
_______________________________________________________________________________________________
2.5 Applying the Power Rule
LEARNING OBJECTIVE 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 |
Blog Posts:
Differentiation Techniques The definition, basics, product rule, Quotient Rule
_______________________________________________________________________________________________
2.6 Derivative Rules: Constant, Sum. Difference, and Constant Multiple
LEARNING OBJECTIVE 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. |
Blog Posts:
Differentiation Techniques The definition, basics, product rule, Quotient Rule
The Derivative Rules 1 (9-14-2012) Constants, sums and differences, powers.
_______________________________________________________________________________________________
ENDURING UNDERSTANDINGS
FUN-3 Recognizing opportunities to apply derivative rules can simplify differentiation.
LIM-3 Reasoning with definitions, theorems, and properties can be used to determine a limit.
_______________________________________________________________________________________________
2.7 Derivatives of cos x, sin x, e^{x}, and ln x
LEARNING OBJECTIVES FUN-3.A Calculate derivatives of familiar functions. LIM-3.A Interpret a limit as a definition of a derivative. |
FUN-3.A.4 Specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions. |
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. |
Blog Posts:
none
_______________________________________________________________________________________________
2.8 The Product Rule
LEARNING OBJECTIVE 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. |
Blog Posts:
Derivative Rules II the Product Rule
_______________________________________________________________________________________________
2.9 The Quotient Rule
LEARNING OBJECTIVE 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. |
Blog Posts:
Derivative Rules III the Quotient Rule
_______________________________________________________________________________________________
2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
LEARNING OBJECTIVE 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. |
Blog Posts:
none