K12MATH013: Calculus AB

Upon successful completion of this unit, you will be able to:

• Demonstrate understanding of the derivative in the following ways: graphically, numerically, and analytically.
• Demonstrate understanding of the derivative as a rate of change
• Find the derivative by using the limit of a difference quotient.
• Solve problems that involve the relationship between differentiability and continuity.
• Compute derivatives of basic functions (power, exponential, logarithmic, trigonometric, inverse trigonometric, etc.).
• Compute derivatives of sums, products, and quotients of functions.
• Compute more complex derivatives by use of the Chain Rule and implicit differentiation.
• Find the slope of a curve at a point using the derivative.
• Find the tangent line to a curve at a point.
• Use local linear approximation to estimate the value of a function near a given point.
• Demonstrate that the instantaneous rate of change is the limit of the average rate of change of a function.
• Obtain an approximate rate of change when given either graphs or tables of values.
• Explain the relationship between characteristics of the graphs of f and its derivative f'.
• Define and apply the relationship between the increasing/decreasing behavior of f and the sign of f'.
• Explain the Mean Value Theorem verbally and interpret it geometrically.
• Write and solve equations that model word problems involving change, using the derivative to represent change.
• Explain the relationship between characteristics of the graphs of f, its derivative f', and its second derivative f''.
• Explain the relationship between the concavity of a function f and the sign of its second derivative f''.
• Explain points of inflection and how they relate to concavity, and find points of inflection given certain information.
• Use derivatives to analyze curves, specifically using the concepts of monotonicity and concavity.
• Use derivatives to solve optimization problems involving both absolute and relative extrema.
• Use derivatives to model rates of change, given certain information, and to solve related rates problems
• Find the derivative of an inverse function by using implicit differentiation.
• Interpret and solve problems involving rates of change using derivatives in various applied contexts, including velocity, speed, and acceleration.
• Interpret the derivative in a geometric context and apply this interpretation to slope fields and solution curves for differential equation.