Essential Understanding 2.3

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

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Applications of derivatives are really fun.

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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.