Chapter 3 - Applications of Differentiation

u3.1: Maximum and Minimum Values

• Geometric understanding of graphs of continuous functions.
• Optimization, both global, and relative extrema.
• Critical values and closed interval method. Extreme Value Theorem, and Fermat's Theorem.

We learn how to analyze graphs of functions in great detail, and interpret the results. We’ll learn how calculus lets us determine exactly where graphs reach minimum and maximum values, and why these values are important.

Absolute and Relative Extrema

KHW: Extreme values from graphs

KHW: Extreme Value Theorem

Relative Extrema and Critical Values

 KHW: Critical Values

u3.2: The Mean Value Theorem

• The MVT and its geometric consequences.
• Rolle's Theorem, what it says, and when it's applicable.
• How to obtain information about a function from its derivative.

We’ll learn how to determine the average rate of change of a function using calculus, and what it means numerically and graphically.

Mean Value Theorem and Rolle's Theorem

 KHW: Mean Value Theorem

u3.3: How Derivatives Affect the Shape of a Graph

·    Relationship between increasing and decreasing behavior of f and the sign of f'.

·    Relationship between the concavity of f and sign of f". Points of Inflection as places where concavity changes.

·    Analysis of curves, including monotonicity and concavity.

·    Optimization.

We’ll explore how the shape of a graph and its rate of growth or decay are related. We’ll learn how to do this by just looking at the graph, and find out how to exactly determine where the rate of change itself changes.

The First Derivative Test

Concavity

KHW: Recognizing Concavity

Graphing Using Derivatives

KHW: Concavity and the Second Derivative

KHW: The Second Derivative Test

u3.4 & 3.5 : Limits at Infinity

·    Understanding horizontal and slant asymptotes in terms of graphical behavior.

·    Describing asymptotic behavior in terms of limits involving infinity.

We’ll study how functions behave as both the dependent and independent variables approach infinity.

u3.6: Graphing with Calculus and Calculators

·    With the aid of technology, graphs of functions are easy to produce.

·    The emphasis on the interplay between geometric and analytic information and the use of calculus both predict and explain the local and global behavior of a function.

·    The use of graphing calculators for estimation of local extrema and inflection points, contrasted with calculus for precise computation of such points.

·    The need for special care when using graphing technology.

We’ll learn how to use technology with calculus to find solutions to interesting problems. We’ll learn how to use technology appropriately, and understand common problems we can run into if we simply rely on the machine for everything.

u3.7: Optimization Problems

·    Setting up and solving optimization problems.

Having studied maxima and minima of functions and graphs, we’ll now apply these skills to finding the optimum solutions to real world problems. We will re-interpret min-max problems in the context of the real world.

Modeling and Optimizing Functions

KHW: Optimization