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Calculus 1

Module 8:

Contextual Applications of Derivatives

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Affirmations

My time is valuable
Failure is great feedback
I am a good role model for others

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Limits at Infinity and Asymptotes

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Learning Goals

Deepen your understanding and form connections within these skills:

Determine limits and predict how functions behave as x increases or decreases indefinitely
Identify and distinguish horizontal and slanting lines that a graph approaches but never touches
Use a function’s derivatives to accurately sketch its graph

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Limit at Infinity (Informal)

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Limit at Infinity (Formal)

We say a function f has a limit at infinity, if there exists a real number L such that for all ε > 0, there exists N > 0 such that | f ( x ) − L | < ε for all x > N. In that case, we write

We say a function f has a limit at negative infinity if there exists a real number L such that for all ε > 0 , there exists N < 0 such that | f ( x ) − L | < ε for all x < N. In that case, we write

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Infinite Limit at Infinity (Informal)

We say a function f has an infinite limit at infinity and write

if f ( x ) becomes arbitrarily large for x sufficiently large. We say a function has a negative infinite limit at infinity and write:

if f ( x ) < 0 and | f ( x ) | becomes arbitrarily large for x sufficiently large. Similarly we can define limits as x→−∞.

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Infinite Limit at Infinity (Formal)

We say a function f has an infinite limit at infinity and write

if for all M > 0, there exists an N > 0 such that f ( x ) > M for all x > N. We say a function has a negative infinite limit at infinity and write

if for all M < 0, there exists an N > 0 such that f ( x ) < M for all x > N. Similarly we can define limits as x→−∞.

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Horizontal Asymptote

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Determine the Horizontal Asymptote

Example:

Solution:

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End Behavior

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Evaluating Limits of Power Functions

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Try It!

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End Behavior for Rational Functions

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Try It!

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Guidelines for graphing a function

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Applied Optimization Problems

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Learning Goals

1 Tackle business problems to find the best ways to increase profits, minimize costs, or maximize revenue

2 Use optimization methods to solve problems involving geometry

Deepen your understanding and form connections within these skills:

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Optimization

Optimization often involves finding the maximum or minimum value of a function under certain constraints.

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Try It!

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Solving Optimization Problems

Identify All Variables: If applicable, sketch the problem scenario and label all variables.
Determine Objective: Identify which quantity needs to be maximized or minimized, and specify the range of values for any other relevant variables.
Develop a Formula: Write a formula for the objective quantity in terms of the variables, which may involve multiple variables.
Formulate Equations: Relate independent variables with any equations necessary to express the objective quantity as a function of one variable.
Set Domain: Determine the domain of consideration for the function based on the practical constraints of the problem.
Find Extremes: Calculate the maximum or minimum value of the function, typically by identifying critical points and evaluating the function at endpoints.

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L’Hôpital’s Rule

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Learning Goals

Deepen your understanding and form connections within these skills:

Spot indeterminate forms like in calculations, and use L’Hôpital’s rule to find precise values
Explain how quickly different functions increase or decrease compared to each other

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L’Hôpital’s Rule (0/0 case)

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Try It!

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L’Hôpital’s Rule (∞/∞ case)

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Try It!

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Newton’s Method

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Learning Goals

Explain how Newton’s method uses repetition to find roots of equations
Recognize when Newton’s method does not work
Apply methods that repeat steps to solve different types of mathematical problems

Deepen your understanding and form connections within these skills:

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Newton’s Method

Newton’s Method is an efficient numerical technique used to find approximately accurate roots of a real-valued function. By starting from an initial guess, the method iteratively refines this guess using the function and its derivative, quickly converging to a root where the function value is zero.

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Try It!

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Failures of Newton’s Method

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Try It!

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Antiderivatives

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Learning Goals

1 Understand indefinite integrals and learn how to find basic antiderivatives for functions

2 Use the rule for integrating functions raised to a power

3 Use antidifferentiation to solve simple initial-value problems

Deepen your understanding and form connections within these skills:

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Antiderivative

A function F is an antiderivative of the function f if F′ ( x ) = f ( x ) for all x in the domain of f.

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General Form of an Antiderivative

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Try It!

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Indefinite Integral

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Power Rule for Integrals

For n≠−1,

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Properties of Indefinite Untegrals

Let F and G be antiderivatives of f and g, respectively, and let k be any real number.

Sums and Differences:

Constant Multiples:

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Evaluating Indefinite Integrals

Example:

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Powering the Future with Calculus Activity

Work in groups to optimize the design and placement of a solar farm using calculus.

Apply concepts like limits, optimization, L'Hôpital's Rule, Newton's Method, and antiderivatives to solve the problems.

Collaborate with your group to solve each part of the challenge, and prepare to present your solutions.

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