WEEK 25-CAPSTONE-PROJECT:-CALCULUS

Capstone Project: Calculus

Objectives

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Motion Analysis Lab

Profit Analysis Function

Growth Population

Summary

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Motion Analysis Lab

Let's analyze motion under friction

Set up:

150 newtons

Try it by yourself:

Try it by yourself:

Profit Analysis Function

ManuTech Industries Production

The company's accounting department discovered their profit follows a nonlinear pattern based on production volume, represented by the function:

ManuTech Industries, a mid-sized chemical manufacturing firm, has been struggling to optimize production levels.

P(x)=-2x³+150x²-3600x+50,000

Units produced (Hundreds)

x:

Try it by Yourself:

Graph the function and its derivative. Observe where the derivative equals zero. Zoom the graph.

f(x)=-2x³+150x²-3600x+50,000

Try it by yourself:

Growth Population Modeling

Let's analyze how rabbit population growth under fixed and varying conditions

Set up:

Try it by Yourself:

In a Table, create a table of each generation vs population of rabbits.How many times does the population growth for each generation?

Try it by yourself:

Summary: Solving Problem in Calculus

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MA.912.C.3 Apply derivatives to solve problems. MA.912.C.3.6 Sketch graphs by using first and second derivatives. Compare the corresponding characteristics of the graphs of f, f' and f". MA.912.C.3.8 Find average and instantaneous rates of change. Explain the instantaneous rate of change as the limit of the average rate of change. Interpret a derivative as a rate of change in applications, including velocity, speed and acceleration. MA.912.C.5 Apply integrals to solve problems. MA.912.C.5.1 Find specific antiderivatives using initial conditions, including finding velocity functions from acceleration functions, finding position functions from velocity functions and solving applications related to motion along a line. MA.912.C.5.4 Display a graphic representation of the solution to a differential equation by using slope fields, and locate particular solutions to the equation. MA.K12.MTR.1: Actively participate in effortful learning both individually and collectively. ELD.K12.ELL.MA: Language of Mathematics ELD.K12.ELL.MA.1 English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. MA.K12.MTR.7: Apply mathematics to real-world contexts.

MATERIAL

It is highly advised to have:

• Grid paper.
• Pencils of different colors.
• Eraser.
• A rule.
• A calculator.
• Geogebra installed on your phone/tablet/computer (or use online version).

Define the Problem

• Identify the physical, business, or biological scenario.
• Specify what needs to be analyzed (e.g., motion, profit, population).

Model the Situation Mathematically

Express the scenario as a function or equation (e.g., position vs. time, profit vs. production, population vs. generation).

Apply Calculus Tools

• Use derivatives to analyze rates of change (velocity, acceleration, profit change, population growth).
• Use integrals if total change or accumulation is needed.

Collect or Simulate Data

Gather relevant data points or use simulations to visualize the system.

Interpret Results

• Relate mathematical findings to the real-world scenario
• Draw conclusions about optimal values, growth rates, or system dynamics.

Graph and Analyze

• Plot the function and its either derivative(s) or integral(s).
• Identify critical points (where derivative = 0), inflection points, and behavior over time.