Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

Key Concepts

This exercise focuses on the following concepts:

• Continuity: Determining parameters to ensure a function is continuous at a point.

• Related Rates: Calculating the rate of change of one quantity based on the rates of change of related variables.

• Differentials: Using $dy\approx f^{\prime }(x)dx$ to estimate changes and propagate errors in measurements.

• Optimization: Finding maximum or minimum values of functions (area, volume, product) under given constraints.

• Applications in Physics: Kinematics (velocity/acceleration) and Einstein's theory of relativity.

• Applications in Economics: Revenue, cost functions, and continuous compounding of inflation.

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Important Formulas

Below are the key formulas used in this exercise:

| Pythagorean Theorem | $x^2+y^2=z^2⟹2x\frac{dx}{dt}+2y\frac{dy}{dt}=2z\frac{dz}{dt}$ |

| Equilateral Triangle Area | $A=\frac{\sqrt{3}}{4}{s}^2$ |

| Cube Diagonal | $D=\sqrt{3}s$ |

| Continuous Compounding | $A=P{e}^{rt}$ | | Optimization | Set $f^{\prime }(x)=0$ and check boundaries or second derivative |

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Summary

This exercise provides a comprehensive review of the practical applications of derivatives. Key learnings include the use of Related Rates to solve geometric problems involving moving objects, the application of Optimization to find efficient dimensions in construction and mathematics, and the use of Differentials for error analysis.

Strategies involve identifying the primary equation relating variables, differentiating with respect to time or a specific variable, and substituting known values to find the desired rate or extremum.