Calculus: Particle Motion and Theorems

Particle Motion

• Position, velocity, and acceleration are related through derivatives.
• Velocity is the derivative of position ($x'(t) = v(t)$).
• Acceleration is the derivative of velocity ($x''(t) = v'(t) = a(t)$).
• A particle is at rest when its velocity is zero.
• Average acceleration is calculated using the average rate of change of velocity.

Mean Value Theorem (MVT)

• The MVT states that for a function continuous on a closed interval [a, b], there exists a value 'c' where the instantaneous rate of change (tangent line) at 'c' is equal to the average rate of change (secant line) between points 'a' and 'b'.
• The MVT can be applied to find a specific value 'c' within a given interval.

Rolle's Theorem

• A special case of the MVT.
• If a function has two points with the same y-value, then there must be at least one point between them where the derivative (and therefore the tangent line) is zero.

Local Linear Approximation

• Uses a tangent line to approximate the value of a function close to a known point.
• The approximation is more accurate for points closer to the known point.
• Formula: $f(x) \approx f(a) + f'(a)(x-a)$, where 'a' is the known point and 'x' is the point being approximated.

L'Hopital's Rule

• Used to evaluate limits that result in indeterminate forms (0/0, ∞/∞).
• States: If the limit as x approaches c of f(x)/g(x) is of the form 0/0 or ∞/∞, then the limit is equal to the limit as x approaches c of f'(x)/g'(x).
• L'Hopital's rule is not the quotient rule.

Related Rates

• Relates different rates of change to each other.
• Often involves finding a relationship between variables and differentiating it with respect to time (t).
• Understanding differentials (infinitely small changes in variables) is crucial.
• Related Rates problems typically involve finding rates of change of variables related by a geometric formula.
• Differentiation in Related Rates problems is always with respect to time (t), not with respect to other variables like x or y.
• Derivatives represent rates of change.

Geometric Formulas

• Memorize geometric formulas for various shapes:
  • Area of a circle
  • Area of a sphere
  • Surface area of a sphere
  • Circumference of a circle
  • Area of a cone
  • Surface area of a cone
  • Area of a cylinder
  • Pythagorean theorem: a² + b² = c²
  • SOHCAHTOA: sine = opposite / hypotenuse
• Lamppost Problem: A man walks away from a lamppost, casting a shadow. The problem finds how fast the shadow lengthens.
  • Variables: Distance (A) from lamppost, rate of change of distance (dA/dt) (man's speed), shadow length (S), rate of change of shadow length (dS/dt).
  • Solution approach: Draw a diagram, recognize similar triangles, find the proportionality constant, and use rates being proportional. Solve for dS/dt.
• Circle Problem: A circle's radius increases. The problem finds how fast the circumference changes.
  • Variables: Radius (r), rate of change of radius (dr/dt), circumference (C), rate of change of circumference (dC/dt).
  • Solution approach: Write the equation (C = 2πr), differentiate wrt t, plug in given values, solve for dC/dt.
• General Tips:
  • Include units in answers.
  • Establish equations relating variables.
  • Practice problems to master related rates and the connections between derivatives and rates of change.

Explore the concepts of particle motion, derivatives, and key theorems in calculus such as the Mean Value Theorem and Rolle's Theorem. Understand how velocity, acceleration, and average rate of change relate to particle motion. This quiz also covers local linear approximation and its significance in calculus.