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Motion and Trigonometric Derivatives

Sep 10, 2025

Overview

This lecture covers motion problems using derivatives, focusing on when acceleration is zero and when a particle is at rest. It then introduces the derivatives of trigonometric functions, explores their patterns, and demonstrates their applications, including examples and connections to physical phenomena like simple harmonic and damped motion.

Motion and Derivatives

  • The position of a particle is given by ( S(T) = T^D - 11T^3 + 27T^2 ), where ( D > 0 ) is a parameter to be determined.
  • Acceleration is the second derivative of position: ( S''(T) ).
  • Given ( S''(1) = 0 ), substitute ( T = 1 ) into the second derivative to solve for ( D ):
    • ( S'(T) = D T^{D-1} - 33T^2 + 54T )
    • ( S''(T) = D(D-1)T^{D-2} - 66T + 54 )
    • At ( T = 1 ): ( D(D-1) - 66 + 54 = 0 ) → ( D^2 - D - 12 = 0 )
    • Factor: ( (D-4)(D+3) = 0 ) → ( D = 4 ) (since ( D > 0 ))
  • Substitute ( D = 4 ) into the derivatives:
    • ( S'(T) = 4T^3 - 33T^2 + 54T )
    • ( S''(T) = 12T^2 - 66T + 54 )
  • To find when acceleration is zero, set ( S''(T) = 0 ):
    • ( 12T^2 - 66T + 54 = 0 )
    • Factor out 6: ( 2T^2 - 11T + 9 = 0 )
    • Factor: ( (T-1)(2T-9) = 0 ) → ( T = 1 ) and ( T = \frac{9}{2} )
  • To find when the particle is at rest, set ( S'(T) = 0 ):
    • ( 4T^3 - 33T^2 + 54T = 0 )
    • Factor out ( T ): ( T(4T^2 - 33T + 54) = 0 )
    • Factor quadratic: ( (4T-9)(T-6) = 0 )
    • Solutions: ( T = 0, \frac{9}{4}, 6 )
  • Summary:
    • Acceleration is zero at ( T = 1 ) and ( T = \frac{9}{2} ).
    • The particle is at rest at ( T = 0, \frac{9}{4}, 6 ).
  • The process demonstrates how to use derivatives to analyze motion, find critical points, and interpret physical meaning (e.g., rest and acceleration).

Derivatives of Trigonometric Functions

  • The derivative of ( \sin(x) ) is ( \cos(x) ).
  • The derivative of ( \cos(x) ) is ( -\sin(x) ).
  • Higher-order derivatives of sine and cosine repeat every four derivatives:
    • 1st: ( \sin(x) \to \cos(x) )
    • 2nd: ( \cos(x) \to -\sin(x) )
    • 3rd: ( -\sin(x) \to -\cos(x) )
    • 4th: ( -\cos(x) \to \sin(x) ) (cycle repeats)
  • This four-step pattern means that, for example, the 107th derivative of ( \sin(x) ) is the same as the 3rd derivative, which is ( -\cos(x) ).
  • The periodic nature of these derivatives reflects the periodicity of sine and cosine themselves, which is important in modeling real-world periodic phenomena.
  • The derivative of ( \sin(2x) ) is ( 2\cos(2x) ), using the double angle identity and the product rule.
  • For products involving trigonometric functions, use the product rule:
    • Example: ( \frac{d}{dx}[x^2\sin(x)] = 2x\sin(x) + x^2\cos(x) )
  • When differentiating sums or more complex expressions, apply the product rule and combine like terms as needed. Sometimes, terms will cancel or combine in a way that simplifies the result.

Other Trigonometric Derivatives

  • The derivative of ( \tan(x) ) is ( \sec^2(x) ).
    • Derived using the quotient rule: ( \tan(x) = \frac{\sin(x)}{\cos(x)} ).
  • The derivative of ( \cot(x) ) is ( -\csc^2(x) ).
    • ( \cot(x) = \frac{\cos(x)}{\sin(x)} ).
  • The derivative of ( \sec(x) ) is ( \sec(x)\tan(x) ).
    • ( \sec(x) = \frac{1}{\cos(x)} ).
  • The derivative of ( \csc(x) ) is ( -\csc(x)\cot(x) ).
    • ( \csc(x) = \frac{1}{\sin(x)} ).
  • Among these, the derivatives of tangent and secant are most commonly used and should be memorized.
  • These derivatives are often derived using the quotient rule or by expressing the function in terms of sine and cosine and then applying the appropriate rules.

Application Examples

  • To find higher-order derivatives, repeatedly apply the product, chain, and quotient rules as needed.
    • Example: The third derivative of ( \tan(\theta) ) involves differentiating ( \tan(\theta) ) to get ( \sec^2(\theta) ), then differentiating again using the product rule, and so on:
      • ( f(\theta) = \tan(\theta) )
      • ( f'(\theta) = \sec^2(\theta) )
      • ( f''(\theta) = 2\sec^2(\theta)\tan(\theta) )
      • ( f'''(\theta) = 4\sec^2(\theta)\tan^2(\theta) + 2\sec^4(\theta) )
  • To find the tangent line to a curve involving trigonometric functions:
    • Find the y-coordinate by evaluating the function at the given x-value.
    • Find the slope by evaluating the derivative at the same x-value.
    • Example: For ( y = 6\tan(x) - \sqrt{2}\sec(x) ) at ( x = \frac{\pi}{4} ):
      • ( \tan(\frac{\pi}{4}) = 1 ), ( \sec(\frac{\pi}{4}) = \sqrt{2} )
      • ( y = 6 \times 1 - \sqrt{2} \times \sqrt{2} = 6 - 2 = 4 )
      • Derivative: ( y' = 6\sec^2(x) - \sqrt{2}\sec(x)\tan(x) )
      • At ( x = \frac{\pi}{4} ): ( y' = 6 \times 2 - \sqrt{2} \times \sqrt{2} \times 1 = 12 - 2 = 10 )
      • Equation of tangent line: ( y - 4 = 10(x - \frac{\pi}{4}) )
  • When differentiating more complex expressions, such as sums or products of trigonometric and polynomial terms, use the product rule for each term and combine like terms. Sometimes, terms will cancel, simplifying the result.

Simple Harmonic Motion and Damped Motion

  • Hooke's Law for springs: ( S'' = -kS ), where ( S ) is displacement and ( k ) is a constant.
    • The general solution is sinusoidal: ( S(t) = A\sin(\sqrt{k}t) + B\cos(\sqrt{k}t) ).
    • The second derivative is the negative of the function times a constant, leading to oscillatory (periodic) motion.
    • The position, velocity, and acceleration are all sinusoidal and related by phase shifts.
  • Damped motion occurs when an additional factor (like friction) causes the amplitude to decrease over time.
    • Example: ( S(t) = e^{-t}\sin(t) )
      • The exponential factor ( e^{-t} ) causes the oscillations to decrease in amplitude over time.
      • The relationship between position and acceleration is no longer exactly opposite, leading to the "damping" effect.
      • In graphs, the motion (position) and acceleration are out of sync, and the oscillations diminish over time.
    • The derivatives of damped motion functions involve both the product and chain rules, and the resulting expressions show how the amplitude decreases and the phase relationship changes.
  • These models are important for understanding real-world systems where periodic motion is affected by resistance or friction, such as springs, pendulums, and other oscillating systems.

Key Terms & Definitions

  • Derivative: Measures the rate of change of a function.
  • Second Derivative (Acceleration): The derivative of velocity; indicates acceleration in motion problems.
  • Product Rule: The derivative of ( f(x)g(x) ) is ( f'(x)g(x) + f(x)g'(x) ).
  • Quotient Rule: The derivative of ( \frac{f(x)}{g(x)} ) is ( \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2} ).
  • Periodic Function: A function that repeats its values at regular intervals.
  • Simple Harmonic Motion: Motion described by sine or cosine functions, resulting from a restoring force proportional to displacement.
  • Damped Motion: Oscillatory motion where amplitude decreases over time due to an external factor (e.g., friction).
  • Tangent Line: A line that touches a curve at a point and has the same slope as the curve at that point.
  • Critical Point: A point where the derivative is zero or undefined, often corresponding to rest or a change in direction in motion problems.

Action Items / Next Steps

  • Practice finding first and second derivatives of trigonometric and composite functions.
  • Review and memorize the derivatives of sine, cosine, tangent, and secant, as well as their patterns and cycles.
  • Work through assigned problems on motion and trigonometric derivatives, including higher-order derivatives and tangent lines.
  • Explore more examples involving tangent lines, higher-order derivatives, and applications to physical systems.
  • Prepare for future topics, such as anti-derivatives and more complex applications in calculus and differential equations, including modeling with periodic and damped functions.

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