Math Problem Statement

Find the second derivative of the function given parametrically: x = a (cos t - ln(cot(t/2))) y = a sin t

Solution

To solve for the second derivative yxxy''_{xx} of the parametric equations provided:

Given: x=a(costln(cott2)),x = a \left( \cos t - \ln \left( \cot \frac{t}{2} \right) \right), y=asint,y = a \sin t,

we'll follow these steps:

  1. Find dx/dtdx/dt and dy/dtdy/dt:

    • Differentiate xx and yy with respect to tt.
  2. Find dydx\frac{dy}{dx}:

    • Use the chain rule: dydx=dydtdxdt\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}.
  3. Find d2ydx2\frac{d^2y}{dx^2}:

    • Differentiate dydx\frac{dy}{dx} with respect to xx, using the parametric relation.

Let me calculate this systematically. sympy as sp

Define variables

t, a = sp.symbols('t a') x = a * (sp.cos(t) - sp.ln(sp.cot(t / 2))) y = a * sp.sin(t)

First derivatives with respect to t

dx_dt = sp.diff(x, t) dy_dt = sp.diff(y, t)

dy/dx

dy_dx = dy_dt / dx_dt

Second derivative: d^2y/dx^2

d2y_dx2 = sp.diff(dy_dx, t) / dx_dt

Simplify the second derivative

d2y_dx2_simplified = sp.simplify(d2y_dx2)

d2y_dx2_simplifiedThe second derivative yxxy''_{xx} of the parametric equations is:

yxx=2(2sin(t)/tan(t/2)+2sin(t)/tan3(t/2)+4+2cos(t)/sin2(t/2)4/sin2(t/2)cos(t)/sin4(t/2))cot(t/2)a(2sin(t)cot(t/2)cot2(t/2)1)3y''_{xx} = \frac{2 \cdot \left( 2\sin(t)/\tan(t/2) + 2\sin(t)/\tan^3(t/2) + 4 + 2\cos(t)/\sin^2(t/2) - 4/\sin^2(t/2) - \cos(t)/\sin^4(t/2) \right) \cot(t/2)}{a \cdot \left( 2\sin(t)\cot(t/2) - \cot^2(t/2) - 1 \right)^3}

Would you like a step-by-step breakdown of this derivation? Here are some related questions to deepen your understanding:

  1. How is the chain rule applied to parametric equations?
  2. What is the importance of expressing the second derivative in terms of tt?
  3. Could simplifications make this formula easier to interpret for specific tt values?
  4. How does ln(cot(t/2))\ln(\cot(t/2)) influence the second derivative's complexity?
  5. Can this be plotted for specific aa values and ranges of tt?

Tip: When dealing with parametric equations, always confirm the domains of trigonometric and logarithmic functions to avoid undefined results.

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Math Problem Analysis

Mathematical Concepts

Parametric Equations
Derivatives
Trigonometric Functions
Logarithmic Functions

Formulas

dx/dt = derivative of x with respect to t
dy/dt = derivative of y with respect to t
dy/dx = (dy/dt) / (dx/dt)
d^2y/dx^2 = derivative of (dy/dx) with respect to t divided by (dx/dt)

Theorems

Chain Rule
Quotient Rule

Suitable Grade Level

University Level - Calculus II