Math Problem Statement
Solution
To solve for the second derivative of the parametric equations provided:
Given:
we'll follow these steps:
-
Find and :
- Differentiate and with respect to .
-
Find :
- Use the chain rule: .
-
Find :
- Differentiate with respect to , 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 of the parametric equations is:
Would you like a step-by-step breakdown of this derivation? Here are some related questions to deepen your understanding:
- How is the chain rule applied to parametric equations?
- What is the importance of expressing the second derivative in terms of ?
- Could simplifications make this formula easier to interpret for specific values?
- How does influence the second derivative's complexity?
- Can this be plotted for specific values and ranges of ?
Tip: When dealing with parametric equations, always confirm the domains of trigonometric and logarithmic functions to avoid undefined results.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
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
Related Recommendation
Find Second Derivative of Parametric Function x = e^t(sin t + cos t), y = e^t(cos t - sin t)
Finding the Second Derivative of a Parametric Function: x = 2sin(t), y = -3cos(t)
Calculate Second Derivative of Parametric Equations x(t) = t^3 + 2t^2, y(t) = 1 - 3t^2
Finding First and Second Derivatives of Parametric Trigonometric Equations
Second Derivative of Parametric Equations: ln(t) and t^3