Second derivatives of parametric curves practice problems
dy/dx and d²y/dx² for a parametric curve, without eliminating the parameter. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
For \( \displaystyle x = e^{t} \), \( \displaystyle y = t e^{t} \), find \( \displaystyle \frac{dy}{dx} \) and \( \displaystyle \frac{d^2y}{dx^2} \) at \( \displaystyle t = 1 \).
For \( \displaystyle x = 2 \cos{\left(t \right)} \), \( \displaystyle y = 3 \sin{\left(t \right)} \), find \( \displaystyle \frac{dy}{dx} \) and \( \displaystyle \frac{d^2y}{dx^2} \) at \( \displaystyle t = \frac{\pi}{4} \).
For \( \displaystyle x = - t^{2} + t \), \( \displaystyle y = t^{2} + 1 \), find \( \displaystyle \frac{dy}{dx} \) and \( \displaystyle \frac{d^2y}{dx^2} \) at \( \displaystyle t = 1 \).
For \( \displaystyle x = \sec{\left(t \right)} \), \( \displaystyle y = \tan{\left(t \right)} \), find \( \displaystyle \frac{dy}{dx} \) and \( \displaystyle \frac{d^2y}{dx^2} \) at \( \displaystyle t = \frac{\pi}{3} \).
For \( \displaystyle x = - t^{2} + t \), \( \displaystyle y = t^{2} + 1 \), find \( \displaystyle \frac{dy}{dx} \) and \( \displaystyle \frac{d^2y}{dx^2} \) at \( \displaystyle t = -1 \).
For \( \displaystyle x = t^{3} \), \( \displaystyle y = t^{2} + 2 t \), find \( \displaystyle \frac{dy}{dx} \) and \( \displaystyle \frac{d^2y}{dx^2} \) at \( \displaystyle t = 3 \).
For \( \displaystyle x = - t^{2} + t \), \( \displaystyle y = t^{2} + 1 \), find \( \displaystyle \frac{dy}{dx} \) and \( \displaystyle \frac{d^2y}{dx^2} \) at \( \displaystyle t = 2 \).
For \( \displaystyle x = \sec{\left(t \right)} \), \( \displaystyle y = \tan{\left(t \right)} \), find \( \displaystyle \frac{dy}{dx} \) and \( \displaystyle \frac{d^2y}{dx^2} \) at \( \displaystyle t = \frac{\pi}{4} \).
For \( \displaystyle x = e^{t} \), \( \displaystyle y = t e^{t} \), find \( \displaystyle \frac{dy}{dx} \) and \( \displaystyle \frac{d^2y}{dx^2} \) at \( \displaystyle t = -1 \).
For \( \displaystyle x = - t^{2} + t \), \( \displaystyle y = t^{2} + 1 \), find \( \displaystyle \frac{dy}{dx} \) and \( \displaystyle \frac{d^2y}{dx^2} \) at \( \displaystyle t = 3 \).