Mathematics, 07.04.2020 21:59 george8396
N exponential spiral has parametric equationr(t) =〈etcost, etsint〉.(a) Plotr(t). Make sure your plot includeslimt→−[infinity]r(t).(b) Find the unit tangent vectorT(t)tor(t).(c) Find the length ofr(t)for−[infinity]< t≤0.(d) Letf(x, y) =−12ln(x2+y2)−arctanxy. Find∇f(x, y).(e) Compute the angle between∇f(r(t))andr′(t).
Answers: 2
![L=\int_{-\inf}^{0}\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2+(\frac{dz}{dt})^2}dt\\\\L=\int_{-inf}^0e^t\sqrt{2}=\sqrt{2}[e^{0}-e^{-inf}]=\sqrt{2}](/tpl/images/0587/5084/abd0e.png)
![|\bigtriangledown f(x,y)|=\sqrt{(\frac{-24}{x^2+y^2}-\frac{y}{1+x^2y^2})^2+(\frac{-24}{x^2+y^2}-\frac{x}{1+x^2y^2})^2}\\\\\theta=cos^{-1}[\frac{\bigtriangledown f(x,y) \cdot r'(t)}{|\bigtriangledown f(x,y)||r'(t)|}]](/tpl/images/0587/5084/9d266.png)
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N exponential spiral has parametric equationr(t) =〈etcost, etsint〉.(a) Plotr(t). Make sure your plot...
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