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Magazine Problem 46
Graph the given curves on the same coordinate axes and describe the shape of the resulting figure. $$ \begin{array}{lll} C_{1}: & x=\frac{3}{2} \cos t+1, y=\sin t-1 ; & -\pi / 2 \leq t \leq \pi / 2 \\\ C_{2}: & x=\frac{3}{2} \cos t+1, y=\sin t+1 ; & -\pi / 2 \leq t \leq \pi / 2 \\\ C_{3}: & x=1, & y=2 \tan t: & -\pi / 4 \leq t \leq \pi / 4 \end{array} $$
Problem 47
Find an equation in \(x\) and \(y\) that has the same graph as the polar equation and use it to help sketch the graph in an \(r \theta\) -plane. $$ r=8 \sin \theta-2 \cos \theta $$
Problem 47
Graph the given curves on the same coordinate axes and describe the shape of the resulting figure. $$ \begin{array}{llll} C_{1}: & x=\tan t, & y=3 \tan t ; & & 0 \leq t \leq \pi / 4 \\ C_{2}: & x=1+\tan t, & y=3-3 \tan t ; & 0 \leq t \leq \pi / 4 \\ C_{3} ; & x=\frac{1}{2}+\tan t, & y=\frac{3}{2} ; & 0 \leq t \leq \pi / 4 \end{array} $$
Problem 48
Find an equation in \(x\) and \(y\) that has the same graph as the polar equation and use it to help sketch the graph in an \(r \theta\) -plane. $$ r=2 \cos \theta-4 \sin \theta $$
Problem 48
Graph the given curves on the same coordinate axes and describe the shape of the resulting figure. $$ \begin{array}{lll} C_{1}: & x=1+\cos t, & y=1+\sin t ; & \pi / 3 \leq t \leq 2 \pi \\ C_{2}: & x=1+\tan t, \quad y=1 ; & 0 \leq t \leq \pi / 4 \end{array} $$
Problem 49
Find an equation in \(x\) and \(y\) that has the same graph as the polar equation and use it to help sketch the graph in an \(r \theta\) -plane. $$ r=\tan \theta $$
Problem 50
Find an equation in \(x\) and \(y\) that has the same graph as the polar equation and use it to help sketch the graph in an \(r \theta\) -plane. $$ r=6 \cot \theta $$
Problem 51
Find the slope of the tangent line to the graph of the polar equation at the point corresponding to the given value of \(\theta\). $$ r=2 \cos \theta ; \quad \theta=\pi / 3 $$
Problem 52
Find the slope of the tangent line to the graph of the polar equation at the point corresponding to the given value of \(\theta\). $$ r=-2 \sin \theta ; \quad \theta=\pi / 6 $$
Problem 53
Find the slope of the tangent line to the graph of the polar equation at the point corresponding to the given value of \(\theta\). $$ r=4(1-\sin \theta) ; \quad \theta=0 $$
