Chapter 10 - Section 10.2 - Calculus with Parametric Curves - 10.2 Exercises - Page 655: 28

(a) The slope of the tangent line is $~~-tan~\theta$ (b) The tangent line is horizontal when $~~\theta = \pi~n~~~$ where $n$ is an integer The tangent line is vertical when $~~\theta = \frac{\pi}{2}+\pi~n~~~$ where $n$ is an integer (c) The tangent line has a slope of $1$ when $~~\theta = \frac{3\pi}{4}+\pi~n~~~$ where $n$ is an integer The tangent line has a slope of $-1$ when $~~\theta = \frac{\pi}{4}+\pi~n~~~$ where $n$ is an integer

(a) We can find $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{3asin^2~\theta~cos~\theta}{3a~cos^2~\theta~(-sin~\theta)} = -\frac{sin~\theta}{cos~\theta} = -tan~\theta$ Since the slope of the tangent line is $\frac{dy}{dx}$, the slope of the tangent line is $-tan~\theta$ (b) The slope is $0$ when the tangent line is horizontal. We can find $\theta$ when $\frac{dy}{dx} = 0$: $-tan~\theta = 0$ $sin~\theta = 0$ $\theta = \pi~n~~~$ where $n$ is an integer The slope is undefined when the tangent line is vertical. We can find $\theta$ when $\frac{dy}{dx}$ is undefined: $-tan~\theta$ is undefined $cos~\theta = 0$ $\theta = \frac{\pi}{2}+\pi~n~~~$ where $n$ is an integer (c) We can find $\theta$ when $\frac{dy}{dx} = 1$: $-tan~\theta = 1$ $tan~\theta = -1$ $sin~\theta = -cos~\theta$ $\theta = \frac{3\pi}{4}+\pi~n~~~$ where $n$ is an integer We can find $\theta$ when $\frac{dy}{dx} = -1$: $-tan~\theta = -1$ $tan~\theta = 1$ $sin~\theta = cos~\theta$ $\theta = \frac{\pi}{4}+\pi~n~~~$ where $n$ is an integer The tangent line has a slope of $1$ when $\theta = \frac{3\pi}{4}+\pi~n~~~$ where $n$ is an integer The tangent line has a slope of $-1$ when $\theta = \frac{\pi}{4}+\pi~n~~~$ where $n$ is an integer