You will have to show that you can complete calculus with parametric equations without eliminating the parameter. True False
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You can perform calculus operations on parametric equations without eliminating the parameter. Show more…
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True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. (a) True or False: Every function $y=f(x)$ can be written in terms of parametric equations. (b) True or False: Given parametric equations $x=x(t)$ and $y=y(t),$ the parameter can be eliminated to obtain the form $y=f(x)$ (c) True or False: Every parametric curve passes the vertical line test. (d) True or False: Every curve in the plane has a unique expression in terms of parametric equations. (e) True or False: If the functions $x=f(t)$ and $y=g(t)$ are differentiable for every $t \in \mathbb{R}$, then the parametric curve defined by $x$ and $y$ is differentiable for every value of $t$ (f) True or False: A curve parametrized by $x=x(t), y=$ $y(t)$ has a horizontal tangent line at $\left(x\left(t_{0}\right), y\left(t_{0}\right)\right)$ if $y^{\prime}(t)=0$ . (g) True or False: A curve parametrized by $x=x(t), y=$ $y(t)$ has a horizontal tangent line at $\left(x\left(t_{0}\right), y\left(t_{0}\right)\right)$ if $x^{\prime}(t) \neq 0$ and $y^{\prime}(t)=0$. (h) True or False: The cycloid curve associated with a circle of radius $r$ is made up of a series of semicircles of radius $r$.
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Are the statements true of false? Give an explanation for your answer. The curve given parametrically by $x=f(t)$ and $y=$ $g(t)$ has no sharp corners if $f$ and $g$ are differentiable.
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