Answer to Question #263706 in Calculus for Alunsina

Question #263706

In numbers 5-7,

(a) make a table of values of t, x and y based on the specified intervals of the given parametic equations

(b) sketch the plane curve defined by the given parametic equations at the specified interval

(c) find dy/dx without eliminating the parameter t. simplify your answers

(d) find an equation such that the parameter t is eliminated


5. x=t²+1 and y=t³-2 for 0≤t≤2

6. x=sqrt(5t) and y=2t+2 for 0≤t≤5

7. x=cost and y=2-cos²t for 0≤t≤π


1
Expert's answer
2021-11-17T00:34:24-0500

5.

(a)


txy0120.21.041.9920.41.161.9360.61.361.7840.81.641.4881211.22.440.2721.42.960.7441.63.562.0961.84.243.832256\def\arraystretch{1.5} \begin{array}{c:c:c} t & x & y \\ \hline 0 & 1 & -2 \\ 0.2 & 1.04 & -1.992 \\ 0.4 & 1.16 & -1.936 \\ 0.6 & 1.36 & -1.784 \\ 0.8 & 1.64 & -1.488 \\ 1 & 2 & -1 \\ 1.2 & 2.44 & -0.272 \\ 1.4 & 2.96 & 0.744\\ 1.6 & 3.56 & 2.096 \\ 1.8 & 4.24 & 3.832 \\ 2 & 5 & 6 \\ \end{array}

(b)




(c)


dydx=dy/dtdx/dt=3t22t=32t,t0\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}=\dfrac{3t^2}{2t}=\dfrac{3}{2}t, t\not=0

(d)


t=x1,t0t=\sqrt{x-1}, t\geq0

t=x1,t<0t=-\sqrt{x-1}, t<0

y(x)={(x1)3/22t<0(x1)3/22t0y(x)= \begin{cases} -(x-1)^{3/2}-2 &t<0 \\ (x-1)^{3/2}-2 &t\geq0 \end{cases}

6.


(a)


txy0020.52.531541.57.5521062.512.5731583.517.59420104.522.5115512\def\arraystretch{1.5} \begin{array}{c:c:c} t & x & y \\ \hline 0 & 0 & 2 \\ 0.5 & \sqrt{2.5} & 3 \\ 1 & \sqrt{5} & 4 \\ 1.5 & \sqrt{7.5} & 5 \\ 2 & \sqrt{10} & 6 \\ 2.5 & \sqrt{12.5} & 7 \\ 3 & \sqrt{15} & 8 \\ 3.5 & \sqrt{17.5} & 9\\ 4 & \sqrt{20} & 10 \\ 4.5 & \sqrt{22.5} & 11 \\ 5 & 5 & 12 \\ \end{array}


(b)



(c)


dydx=dy/dtdx/dt=252t=45t,t>0\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}=\dfrac{2}{\dfrac{5}{2\sqrt{t}}}=\dfrac{4}{5}\sqrt{t}, t>0

(d)


t=x25,x0t=\dfrac{x^2}{5}, x\geq0

y(x)=25x2+2,x0y(x)=\dfrac{2}{5}x^2+2, x\geq0

7.

(a)


txy011π/63/25/4π/42/23/2π/31/27/4π/2022π/31/27/43π/42/23/25π/63/25/4π11\def\arraystretch{1.5} \begin{array}{c:c:c} t & x & y \\ \hline 0 & 1 & 1 \\ \pi/6 & \sqrt{3}/2 & 5/4 \\ \pi/4 & \sqrt{2}/2 & 3/2 \\ \pi/3 & 1/2 & 7/4 \\ \pi/2 & 0 & 2 \\ 2\pi/3 & -1/2 & 7/4 \\ 3\pi/4 & -\sqrt{2}/2 & 3/2 \\ 5\pi/6 & -\sqrt{3}/2 & 5/4\\ \pi & -1 & 1 \\ \end{array}

(b)




(c)


dydx=dy/dtdx/dt=2sintcostsint=2cost,tπn,nZ\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}=\dfrac{2\sin t\cos t}{-\sin t}=-2\cos t, t\not=\pi n, n\in \Z

(d)

y(x)=2x2,1x1y(x)=2-x^2, -1\leq x\leq1


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