Answer to Question #263706 in Calculus for Alunsina

Question

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

Answer 1

5.

(a)

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

(b)

(c)

$$\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}=\dfrac{3t^2}{2t}=\dfrac{3}{2}t, t\not=0$$

(d)

$$t=\sqrt{x-1}, t\geq0$$

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

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

6.

(a)

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

(b)

(c)

$$\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}=\dfrac{2}{\dfrac{5}{2\sqrt{t}}}=\dfrac{4}{5}\sqrt{t}, t>0$$

(d)

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

$$y(x)=\dfrac{2}{5}x^2+2, x\geq0$$

7.

(a)

$$\def\arraystretch{1.5} \begin{array}{c:c:c} t & x & y \\ \hline 0 & 1 & 1 \\ \pi/6 & \sqrt{3}/2 & 5/4 \\ \pi/4 & \sqrt{2}/2 & 3/2 \\ \pi/3 & 1/2 & 7/4 \\ \pi/2 & 0 & 2 \\ 2\pi/3 & -1/2 & 7/4 \\ 3\pi/4 & -\sqrt{2}/2 & 3/2 \\ 5\pi/6 & -\sqrt{3}/2 & 5/4\\ \pi & -1 & 1 \\ \end{array}$$

(b)

(c)

$$\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}=\dfrac{2\sin t\cos t}{-\sin t}=-2\cos t, t\not=\pi n, n\in \Z$$

(d)

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