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Mathematics LibreTexts

4.2E:

  • Page ID
    18597
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    Calculus of Parametric Curves

    For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.

    1) \(\displaystyle x=3+t,y=1−t\)

    2) \(\displaystyle x=8+2t,y=1\)

    Solution: 0

    3) \(\displaystyle x=4−3t,y=−2+6t\)

    4) \(\displaystyle x=−5t+7,y=3t−1\)

    Solution: \(\displaystyle \frac{−3}{5}\)

     

    For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter.

    5) \(\displaystyle x=3sint,y=3cost,t=\frac{π}{4}\)

    6) \(\displaystyle x=cost,y=8sint,t=\frac{π}{2}\)

    Solution: \(\displaystyle Slope=0; y=8.\)

    7) \(\displaystyle x=2t,y=t^3,t=−1\)

    8) \(\displaystyle x=t+\frac{1}{t},y=t−\frac{1}{t},t=1\)

    Solution: Slope is undefined; \(\displaystyle x=2\).

    9) \(\displaystyle x=\sqrt{t},y=2t,t=4\)

     

    For the following exercises, find all points on the curve that have the given slope.

    10) \(\displaystyle x=4cost,y=4sint,\) slope = 0.5

    Solution: \(\displaystyle t=arctan(−2); (\frac{4}{\sqrt{5}},\frac{−8}{\sqrt{5}})\).

    11) \(\displaystyle x=2cost,y=8sint,slope=−1\)

    12) \(\displaystyle x=t+\frac{1}{t},y=t−\frac{1}{t},slope=1\)

    Solution: No points possible; undefined expression.

    13) \(\displaystyle x=2+\sqrt{t},y=2−4t,slope=0\)

     

    For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t.

    14) \(\displaystyle x=e^{\sqrt{t}},y=1−lnt^2,t=1\)

    Solution: \(\displaystyle y=−(\frac{2}{e})x+3\)

    15) \(\displaystyle x=tlnt,y=sin^2t,t=\frac{π}{4}\)

    16) \(\displaystyle x=e^t,y=(t−1)^2,at(1,1)\)

    Solution: \(\displaystyle y=2x−7\)

    17) For \(\displaystyle x=sin(2t),y=2sint\) where \(\displaystyle 0≤t<2π.\) Find all values of t at which a horizontal tangent line exists.

    18) For \(\displaystyle x=sin(2t),y=2sint\) where \(\displaystyle 0≤t<2π\). Find all values of t at which a vertical tangent line exists.

    Solution: \(\displaystyle \frac{π}{4},\frac{5π}{4},\frac{3π}{4},\frac{7π}{4}\)

    19) Find all points on the curve \(\displaystyle x=4cos(t),y=4sin(t)\) that have the slope of \(\displaystyle \frac{1}{2}\).

    20) Find \(\displaystyle \frac{dy}{dx}\) for \(\displaystyle x=sin(t),y=cos(t)\).

    Solution: \(\displaystyle \frac{dy}{dx}=−tan(t)\)

    21) Find the equation of the tangent line to \(\displaystyle x=sin(t),y=cos(t)\) at \(\displaystyle t=\frac{π}{4}\).

    22) For the curve \(\displaystyle x=4t,y=3t−2,\) find the slope and concavity of the curve at \(\displaystyle t=3\).

    Solution: \(\displaystyle \frac{dy}{dx}=\frac{3}{4}\) and \(\displaystyle \frac{d^2y}{dx^2}=0\), so the curve is neither concave up nor concave down at \(\displaystyle t=3\). Therefore the graph is linear and has a constant slope but no concavity.

    23) For the parametric curve whose equation is \(\displaystyle x=4cosθ,y=4sinθ\), find the slope and concavity of the curve at \(\displaystyle θ=\frac{π}{4}\).

    24) Find the slope and concavity for the curve whose equation is \(\displaystyle x=2+secθ,y=1+2tanθ\) at \(\displaystyle θ=\frac{π}{6}\).

    Solution: \(\displaystyle \frac{dy}{dx}=4,\frac{d^2y}{dx^2}=−6\sqrt{3};\) the curve is concave down at \(\displaystyle θ=\frac{π}{6}\).

    25) Find all points on the curve \(\displaystyle x=t+4,y=t^3−3t\) at which there are vertical and horizontal tangents.

    26) Find all points on the curve \(\displaystyle x=secθ,y=tanθ\) at which horizontal and vertical tangents exist.

    Solution: No horizontal tangents. Vertical tangents at \(\displaystyle (1,0),(−1,0)\).

     

    For the following exercises, find \(\displaystyle d^2y/dx^2\).

    27)  \(\displaystyle x=t^4−1,y=t−t^2\)

    28) \(\displaystyle x=sin(πt),y=cos(πt)\)

    Solution: \(\displaystyle −sec^3(πt)\)

    29) \(\displaystyle x=e^{−t},y=te^{2t}\)

    For the following exercises, find points on the curve at which tangent line is horizontal or vertical.

    30) \(\displaystyle x=t(t^2−3),y=3(t^2−3)\)

    Solution: Horizontal \(\displaystyle (0,−9)\); vertical \(\displaystyle (±2,−6).\)

    31) \(\displaystyle x=\frac{3t}{1+t^3},y=\frac{3t^2}{1+t^3}\)

     

    For the following exercises, find \(\displaystyle dy/dx\) at the value of the parameter.

    32) \(\displaystyle x=cost,y=sint,t=\frac{3π}{4}\)

    Solution: 1

    33) \(\displaystyle x=\sqrt{t},y=2t+4,t=9\)

    34) \(\displaystyle x=4cos(2πs),y=3sin(2πs),s=−\frac{1}{4}\)

    Solution: 0

     

    For the following exercises, find \(\displaystyle d^2y/dx^2\) at the given point without eliminating the parameter.

    35) \(\displaystyle x=\frac{1}{2}t^2,y=\frac{1}{3}t^3,t=2\)

    36) \(\displaystyle x=\sqrt{t},y=2t+4,t=1\)

    Solution: 4

    37) Find t intervals on which the curve \(\displaystyle x=3t^2,y=t^3−t\) is concave up as well as concave down.

    38) Determine the concavity of the curve \(\displaystyle x=2t+lnt,y=2t−lnt\).

    Solution: Concave up on \(\displaystyle t>0\).

    39) Sketch and find the area under one arch of the cycloid \(\displaystyle x=r(θ−sinθ),y=r(1−cosθ)\).

    40) Find the area bounded by the curve \(\displaystyle x=cost,y=e^t,0≤t≤\frac{π}{2}\) and the lines \(\displaystyle y=1\) and \(\displaystyle x=0\).

    Solution: 1

    41) Find the area enclosed by the ellipse \(\displaystyle x=acosθ,y=bsinθ,0≤θ<2π.\)

    42) Find the area of the region bounded by \(\displaystyle x=2sin^2θ,y=2sin^2θtanθ\), for \(\displaystyle 0≤θ≤\frac{π}{2}\).

    Solution: \(\displaystyle \frac{3π}{2}\)

     

    For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter.

    43) \(\displaystyle x=2cotθ,y=2sin^2θ,0≤θ≤π\)

    44) [T] \(\displaystyle x=2acost−acos(2t),y=2asint−asin(2t),0≤t<2π\)

    Solution: \(\displaystyle 6πa^2\)

    45) [T] \(\displaystyle x=asin(2t),y=bsin(t),0≤t<2π\) (the “hourglass”)

    46) [T] \(\displaystyle x=2acost−asin(2t),y=bsint,0≤t<2π\) (the “teardrop”)

    Solution: \(\displaystyle 2πab\)

     

    For the following exercises, find the arc length of the curve on the indicated interval of the parameter.

    47) \(\displaystyle x=4t+3,y=3t−2,0≤t≤2\)

    48) \(\displaystyle x=\frac{1}{3}t^3,y=\frac{1}{2}t^2,0≤t≤1\)

    Soluton: \(\displaystyle \frac{1}{3}(2\sqrt{2}−1)\)

    49) \(\displaystyle x=cos(2t),y=sin(2t),0≤t≤\frac{π}{2}\)

    50) \(\displaystyle x=1+t^2,y=(1+t)^3,0≤t≤1\)

    Solution: 7.075

    51) \(\displaystyle x=e^tcost,y=e^tsint,0≤t≤\frac{π}{2}\) (express answer as a decimal rounded to three places)

    52) \(\displaystyle x=acos^3θ,y=asin^3θ\) on the interval \(\displaystyle [0,2π)\) (the hypocycloid)

    Solution: \(\displaystyle 6a\)

    53) Find the length of one arch of the cycloid \(\displaystyle x=4(t−sint),y=4(1−cost).\)

    54) Find the distance traveled by a particle with position \(\displaystyle (x,y)\) as t varies in the given time interval: \(\displaystyle x=sin^2t,y=cos^2t,0≤t≤3π\).

    Solution: \(\displaystyle 6\sqrt{2}\)

    55) Find the length of one arch of the cycloid \(\displaystyle x=θ−sinθ,y=1−cosθ\).

    56) Show that the total length of the ellipse \(\displaystyle x=4sinθ,y=3cosθ\) is \(\displaystyle L=16∫^{π/2}_0\sqrt{1−e^2sin^2θ}dθ\), where \(\displaystyle e=\frac{c}{a}\) and \(\displaystyle c=\sqrt{a^2−b^2}\).

    57) Find the length of the curve \(\displaystyle x=e^t−t,y=4e^{t/2},−8≤t≤3.\)

     

    For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis.

    58) \(\displaystyle x=t^3,y=t^2,0≤t≤1\)

    Solution: \(\displaystyle \frac{2π(247\sqrt{13}+64)}{1215}\)

    59) \(\displaystyle x=acos^3θ,y=asin^3θ,0≤θ≤\frac{π}{2}\)

    60) [T] Use a CAS to find the area of the surface generated by rotating \(\displaystyle x=t+t^3,y=t−\frac{1}{t^2},1≤t≤2\) about the x-axis. (Answer to three decimal places.)

    Solution: 59.101

    61) Find the surface area obtained by rotating \(\displaystyle x=3t^2,y=2t^3,0≤t≤5\) about the y-axis.

    62) Find the area of the surface generated by revolving \(\displaystyle x=t^2,y=2t,0≤t≤4\) about the x-axis.

    Solution: \(\displaystyle \frac{8π}{3}(17\sqrt{17}−1)\)

    63) Find the surface area generated by revolving \(\displaystyle x=t^2,y=2t^2,0≤t≤1\) about the y-axis.