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a)

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To find the coordinates of the point corresponding to $t=\\var{e1}$, substitute $t=\\var{e1}$ into the expression for the curve, i.e.

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\$\\pmatrix{x,y}=\\pmatrix{\\simplify{{a1}*cos({b1*e1})},\\simplify{{c1}*cos({d1*e1})}}=\\pmatrix{\\var{x},\\var{y}}.\$

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b)

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Differentiate each component of the vector in part a) to find the tangent vector $\\boldsymbol{u}$, i.e.

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\$\\boldsymbol{u}=\\pmatrix{\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left(\\simplify{{a1}*cos({b1}*t)}\\right),\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left(\\simplify{{c1}*sin({d1}*t)}\\right)}=\\pmatrix{\\simplify{{-a1*b1}*sin({b1}*t)},\\simplify{{c1*d1}*cos({d1}*t)}}.\$

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The tangent vector at $t=\\var{e1}$ is found by substituting $t=\\var{e1}$ into the tangent vector $\\boldsymbol{u}$, i.e.

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\$\\boldsymbol{u}\\vert_{t=\\var{e1}}=\\pmatrix{\\simplify{{-a1*b1}*sin({b1*e1})},\\simplify{{c1*d1}*cos({d1*e1})}}=\\pmatrix{\\var{dxdte1},\\var{dydte1}}.\$

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c)

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The velocity $u$ is given by $u=\\lvert\\boldsymbol{u}\\rvert=\\sqrt{\\left(\\frac{\\mathrm{d}x}{\\mathrm{d}t}\\right)^2+\\left(\\frac{\\mathrm{d}y}{\\mathrm{d}t}\\right)^2}$.  We must calculate the speed at $t=\\var{f1}$, however, therefore

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\$u\\vert_{t=\\var{f1}}=\\sqrt{\\left(\\simplify{{-a1*b1}*sin({b1*f1})}\\right)^2+\\left(\\simplify{{c1*d1}*cos({d1*f1})}\\right)^2}=\\sqrt{\\var{dxdtf1}^2+\\var{dydtf1}^2}=\\var{speed} \\; \\text{to 3d.p.}\$

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Find the coordinates $\\pmatrix{x,y}$ of the point corresponding to $t=\\var{e1}$.

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$\\pmatrix{x,y}=($[[0]]$,$[[1]]$)$.  (Enter your answers to 2d.p.)

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Find the gradient function of the curve, and hence the gradient at the point $t=\\var{e1}$.

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Hence find the equation of the tangent to C at the point $t=\\var{e1}$, giving your answer in the form $y=mx+c$ where $m$ and $c$ are to be found.

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$y =$[[1]]$x +$[[0]]$\n (Enter your answers to 2d.p.) ", "scripts": {}, "sortAnswers": false, "unitTests": [], "useCustomName": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true}], "functions": {}, "ungrouped_variables": ["f1", "dxdte1", "dxdtf1", "dydte1", "dydtf1", "a1", "b1", "y", "x", "c1", "e1", "speed", "d1", "m"], "statement": " The curve C is defined parametrically by \n \$x = \\simplify{{a1}cos({b1}t)}, y = \\simplify{{c1}sin({d1}t)} \$ \n with respect to the parameter$t\$.

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Parametric form of a curve, cartesian points, tangent vector, and speed.

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