1. (i) Obtain the differential equation for the family of curves of ellipse centered at the origin with eccentricity of 1/2. Note: use the following formulae: 2+2= = 1; e == ; c²=a² - b² a
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- b.) Find an equation of the tangent line to the curve at the given point. y = 4x + 2 cos(x), P = (0, 2)Consider the following. x = e', X = y = e-3t (a) Eliminate the parameter to find a Cartesian equation of the curve.Obtain the differential equation of the family of curves described and sketch some members of the family. A.) Ellipses having its centers at the origin and transverse axis x.
- The cables of the suspension bridge shown in the figure carry a uniform load of w-Newton per horizontal meter. If the horizontal tension of the cable at the origin is H, the cable curve is; It provides the Equation dy Solve this equation with initial condition y = 0 when x = 0 and show that the cable is in the dx H form of a parabola. Bridge cableIf a projectile is fired with an initial speed of v0 ft/s at an angle ? above the horizontal, then its position after t seconds is given by the parametric equations x = (v0 cos(?))t y = (v0 sin(?))t − 16t2 (where x and y are measured in feet).Suppose a gun fires a bullet into the air with an initial speed of 1856 ft/s at an angle of 30° to the horizontal. (b) How far from the gun will the bullet hit the ground? (Round your answer to one decimal place.) mi(c) What is the maximum height attained by the bullet? (Round your answer to one decimal place.) mi12. Determine an equation for the sinusoidal function shown: (-4T/3, 2) + -47 WHT (5/3, -4) (14T/3, 2) 145L 3 x
- . Given dy/dx = −2x + 6, find the equation of the curve that passes through the point (2, −2)Consider the following. x = sinh(t), y = cosh(t) (a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.Find an equation of the tangent to the curve (in cartesian coordinates) at the point corresponding to the given value of the parameter. x = cose + sin 20+2, y = sin0+ cos20 +2,0 = π O -=-=-=-1 2 v = 7²/4 X y= 22 +2²2/2 y=2+2 None of these