Question

Your friend Lisa from California is currently taking calculus and is having a difficult time understanding what a derivative is. In addition, she does not understand why the tangent line to a function f(x) at r - a is the line which "best-fits" the curve y-f(x) at the point (a, f(a)). Write Lisa an email explaining what a derivative is and why the tangent line is the "best-fit" line to the curve y-f(x) at the point (a, f(a)). To make these ideas even more concrete for Lisa, send Lisa the following problem (along with your solution!): A car is traveling at night along a highway shaped like a parabola with its verter at the origin. The car starts at a point 100mi west and 100mi north of the origin and travels in an easterly direction. There is a statue located 100mi east and 50mi north of the origin. At what point on the highway will the car's headlights illuminate the statue? Car Statue The solution to this problem makes use of the tangent line in a key way

Answer 1

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Equation of parabola: from the given diagram & information we can get, The parabola has the at (0, 0) and has the y - axis of its axis i.e y (0) = 0 Also, the point (plusminus 10, 100) is on the parabola value i.e y (100) = 100 (second quadrant) y (100) = 100 (first quadrant) Thus, the general parabola eqn can be obtained, y = ax^2 + bx + c using initial condition 1, y (0) = 0 0 = 0 + 0 + c therefore c = 0 using second initial condition 2, y (-100) = 100 100 = a (-100)^2 + b (-100) 100 = 10000 a - 100 b therefore b = 100 a - 1 Using initial condition 3 y (100) = 100 100 = 10000 a + 100 b therefore 100 a + b = 1 put b = 100 a.1 therefore 100a + 100a - 1 = 1 therefore 200 a = 2 a