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math math-format graphie polynomials
Finding Velocity graphically
d-XX-02
number
54
randRange(5,10) 0.125*n randRange(2,4) randRangeExclude(1,2,[a]) b/a randRange(1,6) c/x0 function( t ) { return [x0, t]; }

Frida and Gustav are running a race and enter the final lab.

At which point in time t = 1,2,3, \ldots are the two on this equally fast?

With {\color{red}f},{\color{blue}g} we give the position as a function in time t.


graphInit({ range: [[-1, 10],[-0.5, 9.5]], scale: [45,45], tickStep: [1,10], labelStep: 1, gridOpacity: 0.1, axisOpacity: 0.8, tickOpacity: 0.6, labelOpacity: 0.8 }); label( [ 0, 9 ], "\\color{red}{f(t)}, \\color{blue}{g(t)}", "above" ); label( [ 10, 0 ], "t", "right" ); style({stroke: "black", strokeWidth: 2}); plot(function(x) {return y0 + c*pow(2.718,(x-x0)*m/c);}, [-0.75, 9], {stroke: "red"}); plot(function(x) {return m*x+3;}, [-0.75, 9], {stroke: "blue"});

x0

Gustav's velocity is constant over the time and equal to the slope of the blue line.

Frida's velocity at time t_0 is the slope of the tangent at the graph of \color{red}f in (t_0,{\color{red}f(t_0)}).

As we do not know the function term for {\color{red}f(t)},{\color{blue}g(t)} we are seeking the tangent graphically at the red graph with the slope of the blue line.

To achieve this we move the blue line parallel till it touches the red graph.

plot(function(x) {return m*x+c-m*x0+y0;}, [-0.75, 9], {stroke: "orange"});

style({ fill: "orange", stroke: "orange" }, function() { circle( [x0, 0], 0.1 ); }); style({ fill: "red", stroke: "red" }, function() { circle( [x0, y0 +c ], 0.1 ); }); style({ "stroke-dasharray":"."}, function() { line( [x0, y0+0.2 +c-0.2 ], [x0, 0 ] ); });

We sese that both coincide at \color{orange}{t_g = x0}, therefore Frida and Gustav are equally fast at this point.