{"version":9,"randomSeed":"6166b5bbc3bad93adcfaa3674ce21005","graph":{"viewport":{"xmin":-4.220303853158927,"ymin":-13.823966631530624,"xmax":8.093677337632883,"ymax":17.13734856817469}},"expressions":{"list":[{"type":"expression","id":"1","color":"#c74440","latex":"a=3.4","labelSize":"medium"},{"type":"expression","id":"3","color":"#388c46","latex":"a^{2}"},{"type":"text","id":"22","text":"a + a + a + ... + a? the repeat is discrete, so that equals a * floor(a), not a^2"},{"type":"text","id":"14","text":"fractional component function"},{"type":"expression","id":"6","color":"#c74440","latex":"F\\left(x\\right)=x-\\operatorname{floor}\\left(x\\right)"},{"type":"text","id":"16","text":"same as a^2"},{"type":"expression","id":"5","color":"#000000","latex":"a\\operatorname{floor}\\left(a\\right)+aF\\left(a\\right)","labelSize":"medium"},{"type":"text","id":"18","text":"differentiate and find that it works"},{"type":"expression","id":"7","color":"#2d70b3","latex":"\\frac{d}{da}\\left(a\\operatorname{floor}\\left(a\\right)+aF\\left(a\\right)\\right)"},{"type":"expression","id":"8","color":"#388c46","latex":"\\frac{d}{da}\\left(a\\operatorname{floor}\\left(a\\right)\\right)+\\frac{d}{da}\\left(aF\\left(a\\right)\\right)"},{"type":"expression","id":"9","color":"#6042a6","latex":"a\\cdot\\frac{d}{da}\\left(\\operatorname{floor}\\left(a\\right)\\right)+\\operatorname{floor}\\left(a\\right)\\cdot\\frac{d}{da}\\left(a\\right)+a\\cdot\\frac{d}{da}\\left(F\\left(a\\right)\\right)+F\\left(a\\right)\\cdot\\frac{d}{da}\\left(a\\right)"},{"type":"expression","id":"10","color":"#000000","latex":"1\\cdot0+\\operatorname{floor}\\left(a\\right)\\cdot1+a\\cdot1+F\\left(a\\right)\\cdot1"},{"type":"expression","id":"11","color":"#c74440","latex":"a+\\operatorname{floor}\\left(a\\right)+F\\left(a\\right)","labelSize":"medium"},{"type":"expression","id":"12","color":"#2d70b3","latex":"2a"}]}}