normal of y=x^2-x^3+x,(-2,10)

{ "query": { "display": "normal of $$y=x^{2}-x^{3}+x,\\:\\left(-2,\\:10\\right)$$", "symbolab_question": "PRE_CALC#normal y=x^{2}-x^{3}+x,(-2,10)" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivative Applications", "subTopic": "Normal", "default": "y=\\frac{1}{15}x+\\frac{152}{15}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Normal line to $$y=x^{2}-x^{3}+x$$, with a tangent point $$\\left(-2,\\:10\\right):{\\quad}y=\\frac{1}{15}x+\\frac{152}{15}$$", "steps": [ { "type": "interim", "title": "Find the slope of $$y=x^{2}-x^{3}+x:{\\quad}\\frac{dy}{dx}=2x-3x^{2}+1$$", "input": "y=x^{2}-x^{3}+x", "steps": [ { "type": "step", "primary": "In order to find the slope of the function, take the derivative of $$x^{2}-x^{3}+x$$" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}-x^{3}+x\\right)=2x-3x^{2}+1$$", "input": "\\frac{d}{dx}\\left(x^{2}-x^{3}+x\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(x^{2}\\right)-\\frac{d}{dx}\\left(x^{3}\\right)+\\frac{dx}{dx}" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{3}\\right)=3x^{2}$$", "input": "\\frac{d}{dx}\\left(x^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=3x^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=3x^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYtb6j95rHG7YtZ73Xx2qCjqk3hxk9aCfAWodBRxXgUexf7nh0v5ML3fMP9GgRVbRX/8//6/nV5O4fb8Xgwi7maqO95Is7YBdcRqXmTH8euc2MRY7LLv3QukQErzdJ9wdtQ==" } }, { "type": "interim", "title": "$$\\frac{dx}{dx}=1$$", "input": "\\frac{dx}{dx}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYko/29fz701XcRtz4b42RqRjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIdz2XHFZ6BxfaHSMA6lT+lbVmoiKRd+ttkZ9NIrGodT+" } }, { "type": "step", "result": "=2x-3x^{2}+1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "2x-3x^{2}+1" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWS7XRmhMa17/yVjNhC3DJG4M/y9DKGIPglJ+qMi9xDu2K3hU5AaFwz4+vUEs4LPkLXfSA1zF/KHdtRDTyLuSh/sZTQA4rQ4m9b7gJNieEMv5I" } }, { "type": "interim", "title": "$$EN:\\:Title\\:General\\:Equation\\:Slope\\:At\\:Point\\:2Eq:{\\quad}m=-15$$", "steps": [ { "type": "step", "primary": "Plug $$x=-2$$ into the equation $$2x-3x^{2}+1$$", "result": "2\\left(-2\\right)-3\\left(-2\\right)^{2}+1" }, { "type": "interim", "title": "Simplify $$2\\left(-2\\right)-3\\left(-2\\right)^{2}+1:{\\quad}-15$$", "input": "2\\left(-2\\right)-3\\left(-2\\right)^{2}+1", "result": "=-15", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-2\\cdot\\:2-3\\left(-2\\right)^{2}+1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=-4-\\left(-2\\right)^{2}\\cdot\\:3+1" }, { "type": "interim", "title": "$$\\left(-2\\right)^{2}=2^{2}$$", "input": "\\left(-2\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-2\\right)^{2}=2^{2}$$" ], "result": "=2^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sNuhSfBo+/I8oqMceBlhcs0ag8T1MwTer44+aCS/ZFBDeoKWfP4f0hW8hp+DjlqkWG48kfKlXwh1JXHkPaftrOeZImDuB9kLWbJJECF6RjY=" } }, { "type": "step", "result": "=-4-2^{2}\\cdot\\:3+1" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-4+1=-3$$", "result": "=-2^{2}\\cdot\\:3-3" }, { "type": "interim", "title": "$$2^{2}\\cdot\\:3=12$$", "input": "2^{2}\\cdot\\:3", "steps": [ { "type": "step", "primary": "$$2^{2}=4$$", "result": "=4\\cdot\\:3" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:3=12$$", "result": "=12" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YlMCdSu9defMDx6stnf0hS061ljBSPJeENOw2efoSWvRhv/4tiXq5Z5AYo1OPkf//COFlUvA93NcQfHx1F5YKhX7Z8MQIFQnyR+DcLAqjH0kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=-12-3" }, { "type": "step", "primary": "Subtract the numbers: $$-12-3=-15$$", "result": "=-15" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oNJ9Ye9MavO9AB2FogugaKtvA9LzOqVE0SxHsPBFH0CjkVi15I8rBefLi4Iyt2wrVgpO0S1QWr9qttkqKc5t1O5AIz++qluupTlLFEcE9J16c9VEWKVs8zErVHS2EWvNjP0/PzV317q7ttYVEJWaiQ==" } }, { "type": "step", "result": "m=-15" } ], "meta": { "interimType": "General Equation Slope At Point 2Eq" } }, { "type": "interim", "title": "Compute the slope of the perpendicular line:$${\\quad}m_{p}=\\frac{1}{15}$$", "steps": [ { "type": "step", "primary": "The perpendicular slope is the negative reciprocal of the given slope" }, { "type": "interim", "title": "$$\\left(-15\\right)m_{p}=-1{\\quad:\\quad}m_{p}=\\frac{1}{15}$$", "input": "\\left(-15\\right)m_{p}=-1", "steps": [ { "type": "interim", "title": "Divide both sides by $$-15$$", "input": "\\left(-15\\right)m_{p}=-1", "result": "m_{p}=\\frac{1}{15}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-15$$", "result": "\\frac{\\left(-15\\right)m_{p}}{-15}=\\frac{-1}{-15}" }, { "type": "step", "primary": "Simplify", "result": "m_{p}=\\frac{1}{15}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "m_{p}=\\frac{1}{15}" } ], "meta": { "interimType": "Line Equation Slope Perpendicular 0Eq" } }, { "type": "interim", "title": "Find the line with slope m=$$\\frac{1}{15}$$ and passing through $$\\left(-2,\\:10\\right):{\\quad}y=\\frac{1}{15}x+\\frac{152}{15}$$", "steps": [ { "type": "step", "primary": "Compute the line equation $$\\mathbf{y=mx+b}$$ for slope m=$$\\frac{1}{15}$$ and passing through $$\\left(-2,\\:10\\right)$$" }, { "type": "interim", "title": "Compute the $$y$$ intercept:$${\\quad}b=\\frac{152}{15}$$", "steps": [ { "type": "step", "primary": "Plug the slope $$\\frac{1}{15}$$ into $$y=mx+b$$", "result": "y=\\frac{1}{15}x+b" }, { "type": "step", "primary": "Plug in $$\\left(-2,\\:10\\right)$$: $$\\quad\\:x=-2,\\:y=10$$", "result": "10=\\frac{1}{15}\\left(-2\\right)+b" }, { "type": "step", "primary": "Isolate $$b$$" }, { "type": "interim", "title": "$$10=\\frac{1}{15}\\left(-2\\right)+b{\\quad:\\quad}b=\\frac{152}{15}$$", "input": "10=\\frac{1}{15}\\left(-2\\right)+b", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\frac{1}{15}\\left(-2\\right)+b=10" }, { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "-\\frac{1}{15}\\cdot\\:2+b=10" }, { "type": "interim", "title": "$$\\frac{1}{15}\\cdot\\:2=\\frac{2}{15}$$", "input": "\\frac{1}{15}\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{15}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2}{15}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PFtmKDYOqrdPl+AUuM9DAv4CsFtYCGVpnwQ6GR+/iwQJQJZuTAY5js+oqjdT8ksl+zLwo/o5iAJf0YEhVGjNYP8//6/nV5O4fb8Xgwi7maqYhoA+EEHjhE5EMIcRBpYYx2eCxF1gSdx+eiMLlSeIG8T/HN4EYnnyCFtwyxTkU8M=" } }, { "type": "step", "result": "-\\frac{2}{15}+b=10" }, { "type": "interim", "title": "Move $$\\frac{2}{15}\\:$$to the right side", "input": "-\\frac{2}{15}+b=10", "result": "b=\\frac{152}{15}", "steps": [ { "type": "step", "primary": "Add $$\\frac{2}{15}$$ to both sides", "result": "-\\frac{2}{15}+b+\\frac{2}{15}=10+\\frac{2}{15}" }, { "type": "interim", "title": "Simplify", "input": "-\\frac{2}{15}+b+\\frac{2}{15}=10+\\frac{2}{15}", "result": "b=\\frac{152}{15}", "steps": [ { "type": "interim", "title": "Simplify $$-\\frac{2}{15}+b+\\frac{2}{15}:{\\quad}b$$", "input": "-\\frac{2}{15}+b+\\frac{2}{15}", "steps": [ { "type": "step", "primary": "Add similar elements: $$-\\frac{2}{15}+\\frac{2}{15}=0$$" }, { "type": "step", "result": "=b" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$10+\\frac{2}{15}:{\\quad}\\frac{152}{15}$$", "input": "10+\\frac{2}{15}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$10=\\frac{10\\cdot\\:15}{15}$$", "result": "=\\frac{10\\cdot\\:15}{15}+\\frac{2}{15}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{10\\cdot\\:15+2}{15}" }, { "type": "interim", "title": "$$10\\cdot\\:15+2=152$$", "input": "10\\cdot\\:15+2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$10\\cdot\\:15=150$$", "result": "=150+2" }, { "type": "step", "primary": "Add the numbers: $$150+2=152$$", "result": "=152" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YzelT2lRbLo1OVu2qSqsiy061ljBSPJeENOw2efoSWtGhBrsGVlTgtDZhGXySGVbjFF+Grhte/2UqFkzsPs4p40cc3puy+RRKTsPrQ4F5+SwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{152}{15}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7k3S3l9l89KMNodNktUCTxl+rVXIYCl1jsDU19dobIgH9ovYKijQYhJDCbxu/nAOJRpEvYIC+wAueWOh+K5roYaN6Hv6MoTMtvtU0IQwXdn9szOhN37mcRdV5CgGGkVwg9Ufz6Z4shxA/EtQsZDYxwompXFf3SOUx+H18qfp3MLg=" } }, { "type": "step", "result": "b=\\frac{152}{15}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "b=\\frac{152}{15}" } ], "meta": { "interimType": "Line Equation Find Intersection From Point 0Eq" } }, { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ where $$\\mathbf{m}=\\frac{1}{15}$$ and $$\\mathbf{b}=\\frac{152}{15}$$", "result": "y=\\frac{1}{15}x+\\frac{152}{15}" } ], "meta": { "interimType": "Line Equation Slope Point 6Eq" } }, { "type": "step", "result": "y=\\frac{1}{15}x+\\frac{152}{15}" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=x^{2}-x^{3}+x", "displayFormula": "y=x^{2}-x^{3}+x", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=\\frac{x+152}{15}", "displayFormula": "y=\\frac{x+152}{15}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-2,10)" ], "pointsDecimal": [ { "fst": -2, "snd": 10 } ], "attributes": [ { "color": "BLACK", "labels": [ null ], "labelTypes": [ "SOLUTION" ], "labelColors": [ "BLACK" ] } ] }, "functionChanges": [ { "origFormulaLatex": [ "\\frac{1}{15}x+\\frac{152}{15}" ], 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