tangent of 4x^3-13x^2+4

{ "query": { "display": "tangent of $$4x^{3}-13x^{2}+4$$", "symbolab_question": "PRE_CALC#tangent 4x^{3}-13x^{2}+4" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Derivative Applications", "subTopic": "Tangent", "default": "y=(12a_{0}^{2}-26a_{0})x-8a_{0}^{3}+13a_{0}^{2}+4", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Tangent line to $$y=4x^{3}-13x^{2}+4:{\\quad}y=\\left(12a_{0}^{2}-26a_{0}\\right)x-8a_{0}^{3}+13a_{0}^{2}+4$$", "steps": [ { "type": "step", "primary": "Compute the tangent line to the general point $$x=a_{0}$$" }, { "type": "interim", "title": "Find the tangent point:$${\\quad}\\left(a_{0},\\:4a_{0}^{3}-13a_{0}^{2}+4\\right)$$", "steps": [ { "type": "step", "primary": "Plug $$x=a_{0}$$ into the equation $$y=4x^{3}-13x^{2}+4$$", "result": "y=4a_{0}^{3}-13a_{0}^{2}+4" } ], "meta": { "interimType": "Tangent Find Tangent Point Title 0Eq" } }, { "type": "interim", "title": "Find the slope of $$y=4x^{3}-13x^{2}+4:{\\quad}\\frac{dy}{dx}=12x^{2}-26x$$", "input": "y=4x^{3}-13x^{2}+4", "steps": [ { "type": "step", "primary": "In order to find the slope of the function, take the derivative of $$4x^{3}-13x^{2}+4$$" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4x^{3}-13x^{2}+4\\right)=12x^{2}-26x$$", "input": "\\frac{d}{dx}\\left(4x^{3}-13x^{2}+4\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(4x^{3}\\right)-\\frac{d}{dx}\\left(13x^{2}\\right)+\\frac{d}{dx}\\left(4\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4x^{3}\\right)=12x^{2}$$", "input": "\\frac{d}{dx}\\left(4x^{3}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=4\\frac{d}{dx}\\left(x^{3}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4\\cdot\\:3x^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=12x^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYuytnrDSBQVECiAN7hMFdqKTdaV09PMxEKZ9FieghTFwbVQDmNnvMzBhKnFOUzUT515NkzKQgtswLlLi9MgL+gq5QV7agSZLIzF7D9vX0CHvx8XaWXbYXWiYPXxbVFoLirCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(13x^{2}\\right)=26x$$", "input": "\\frac{d}{dx}\\left(13x^{2}\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=13\\frac{d}{dx}\\left(x^{2}\\right)" }, { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=13\\cdot\\:2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=26x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYsGk3JMrmRJlgfF2jxt0kHYcjlLRK1jUV206qo4+vRN755Gj06uB8gi6/oDWYJqDhz/L0MoYg+CUn6oyL3EO7YrjEV6JR4edQWwqf3Oe/gdjE6VRTRdoT3bpoO611Q9PbQ==" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(4\\right)=0$$", "input": "\\frac{d}{dx}\\left(4\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYjVwwDW+HeFUFiKZ8J+l8XpJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTt4/nDM7CraQVY2V0O4nKcI" } }, { "type": "step", "result": "=12x^{2}-26x+0" }, { "type": "step", "primary": "Simplify", "result": "=12x^{2}-26x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "result": "12x^{2}-26x" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWSxSnk+7QxHR30gHiIbDEKxtLZK6RYGwvVo+GijtM4oSqbkoH6MHGg0mtOsg27VY89vapQ8ht09fkDiBiSor1ec0InMWGpSPLPYURGOlGWUJr5P7iAecl41dEBOeztixdPg==" } }, { "type": "interim", "title": "$$EN:\\:Title\\:General\\:Equation\\:Slope\\:At\\:Point\\:2Eq:{\\quad}m=12a_{0}^{2}-26a_{0}$$", "steps": [ { "type": "step", "primary": "Plug $$x=a_{0}$$ into the equation $$12x^{2}-26x$$", "result": "12a_{0}^{2}-26a_{0}" } ], "meta": { "interimType": "General Equation Slope At Point 2Eq" } }, { "type": "interim", "title": "Find the line with slope m=$$12a_{0}^{2}-26a_{0}$$ and passing through $$\\left(a_{0},\\:4a_{0}^{3}-13a_{0}^{2}+4\\right):{\\quad}y=\\left(12a_{0}^{2}-26a_{0}\\right)x-8a_{0}^{3}+13a_{0}^{2}+4$$", "steps": [ { "type": "step", "primary": "Compute the line equation $$\\mathbf{y=mx+b}$$ for slope m=$$12a_{0}^{2}-26a_{0}$$ and passing through $$\\left(a_{0},\\:4a_{0}^{3}-13a_{0}^{2}+4\\right)$$" }, { "type": "interim", "title": "Compute the $$y$$ intercept:$${\\quad}b=-8a_{0}^{3}+13a_{0}^{2}+4$$", "steps": [ { "type": "step", "primary": "Plug the slope $$12a_{0}^{2}-26a_{0}$$ into $$y=mx+b$$", "result": "y=\\left(12a_{0}^{2}-26a_{0}\\right)x+b" }, { "type": "step", "primary": "Plug in $$\\left(a_{0},\\:4a_{0}^{3}-13a_{0}^{2}+4\\right)$$: $$\\quad\\:x=a_{0},\\:y=4a_{0}^{3}-13a_{0}^{2}+4$$", "result": "4a_{0}^{3}-13a_{0}^{2}+4=\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}+b" }, { "type": "step", "primary": "Isolate $$b$$" }, { "type": "interim", "title": "$$4a_{0}^{3}-13a_{0}^{2}+4=\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}+b{\\quad:\\quad}b=-8a_{0}^{3}+13a_{0}^{2}+4$$", "input": "4a_{0}^{3}-13a_{0}^{2}+4=\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}+b", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}+b=4a_{0}^{3}-13a_{0}^{2}+4" }, { "type": "interim", "title": "Move $$\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}\\:$$to the right side", "input": "\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}+b=4a_{0}^{3}-13a_{0}^{2}+4", "result": "b=-8a_{0}^{3}+13a_{0}^{2}+4", "steps": [ { "type": "step", "primary": "Subtract $$\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}$$ from both sides", "result": "\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}+b-\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}=4a_{0}^{3}-13a_{0}^{2}+4-\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}" }, { "type": "interim", "title": "Simplify", "input": "\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}+b-\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}=4a_{0}^{3}-13a_{0}^{2}+4-\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}", "result": "b=-8a_{0}^{3}+13a_{0}^{2}+4", "steps": [ { "type": "interim", "title": "Simplify $$\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}+b-\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}:{\\quad}b$$", "input": "\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}+b-\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}", "steps": [ { "type": "step", "primary": "Add similar elements: $$\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}-\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}=0$$" }, { "type": "step", "result": "=b" } ], "meta": { "interimType": "Generic Simplify Specific 1Eq" } }, { "type": "interim", "title": "Simplify $$4a_{0}^{3}-13a_{0}^{2}+4-\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}:{\\quad}-8a_{0}^{3}+13a_{0}^{2}+4$$", "input": "4a_{0}^{3}-13a_{0}^{2}+4-\\left(12a_{0}^{2}-26a_{0}\\right)a_{0}", "steps": [ { "type": "step", "result": "=4a_{0}^{3}-13a_{0}^{2}+4-a_{0}\\left(12a_{0}^{2}-26a_{0}\\right)" }, { "type": "interim", "title": "Expand $$-a_{0}\\left(12a_{0}^{2}-26a_{0}\\right):{\\quad}-12a_{0}^{3}+26a_{0}^{2}$$", "input": "-a_{0}\\left(12a_{0}^{2}-26a_{0}\\right)", "result": "=4a_{0}^{3}-13a_{0}^{2}+4-12a_{0}^{3}+26a_{0}^{2}", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=-a_{0},\\:b=12a_{0}^{2},\\:c=26a_{0}$$" ], "result": "=-a_{0}\\cdot\\:12a_{0}^{2}-\\left(-a_{0}\\right)\\cdot\\:26a_{0}", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a$$" ], "result": "=-12a_{0}^{2}a_{0}+26a_{0}a_{0}" }, { "type": "interim", "title": "Simplify $$-12a_{0}^{2}a_{0}+26a_{0}a_{0}:{\\quad}-12a_{0}^{3}+26a_{0}^{2}$$", "input": "-12a_{0}^{2}a_{0}+26a_{0}a_{0}", "result": "=-12a_{0}^{3}+26a_{0}^{2}", "steps": [ { "type": "interim", "title": "$$12a_{0}^{2}a_{0}=12a_{0}^{3}$$", "input": "12a_{0}^{2}a_{0}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$a_{0}^{2}a_{0}=\\:a_{0}^{2+1}$$" ], "result": "=12a_{0}^{2+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$2+1=3$$", "result": "=12a_{0}^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7J+m4C2dqkZN5pgLIPhqyZaDkfKJxJZVCmfBUAvjyrEljZrOM+Q8JEbZyOig8Btyy8NvpF8F0rb9ySuzQCHCWf+b+HHiXbwkj2h7DsxgxbrSIJTyWgU69SOTqh8e3OZM1Fa6ojj0Q5FmHNmJWkpJGCg==" } }, { "type": "interim", "title": "$$26a_{0}a_{0}=26a_{0}^{2}$$", "input": "26a_{0}a_{0}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$a_{0}a_{0}=\\:a_{0}^{1+1}$$" ], "result": "=26a_{0}^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=26a_{0}^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/hHxZjzmmEjtumzhoD/Vf1XTSum/z5kLpMzXS1UJIeyRtdViybF3faPv9I9mFeQojxDJoujgV3bUnrA9nLNzoDFmQjQPtnMwX1U7nDIO9Tc9VbC6lsCOncFe2KQJrOSzJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "=-12a_{0}^{3}+26a_{0}^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wqm2G9/Jow3/IytgxlpRD8q6x1XZy/sN6qhdOPxOFv7ehkKrn0era9rz8TlL+x/vDtgs3cAMPERsV43gb7xCcFXsfSbAwo3JJyFwoDP6wIYRztKE552b3ssyeALQtWRjgnLy9o0s8NVZUXBl+JGuqpsfKFZVpe65SfMcK0ivNYyx1fr01sVlYGaMPHczLa6CJLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "Simplify $$4a_{0}^{3}-13a_{0}^{2}+4-12a_{0}^{3}+26a_{0}^{2}:{\\quad}-8a_{0}^{3}+13a_{0}^{2}+4$$", "input": "4a_{0}^{3}-13a_{0}^{2}+4-12a_{0}^{3}+26a_{0}^{2}", "result": "=-8a_{0}^{3}+13a_{0}^{2}+4", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=4a_{0}^{3}-12a_{0}^{3}-13a_{0}^{2}+26a_{0}^{2}+4" }, { "type": "step", "primary": "Add similar elements: $$-13a_{0}^{2}+26a_{0}^{2}=13a_{0}^{2}$$", "result": "=4a_{0}^{3}-12a_{0}^{3}+13a_{0}^{2}+4" }, { "type": "step", "primary": "Add similar elements: $$4a_{0}^{3}-12a_{0}^{3}=-8a_{0}^{3}$$", "result": "=-8a_{0}^{3}+13a_{0}^{2}+4" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7o5oIe+bkCqXL7UJTU3dZ6zfzfRC/ZputISeJLpW04zn/CFdFxiTzVrZAJ9uGcLAmYPviw1+d5PCsChKHIX7xlXCQoYlYQ8U+Tfyx0kyzI8iLDPS6MFSzETxGuRdhSvC1ouOzGWkzw2BvE4mwAcG22IEFMST8lDZxn1Yq5HMKVTtbUZ3Y5M7+CGR1PItOAOfWnjRY4vHI1j2NevRfm3yNfDZoNKU+zhtERqEKwRNCcOWkWooFX7fG9n16NoAskFAdialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "result": "b=-8a_{0}^{3}+13a_{0}^{2}+4" } ], "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=-8a_{0}^{3}+13a_{0}^{2}+4" } ], "meta": { "interimType": "Line Equation Find Intersection From Point 0Eq" } }, { "type": "step", "primary": "Construct the line equation $$\\mathbf{y=mx+b}$$ where $$\\mathbf{m}=12a_{0}^{2}-26a_{0}$$ and $$\\mathbf{b}=-8a_{0}^{3}+13a_{0}^{2}+4$$", "result": "y=\\left(12a_{0}^{2}-26a_{0}\\right)x-8a_{0}^{3}+13a_{0}^{2}+4" } ], "meta": { "interimType": "Line Equation Slope Point 6Eq" } }, { "type": "step", "result": "y=\\left(12a_{0}^{2}-26a_{0}\\right)x-8a_{0}^{3}+13a_{0}^{2}+4" } ], "meta": { "solvingClass": "PreCalc" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "tangent 4x^{3}-13x^{2}+4" }, "showViewLarger": true } }, "meta": { "showVerify": true } }