{
"query": {
"display": "tangente de $$f\\left(x\\right)=\\sin\\left(2x\\right)\\cos\\left(x\\right),\\:\\at\\:x=π$$",
"symbolab_question": "PRE_CALC#tangent f(x)=\\sin(2x)\\cos(x),\\at x=π"
},
"solution": {
"level": "PERFORMED",
"subject": "Calculus",
"topic": "Derivative Applications",
"subTopic": "Tangent",
"default": "y=-2x+2π",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "Recta tangente a la ecuación $$f\\left(x\\right)=\\sin\\left(2x\\right)\\cos\\left(x\\right)$$, cuando $$x=π:{\\quad}y=-2x+2π$$",
"steps": [
{
"type": "interim",
"title": "Encontrar el punto de tangencia:$${\\quad}\\left(π,\\:0\\right)$$",
"steps": [
{
"type": "step",
"primary": "Sustituir $$x=π$$ en la ecuación $$f\\left(x\\right)=\\sin\\left(2x\\right)\\cos\\left(x\\right)$$",
"result": "f\\left(x\\right)=\\sin\\left(2π\\right)\\cos\\left(π\\right)"
},
{
"type": "step",
"primary": "Resolver $$f\\left(x\\right)$$",
"result": "f\\left(x\\right)=0"
}
],
"meta": {
"interimType": "Tangent Find Tangent Point Title 0Eq"
}
},
{
"type": "interim",
"title": "Encuentra la pendiente de $$f\\left(x\\right)=\\sin\\left(2x\\right)\\cos\\left(x\\right):{\\quad}\\frac{df\\left(x\\right)}{dx}=2\\cos\\left(2x\\right)\\cos\\left(x\\right)-\\sin\\left(x\\right)\\sin\\left(2x\\right)$$",
"input": "f\\left(x\\right)=\\sin\\left(2x\\right)\\cos\\left(x\\right)",
"steps": [
{
"type": "step",
"primary": "Para hallar la pendiente de la función, obtenemos la derivada de $$\\sin\\left(2x\\right)\\cos\\left(x\\right)$$"
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(\\sin\\left(2x\\right)\\cos\\left(x\\right)\\right)=2\\cos\\left(2x\\right)\\cos\\left(x\\right)-\\sin\\left(x\\right)\\sin\\left(2x\\right)$$",
"input": "\\frac{d}{dx}\\left(\\sin\\left(2x\\right)\\cos\\left(x\\right)\\right)",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla del producto: $$\\left(f{\\cdot}g\\right)'=f'{\\cdot}g+f{\\cdot}g'$$",
"secondary": [
"$$f=\\sin\\left(2x\\right),\\:g=\\cos\\left(x\\right)$$"
],
"result": "=\\frac{d}{dx}\\left(\\sin\\left(2x\\right)\\right)\\cos\\left(x\\right)+\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)\\sin\\left(2x\\right)",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Product%20Rule",
"practiceTopic": "Product Rule"
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(\\sin\\left(2x\\right)\\right)=\\cos\\left(2x\\right)\\cdot\\:2$$",
"input": "\\frac{d}{dx}\\left(\\sin\\left(2x\\right)\\right)",
"steps": [
{
"type": "interim",
"title": "Aplicar la regla de la cadena:$${\\quad}\\cos\\left(2x\\right)\\frac{d}{dx}\\left(2x\\right)$$",
"input": "\\frac{d}{dx}\\left(\\sin\\left(2x\\right)\\right)",
"result": "=\\cos\\left(2x\\right)\\frac{d}{dx}\\left(2x\\right)",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la cadena: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\sin\\left(u\\right),\\:\\:u=2x$$"
],
"result": "=\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)\\frac{d}{dx}\\left(2x\\right)",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)=\\cos\\left(u\\right)$$",
"input": "\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\frac{d}{du}\\left(\\sin\\left(u\\right)\\right)=\\cos\\left(u\\right)$$",
"result": "=\\cos\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgerJLn9ae0g0/tUjnRuL1v8zeERICEnv1Ds5A1/BdIwQslTDKxOR/6J+ZOGvUcaugHqnJiEuQ8NpaCSOBx7rI4+YUX37Aa/AAEf1Hkty8FUcUM2sEdv7dIX0bKYOeE19OmgDIY2KBZfpU9cYqvCXz4="
}
},
{
"type": "step",
"result": "=\\cos\\left(u\\right)\\frac{d}{dx}\\left(2x\\right)"
},
{
"type": "step",
"primary": "Sustituir en la ecuación $$u=2x$$",
"result": "=\\cos\\left(2x\\right)\\frac{d}{dx}\\left(2x\\right)"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYh6yUCRk/MnA93xfZq2OhuSQp7tdIFyr1eVqMMLZHDTGOK1n91tyBoBr/ZHP0eNC/RSNU68ZmiYZN//Vg53tMEzREhQBxQGxynSTy6vjYFdXrqk0J72Q7Tc2GvO8AjiU6iGk8vIJisuT2N3pfkW1JpZOOhV8CCi1NFOF64UTCTG343n9TEBKhGpOP+x9QkQNEg6CRM0oxQsCkRwDOoJ6qfk="
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(2x\\right)=2$$",
"input": "\\frac{d}{dx}\\left(2x\\right)",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=2\\frac{dx}{dx}"
},
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\frac{dx}{dx}=1$$",
"result": "=2\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=2",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2sQzwGEAAPyDk8n13Ps8XZGku9zFkxwe1dTH8vycb94wHsFp27x8BxzSfXYcuPllNbbqpyK7JQEZdATEJR51jH4j/fzMjnIhJwos1vPNWw"
}
},
{
"type": "step",
"result": "=\\cos\\left(2x\\right)\\cdot\\:2"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)=-\\sin\\left(x\\right)$$",
"input": "\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\frac{d}{dx}\\left(\\cos\\left(x\\right)\\right)=-\\sin\\left(x\\right)$$",
"result": "=-\\sin\\left(x\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYoTIPsH/5VFEfonU6bvi80j8zeERICEnv1Ds5A1/BdIwwxWDXidEV9CzsGPnUu41zA92cpyjnQxeYFWLLJRXAqymcxh5GfxfsNed5mphvPA8hRo8/W0Hq8WASQDG4Qt2SWMOnp3ra3sqxOu3sWEDlnw="
}
},
{
"type": "step",
"result": "=\\cos\\left(2x\\right)\\cdot\\:2\\cos\\left(x\\right)+\\left(-\\sin\\left(x\\right)\\right)\\sin\\left(2x\\right)"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=2\\cos\\left(2x\\right)\\cos\\left(x\\right)-\\sin\\left(x\\right)\\sin\\left(2x\\right)",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "2\\cos\\left(2x\\right)\\cos\\left(x\\right)-\\sin\\left(x\\right)\\sin\\left(2x\\right)"
}
],
"meta": {
"interimType": "Slope Equation Top 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWS+weOnmaCIjfrej4kJXvYnJx0PMKptP0yG56gJLsJ8/pTeQKHeh69S6dnv9vSoUoFEwzRNIhDp3BIUSY1sxb+u5he8LhSYBf5CbqSAdng8YnBUFkutl0/cVPW7azjj87duIASZeFjDtawNGt9P21GJY="
}
},
{
"type": "interim",
"title": "$$EN:\\:Title\\:General\\:Equation\\:Slope\\:At\\:Point\\:2Eq:{\\quad}m=-2$$",
"steps": [
{
"type": "step",
"primary": "Sustituir $$x=π$$ en la ecuación $$2\\cos\\left(2x\\right)\\cos\\left(x\\right)-\\sin\\left(x\\right)\\sin\\left(2x\\right)$$",
"result": "2\\cos\\left(2π\\right)\\cos\\left(π\\right)-\\sin\\left(π\\right)\\sin\\left(2π\\right)"
},
{
"type": "interim",
"title": "Simplificar $$2\\cos\\left(2π\\right)\\cos\\left(π\\right)-\\sin\\left(π\\right)\\sin\\left(2π\\right):{\\quad}-2$$",
"input": "2\\cos\\left(2π\\right)\\cos\\left(π\\right)-\\sin\\left(π\\right)\\sin\\left(2π\\right)",
"result": "=-2",
"steps": [
{
"type": "interim",
"title": "$$2\\cos\\left(2π\\right)\\cos\\left(π\\right)=-2$$",
"input": "2\\cos\\left(2π\\right)\\cos\\left(π\\right)",
"steps": [
{
"type": "interim",
"title": "$$\\cos\\left(2π\\right)=1$$",
"input": "\\cos\\left(2π\\right)",
"result": "=2\\cdot\\:1\\cdot\\:\\cos\\left(π\\right)",
"steps": [
{
"type": "interim",
"title": "$$\\cos\\left(2π\\right)=\\cos\\left(0\\right)$$",
"input": "\\cos\\left(2π\\right)",
"result": "=\\cos\\left(0\\right)",
"steps": [
{
"type": "step",
"primary": "Reescribir $$2π$$ como $$2π+0$$",
"result": "=\\cos\\left(2π+0\\right)"
},
{
"type": "step",
"primary": "Utilizar la periodicidad de $$\\cos$$: $$\\cos\\left(x+2π\\right)=\\cos\\left(x\\right)$$",
"secondary": [
"$$\\cos\\left(2π+0\\right)=\\cos\\left(0\\right)$$"
],
"result": "=\\cos\\left(0\\right)"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "interim",
"title": "Utilizar la siguiente identidad trivial:$${\\quad}\\cos\\left(0\\right)=1$$",
"input": "\\cos\\left(0\\right)",
"steps": [
{
"type": "step",
"primary": "$$\\cos\\left(x\\right)$$ tabla de valores periódicos con $$2πn$$ intervalos:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$"
},
{
"type": "step",
"result": "=1"
}
],
"meta": {
"interimType": "Trig Trivial Angle Value Title 0Eq"
}
},
{
"type": "step",
"result": "=1"
}
],
"meta": {
"solvingClass": "Trig Evaluate",
"interimType": "Trig Evaluate"
}
},
{
"type": "interim",
"title": "Simplificar $$\\cos\\left(π\\right):{\\quad}-1$$",
"input": "\\cos\\left(π\\right)",
"result": "=2\\cdot\\:1\\cdot\\:\\left(-1\\right)",
"steps": [
{
"type": "step",
"primary": "Utilizar la siguiente identidad trivial:$${\\quad}\\cos\\left(π\\right)=\\left(-1\\right)$$",
"secondary": [
"$$\\cos\\left(x\\right)$$ tabla de valores periódicos con $$2πn$$ intervalos:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$"
],
"result": "=-1"
}
],
"meta": {
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"primary": "Quitar los parentesis: $$\\left(-a\\right)=-a$$",
"result": "=-2\\cdot\\:1\\cdot\\:1"
},
{
"type": "step",
"primary": "Multiplicar los numeros: $$2\\cdot\\:1\\cdot\\:1=2$$",
"result": "=-2"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xufXhDkCP60yL8UVDWDDJQANE3P7ehounOdB47mIWgOrju+5Z51e/ZZSD3gRHwjBy++q5TspDrdoyXuh0DUdTuwHtPTZQgCTaLxxp2ffbNaH4O4IpfGeZyZOD8hwNTnX"
}
},
{
"type": "step",
"result": "=-2-\\sin\\left(π\\right)\\sin\\left(2π\\right)"
},
{
"type": "interim",
"title": "$$\\sin\\left(π\\right)\\sin\\left(2π\\right)=0$$",
"input": "\\sin\\left(π\\right)\\sin\\left(2π\\right)",
"steps": [
{
"type": "interim",
"title": "Simplificar $$\\sin\\left(π\\right):{\\quad}0$$",
"input": "\\sin\\left(π\\right)",
"result": "=0\\cdot\\:\\sin\\left(2π\\right)",
"steps": [
{
"type": "step",
"primary": "Utilizar la siguiente identidad trivial:$${\\quad}\\sin\\left(π\\right)=0$$",
"secondary": [
"$$\\sen\\left(x\\right)$$ tabla de valores periódicos con $$2πn$$ intervalos:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$"
],
"result": "=0"
}
],
"meta": {
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "interim",
"title": "$$\\sin\\left(2π\\right)=0$$",
"input": "\\sin\\left(2π\\right)",
"result": "=0\\cdot\\:0",
"steps": [
{
"type": "interim",
"title": "$$\\sin\\left(2π\\right)=\\sin\\left(0\\right)$$",
"input": "\\sin\\left(2π\\right)",
"result": "=\\sin\\left(0\\right)",
"steps": [
{
"type": "step",
"primary": "Reescribir $$2π$$ como $$2π+0$$",
"result": "=\\sin\\left(2π+0\\right)"
},
{
"type": "step",
"primary": "Utilizar la periodicidad de $$\\sin$$: $$\\sin\\left(x+2π\\right)=\\sin\\left(x\\right)$$",
"secondary": [
"$$\\sin\\left(2π+0\\right)=\\sin\\left(0\\right)$$"
],
"result": "=\\sin\\left(0\\right)"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "interim",
"title": "Utilizar la siguiente identidad trivial:$${\\quad}\\sin\\left(0\\right)=0$$",
"input": "\\sin\\left(0\\right)",
"steps": [
{
"type": "step",
"primary": "$$\\sen\\left(x\\right)$$ tabla de valores periódicos con $$2πn$$ intervalos:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$"
},
{
"type": "step",
"result": "=0"
}
],
"meta": {
"interimType": "Trig Trivial Angle Value Title 0Eq"
}
},
{
"type": "step",
"result": "=0"
}
],
"meta": {
"solvingClass": "Trig Evaluate",
"interimType": "Trig Evaluate"
}
},
{
"type": "step",
"primary": "Multiplicar los numeros: $$0\\cdot\\:0=0$$",
"result": "=0"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7eGdu+WKTow/xGVQ4i5h4qXrVK9CGpzwCGNCiiTAhk65wkKGJWEPFPk38sdJMsyPIc1L1JfkzeAMH8Sv8wAfVX5S35qHLu5GNQPuB3apEJSVKJjcQy0Dt0Qol+oll+/fx"
}
},
{
"type": "step",
"result": "=-2-0"
},
{
"type": "step",
"primary": "Simplificar"
},
{
"type": "step",
"result": "=-2"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq"
}
},
{
"type": "step",
"result": "m=-2"
}
],
"meta": {
"interimType": "General Equation Slope At Point 2Eq"
}
},
{
"type": "interim",
"title": "Encontrar la recta con pendiente m=$$-2$$ que pasa por $$\\left(π,\\:0\\right):{\\quad}y=-2x+2π$$",
"steps": [
{
"type": "step",
"primary": "Calcular la ecuación de la recta $$\\mathbf{y=mx+b}$$ con pendiente m=$$-2$$ que pasa por $$\\left(π,\\:0\\right)$$"
},
{
"type": "interim",
"title": "Calcular la intersección de $$y:{\\quad}b=2π$$",
"steps": [
{
"type": "step",
"primary": "Sustituir la pendiente $$-2$$ en $$y=mx+b$$",
"result": "y=\\left(-2\\right)x+b"
},
{
"type": "step",
"primary": "Sustituir $$\\left(π,\\:0\\right)$$: $$\\quad\\:x=π,\\:y=0$$",
"result": "0=\\left(-2\\right)π+b"
},
{
"type": "step",
"primary": "Despejar $$b$$"
},
{
"type": "interim",
"title": "$$0=\\left(-2\\right)π+b{\\quad:\\quad}b=2π$$",
"input": "0=\\left(-2\\right)π+b",
"steps": [
{
"type": "step",
"primary": "Intercambiar lados",
"result": "\\left(-2\\right)π+b=0"
},
{
"type": "step",
"result": "-2π+b=0"
},
{
"type": "interim",
"title": "Desplace $$2π\\:$$a la derecha",
"input": "-2π+b=0",
"result": "b=2π",
"steps": [
{
"type": "step",
"primary": "Sumar $$2π$$ a ambos lados",
"result": "-2π+b+2π=0+2π"
},
{
"type": "step",
"primary": "Simplificar",
"result": "b=2π"
}
],
"meta": {
"interimType": "Move to the Right Title 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7b+w5sKXXpsGhgRphlc6SFK9OOsSYPyjmaBpxiizP0Aba1kZfguPjcAGqoYBdjmVxm8TC9vlgXtjrQ7UXLN79Hd6OtnruhjmpPeHt9eeGCBqtMfoIfYB+jIoSvT79TF75xmjGSXtcoI1/PQEtNvwvV90Wljz5MlBRGHaUS3SDbfhIqhYGYEq+e9azIAcjkIYntn8PCnJ3zmfownEb/Rvo4Hpbum+kuog8Fan2Kq3Fphl5eXNvF0IdsgpbwTa1L0Ch6UCZIRn+/Q1+0oC7BLtMwMcfQs0dSnegVbt6Hnh1Kq2Gpgo2/xZW4v5OKdb2b6QbUgwmeOGRRlp/iq0RmzJeM6WCLSjpVfJyjFZphSRVDHmKiYd+pdN5bTdugccZFPeCZglPP2JjpHYk1N89fQpXm7b72/Ou2eXr7SZy9I2J4oVt02X/9BW4mNdfUNuMGQ25mJaT0YzFMcIouUXhHqE5CB531MgD8qeqYYEFhEuspObRbcB3EKGfV9fsX9ktKPd5SSi3s9xlL9jqXy3j10d+atbA+zX4bD3u3gx65o2NJhPeRU7XR57ZgIJ+R+CGtujsOOJ6Ho9QEonfXng6YbHLuAJCduYqHSm7ZRKoYUus89E="
}
}
],
"meta": {
"solvingClass": "Equations",
"interimType": "Equations"
}
},
{
"type": "step",
"result": "b=2π"
}
],
"meta": {
"interimType": "Line Equation Find Intersection From Point 0Eq"
}
},
{
"type": "step",
"primary": "Construir la ecuación de la recta $$\\mathbf{y=mx+b}$$ donde $$\\mathbf{m}=-2$$ y $$\\mathbf{b}=2π$$",
"result": "y=-2x+2π"
}
],
"meta": {
"interimType": "Line Equation Slope Point 6Eq"
}
},
{
"type": "step",
"result": "y=-2x+2π"
}
],
"meta": {
"solvingClass": "PreCalc"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"plotRequest": "tangent f(x)=\\sin(2x)\\cos(x),\\at x=π"
},
"showViewLarger": true
}
},
"meta": {
"showVerify": true
}
}
Solución
tangente de
Solución
Pasos de solución
Encontrar el punto de tangencia:
Encuentra la pendiente de
Encontrar la recta con pendiente m= que pasa por