R1

manuscript.json

DetailsActivityApprovals

{
   "voice_prompt": "",
   "manuscript": {
       "title": {
           "text": "How do you understand Functions with Piecewise Definitions?",
           "audio": "How do you understand Functions with Piecewise Definitions?"
       },
       "description": {
           "text": "Piecewise functions are functions defined by different rules on different parts of the x-axis.",
           "audio": "Piecewise functions are functions defined by different rules on different parts of the x-axis."
       },
       "scenes": [
           {
               "text": "A taxi company has a flat fare for any trip up to 2 kilometers of 10 dollars. After 2 kilometers, the cab adds 3 dollars per kilometer.",
               "latex": "a+b",
               "pop_animation_prompt": "Create pop animations for the mathematical expression \"a+11\" as follows: When the transcript mentions \"free for the first hour,\" pop the element \"a\" (S1.a). When \"2 dollars per hour after that\" is referenced, pop the elements \"+\", \"11\" (S1.+, S1.11) together. Ensure the pop animations align with the timing of these references in the transcript."
           },
           {
               "text": "You can write the taxi fare as a piecewise function, $f(x)$, where $x$ is how many kilomteters you travel:",
               "latex": "f(x)= \\begin{cases} 10, & 0 \\le x \\le 2 \\\\ 10 + 3 \\bigl(x-2 \\bigr), & 2 < x  \\end{cases}",
               "pop_animation_prompt": "Create pop animations for the mathematical expression \"a+10\" as follows: When the transcript references the function, pop the entire expression \"a+10\". Ensure that the symbol \"a\", the operator \"+\", and the number \"10\" pop simultaneously to emphasize the complete expression."
           },
           {
               "text": "You write piecewise functions using a left bracket like this, and pay attention to which values of x each rule applies to",
               "latex": "a+71",
               "pop_animation_prompt": "Create pop animations for the mathematical expression \"a+71\". First, pop the symbol \"a\" S3.a to highlight the variable. Next, pop the operator \"+\" S3.+ to emphasize the addition operation. Finally, pop the number \"71\" S3.71 to focus on the constant value. Ensure each element pops as it is referenced in the sequence."
           },
           {
               "text": "If you travel any distance less than 2 kilometers, you pay 10 dollars.",
               "latex": "f(x) = 10, \\text{ for } 0 \\le x \\leq 2",
               "pop_animation_prompt": "Create pop animations for the following elements in sequence as they are referenced in the transcript: First, pop the numbers \"0\" and \"1\" when mentioned as the time range. Next, pop the expression \"C(t)\" when the cost function is introduced. Finally, pop the number \"0\" again when the cost is stated as zero dollars. Ensure each pop animation is clear and highlights the referenced elements effectively."
           },
           {
               "text": "For example, if you travel 1 kilometers, the cost is 10. Since x equals 1 which is less than 2, you use the function expression F of x equals 10.",
               "latex": "f(x) = 10 \\ F(1) = 10",
               "pop_animation_prompt": "Create pop animations for the mathematical expression. First, pop the number \"0.5\" when it is referenced in the transcript. Then, pop the entire expression \"C(0.5)=0\" when it is mentioned. Ensure the number \"0.5\" pops twice, once for each reference, and the expression pops once, highlighting the function's evaluation."
           },
           {
               "text": "If you travel more than 2 kilometers, the cost is calculated by $f(x)$ is 10 plus 3 times x minus 2.",
               "latex": "f(x) = 10 + 3(x-2), \\text{ for } x > 2",
               "pop_animation_prompt": "Create pop animations for the mathematical expression as follows: First, pop the elements \"t\", \">\", and \"1\" together when referenced. Next, pop the elements \"C\", \"(\", \"t\", and \")\" together when referenced. Finally, pop the element \"20\" when it is referenced. Ensure each pop animation occurs precisely at the time of its respective reference in the transcript."
           },
           {
               "text": "For example, if you travel  4 kilometers, the cost is F of 4 equals 10 plus 3 times 4 minus 2, which equals 10 plus 3 times 2. That is 16 dollars:",
               "latex": "F(4) = 10 + 3(4-2) = 10 + 3 \\times 2 = 16",
               "pop_animation_prompt": "Create pop animations for the mathematical expression \"C(2) = 20 \\times 2 = 40\". First, pop the number \"2\" when it is referenced in the transcript. Then, pop the entire expression \"C(2) = 20 \\times 2 = 40\" to emphasize the calculation process."
           },
           {
               "text": "The 'split point' is $x = 2$ kilometers. The rule for calculating the cost changes at that point.",
               "latex": "a+c",
               "pop_animation_prompt": "Create pop animations for the mathematical expression \"a+21\" as follows: When the transcript references the symbol \"a\" with , pop the element \"a\". When the transcript references the digit \"1\" with , pop the digit \"1\" within the number \"21\". Ensure these elements pop at the exact moment they are mentioned in the transcript to enhance understanding."
           },
           {
               "text": "What is the cost for traveling exactly 2 kilometers? Since the rule f of x equals 10 holds for all values of x less than or equal to 2. The cost is calculated like this: F of x equals 10. Insert 2 and you get f of 2 equals 10. Travelling exactly 2 kilometers cost 10 dollars.",
               "latex": "f(x) = 10 \\ F(2)=10",
               "pop_animation_prompt": "Create pop animations for the mathematical expression \"C(1) = 0\" as follows: First, pop the number \"1\" when it is mentioned in the context of \"1 hour\". Next, pop the symbol \"C\" when the rule is referenced. Finally, pop the number \"0\" when \"zero\" is mentioned. Ensure that the number \"1\" pops again when it is referenced a second time."
           },
           {
               "text": "Piecewise functions help model situations where conditions change, leading to different outcomes.",
               "latex": "a+d",
               "pop_animation_prompt": "Create pop animations for the mathematical expression \"a+31\". First, pop the symbol \"a\" S11.a as it is introduced. Next, pop the operator \"+\" S11.+ to highlight the addition operation. Finally, pop the number \"31\" S11.31 to complete the expression. Ensure each element pops in sequence as it is referenced."
           },
           {
               "text": "You'll see this in tiered pricing, motion scenarios, and many other real-world examples.",
               "latex": "a+e",
               "pop_animation_prompt": "Create pop animations for the mathematical expression \"a+41\". First, pop the symbol \"a\" (S12.a) when it is first mentioned. Next, pop the operator \"+\" (S12.+) and the number \"41\" (S12.41) together when they are referenced. Ensure the animations are synchronized with the references in the transcript."
           }
       ],
       "outro": {
           "text": "Piecewise functions are functions defined by different rules on different parts of the x-axis. ",
           "audio": "Piecewise functions are functions defined by different rules on different parts of the x-axis. "
       }
   }
}

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