R1

{
 "voice_prompt": "",
 "manuscript": {
   "title": {
     "text": "How Do You Estimate Growth Rate Numerically?",
     "audio": "How do you estimate growth rate numerically?"
   },
   "description": {
     "text": "You estimate growth rates numerically by analyzing small changes over specific time intervals. First, identify the function that describes the change. Then, calculate the change in quantity, delta h, over a small time interval, delta t.",
     "audio": "You estimate growth rates numerically by analyzing small changes over specific time intervals. First, identify the function that describes the change. Then, calculate the change in quantity, delta h, over a small time interval, delta t."
   },
   "scenes": [
     {
       "text": "Let’s say you have this scenario: The height of a plant, h, at any time, t, in days, is represented by the function: h of t is equal to 0.1 times t squared, plus 2 times t. Now, you want to estimate how fast the plant is growing on day 5.",
       "latex": "h(t) = 0.1t^2 + 2t"
     },
     {
       "text": "So, how do you approach this problem? First, you need to determine the change in height over a small time interval around day 5. Let’s set the time interval, delta t, to 0.01 days, which is equivalent to 14.4 minutes.",
       "latex": "\\Delta t = 0.01"
     },
     {
       "text": "Now, calculate the height at time t equal to 5. h of 5 equals 0.1 times 5 squared, plus 2 times 5, which equals 12.5 centimeters.",
       "latex": "h(5) = 0.1 \\cdot 5^2 + 2 \\cdot 5 = 12.5"
     },
     {
       "text": "Next, calculate the height at time t equal to 5.01. h of 5.01 equals 0.1 times 5.01 squared, plus 2 times 5.01, which equals 12.53001 centimeters.",
       "latex": "h(5.01) = 0.1 \\cdot (5.01)^2 + 2 \\cdot 5.01 = 12.53001"
     },
     {
       "text": "Now, you have the height at two time points. The next step is to find the change in height, delta h. Delta h is calculated as 12.53001 minus 12.5, which equals 0.03001 centimeters.",
       "latex": "\\Delta h = 12.53001 - 12.5 = 0.03001"
     },
     {
       "text": "Now, let’s estimate the growth rate by dividing the change in height, delta h, by the change in time, delta t: In this case, 0.03001 divided by 0.01 which equals 3.001 centimeters per day.",
       "latex": "\\text{Growth Rate} = \\frac{\\Delta h}{\\Delta t} = \\frac{0.03001}{0.01} = 3.001"
     },
     {
       "text": "And there you have it. By calculating the change in height over a small interval, you can numerically estimate the growth rate.",
       "latex": ""
     }
   ],
   "outro": {
     "text": "You estimate growth rates numerically by analyzing small changes over specific time intervals. First, identify the function that describes the change. Then, calculate the change in quantity, delta h, over a small time interval, delta t. This method is especially useful when working with nonlinear functions or when the exact derivative is difficult to calculate.",
     "audio": "You estimate growth rates numerically by analyzing small changes over specific time intervals. First, identify the function that describes the change. Then, calculate the change in quantity, delta h, over a small time interval, delta t. This method is especially useful when working with nonlinear functions or when the exact derivative is difficult to calculate."
   }
 }
}

en_R1_diff_numerical_method.jsonOpen with Text Editor Share

Displaying en_R1_diff_numerical_method.json.