Owlet talons Let L (t) equal the average leng...

Question

Owlet talons Let L (t) equal the average length (in mm) of the middle talon on an Indian spotted owlet that is t weeks old, as shown in the figure.

a. Estimate L' (1.5) and state the physical meaning of this quantity.

Accepted Answer

Welcome back, everyone. Given a function H of T that models the height in centimeters of a plant T days after it was planted, estimate H of 90 from the provided graph. What does this derivative indicate? And here we have our graph below our problem statement. Now how are we going to figure out each of 90 and how are we going to interpret what it means? Well, first, let's locate the point on the graph where T equals 90 days to find the corresponding height of the plant at this time. No notice here on our horizontal axis for T, OK. It increases by 10 centimeters per unit. OK. So that means this is the point where X is 90, and if we go to our curve, OK. And look across then at this point, OK, then. Our height H of T is 180, so the 09 1080 is on our curve, which tells us that at T equals 90 days. OK. The height of the plant, age of 90, is equal to 180 centimeters. Now how are we going to find H 90? Well, first of all, let's think about what the derivative H of T means. Recall that it represents the rate of change of the height of the plant with respect to time at any given day T. So if we draw a tangent line. If you were to draw a tangent line at our point, OK. That is a line that touches our curve only once at X equals 90. Then we should be able to estimate H of 90 using the slope of our tangent line because remember the slope represents the rate of change for our graph. So let's choose two other points that are close by, OK? We could use the points where X is 40, OK. As we can see here that when X is 40, sorry, when T is 40 H of T is 170, and we can also use the point where T is 140, OK? Because at that point we can also tell that H of T would be 190. So now, From the points of 40, uh. From the points 40, 170 and 140, 190 on our graph, OK. Remember that to find the slope of the graph. H T is going to be approximately equal to H of T2. Minus H of T1, OK. The difference in the vertical, uh, the vertical change, sorry, the vertical divided by the horizontal change T2 minus T1. So in this case, if we call 40 T1 and 178 of T1, while we call 140T2 and 1908 of T2, OK, then that means H of 90. And each prime of 90 is going to be approximately equal to 190 minus 170 divided by 140 minus 40. That would be equal to 20 divided by 100, which is approximately 1/5. So the derivative here indicates then that the plant has been growing at T equals 90 days since H 90 is greater than 0. Derivative is greater than 0. And this rate then tells us that at the point T equals 90 days, the plant, OK, let's make a note here. Therefore, the plant. Is growing Yeah. Add a re Of 1/5 of a centimeter per day. In other words, its height is changing by 1/5 of a centimeter for each day. That's what our derivative indicates. Thanks a lot for watching everyone. I hope this video helped.

Owlet talons Let L (t) equal the average length (in mm) of the middle talon on an Indian spotted owlet that is t weeks old, as shown in the figure.<IMAGE>
a. Estimate L' (1.5) and state the physical meaning of this quantity.

Key Concepts

Derivative

Average Length Function

Physical Interpretation

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