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Anywho, we started off the class with a rates of change question in which we were split into three groups. The question was incredibly long to read and it was about a dingo?..named Wallaby. Instead of explaining the question to you, here it is!
Willy the Wallaby is a hard worker, but not devoted enough to work at the circus for almost 70 hours building a tank just so he could be shot out of a cannon like a metal ball. He is convinced there is a better job out there (and he wants to at least keep a shred of dignity). So Willy joins up with his friend Roo and his old coach Spike to seek out a tough guy name Wyman Weasel who has been sending out letter for a covert employment operation offering great pay for the ones with enough "guys and smarts" to find the secret hangout. The instructions began at the intersection of Crime Dr. and Twotough Blvd. The streetlight on that corner is 36 ft off the ground. The only ones who can get the job have to figure out the rate they have to be moving toward the south end of Crime Dr. in order to have their shadow changing length at a rate of 11/35 ft/sec. Once the applicant found the rate, the next instruction was to move at that pace and an escort will approach. The escort will take any animal that moves at the right pace to the clubhouse door. The final password to get in is the rate of change of the tip of the shadow. Willy and his friends want your help to finally get the good paying job they deserve. Find the rate Willy and his friends must move if they are all 4 ft 6 inches tall, and then find the rate of change of the tip of the shadow.
OKAYYY!! First, let's draw a handy dandy diagram. Its important to get the visual first when doing these types of questions to see what you're working with.
Now, we are given the rate the shadow, s, is moving, which is 11/35. We also know the height of Wallaby and his pals, which is 4'6, convert that 6 there into feet and we got 4.5, as shown in the diagram. We now have all the given elements to this question and we are asked to find the rate Willy and his friends are to move with that given height, in other words, we are looking for dd/dt. The second part asks for the rate of change of the tip of their shadows, so the last piece to this puzzle would be da/dt.
From there, we can then take a closer look at the diagram. Notice the relationship between the bases of the two triangles compared to a. That's right!
We can also derive the derivatives from that relationship in respect to time. Now with that, we can then move onto the the next inspection of the diagram. There are similar triangles, so let's make our ratio and solve for the rate of change of d.
If you're wondering how we went from line 2 to line 3, we basically just multiplied both sides of the equation by 36 so that it is easier to solve for dd/dt. This also helps save time, since on the exam, we are only given about 15 minutes to solve these questions (We took more than 30 mins to solve it in class XD).Alright! Now that we have the dd/dt, we can now solve for da/dt. All we have to do is put dd/dt and ds/dt back into the equation dd/dt + ds/dt = da/dt. Voila, we're done! You've now successfully helped Willy and friends :D!
Now, the next part of class consisted of us trying to find the root, using Newton's Method (x0 was -0.5) for the function:
f(x) = x + sinx - 2x + 1
Let's simplify this first...
f(x) = sinx - x + 1
Find the derivative next which is...
f'(x) = cosx - 1
Now we can use Newton's method to find x1. If you don't remember, the formula for Newton's method is as followed:
Using it is as simple as it looks like. Take the negative of x0, in this case it is -0.5, and divide that by the derivative of x0. Then add x0 to that to get x1. As you can probably see, it doesn't equal zero, so we have to keep going. Instead of using x0, now you use x1. If that still doesn't equal zero, then you keep going with x2 and so on till you get zero.
Mr. K also taught us how to use Newton's method using our calculators.
-First you have to graph f(x) into Y1.
-Then, you graph f'(x) into Y2.
-Now 2nd QUIT and press VARS, Function, Y1.
-Put an open bracket from there and input the value for x0.
-Close the bracket, then press STO-> (above the On key) and then Alpha A. This stores that value into A.
Go back to step 2, instead of going to Y1, go to Y2. The rest is the same till the last step, you'll be pressing Alpha B instead to store it into B. We've now got all our elements needed to solve using a calculator.
From there, first press the negative button and then Alpha A. Now divide that by Alpha B. Then add -0.5. Yippee Kayeeh! You've now found x1 using your calculator! Now repeat the steps using the consecutive x values found till you get it to equal zero!
*Pants* Alrighty now, the test is on Wednesday! Be sure to get your BOBs out by then. Also, if you haven't done it by now, Chapter 4.7, questions 11-22 ODD...I believe. Okie dokes, I am done. Oh yeah the next scribe is ...........Yinan. Have a nice day. =)
Using it is as simple as it looks like. Take the negative of x0, in this case it is -0.5, and divide that by the derivative of x0. Then add x0 to that to get x1. As you can probably see, it doesn't equal zero, so we have to keep going. Instead of using x0, now you use x1. If that still doesn't equal zero, then you keep going with x2 and so on till you get zero.Mr. K also taught us how to use Newton's method using our calculators.
-First you have to graph f(x) into Y1.
-Then, you graph f'(x) into Y2.
-Now 2nd QUIT and press VARS, Function, Y1.
-Put an open bracket from there and input the value for x0.
-Close the bracket, then press STO-> (above the On key) and then Alpha A. This stores that value into A.
Go back to step 2, instead of going to Y1, go to Y2. The rest is the same till the last step, you'll be pressing Alpha B instead to store it into B. We've now got all our elements needed to solve using a calculator.
From there, first press the negative button and then Alpha A. Now divide that by Alpha B. Then add -0.5. Yippee Kayeeh! You've now found x1 using your calculator! Now repeat the steps using the consecutive x values found till you get it to equal zero!
*Pants* Alrighty now, the test is on Wednesday! Be sure to get your BOBs out by then. Also, if you haven't done it by now, Chapter 4.7, questions 11-22 ODD...I believe. Okie dokes, I am done. Oh yeah the next scribe is ...........Yinan. Have a nice day. =)
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