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Saturday, March 21, 2009

Slope Fields: Graphically

  • AP Exam Practice Non-Calculator Quiz #3
  • Introduction to Differential Equations (Cont'd)
  • Slope Fields: Graphically


Since f'(1)=0, that means that when we integrate starting at x=1, we should integrate up to the point where the integration yields 0, to satisfy the condition that f'(x)=0. (Where else would f'(x)=0 besides at x=1?) We can see through symmetry that occurs at x=3.

To find P'(x), we differentiate P(x) via chain rule. Plug the numbers in to get P'(3) = 12. Straight-up application of chain rule.

It's given in the question that H(x)=f^-1(x). If we insert both sides of the equation into f(x), we get f(H(x)) = x, which is the key idea to solving the question. Use implicit differentiation to find H'(x). Solve for H'(x). Plug and chug.

Let's visualize the graph. sqrt(1-x^2) is the equation of a semi-circle, derived from the equation of a circle, x^2 + y^2 = 1, and x is a factor that distorts the perfect semi-circle when x is multiplied by sqrt(1-x^2).

Because the function given contains a square root function, we know that the function has a limited domain. (The radicand can't be negative.) So, the interval of the function is [-1,1].

Note that the question is asking for the total area, which means we make sure that the top area (above the x-axis) doesn't cancel the bottom area (below the x-axis). That can be avoided by taking the absolute value of the equation. But, in this case, if we graph x(1-x^2)^0.5, we see that the graph is symmetrical about the y-axis. So, we can instead find twice the area of one-half the semi-circle-like figure. Either way yields the same result.

To find the area underneath a function (integration), we antidifferentiate the function, in this case by integration by parts, and apply the fundamental theorem of calculus to get 2/3. This is shown on the slide in blue font.


Continuing from our previous class' introduction to differential equations...

a) We know that a(t)=-32 ft/s^2 is the acceleration due to gravity. Antidifferentiating a(t) yields velocity. We know v(0)=-16. So, we know from the family of velocity functions (-32t+C) that we're pinpointing the -32t-16 function. Antidifferentiating v(t) yields s(t), the position function. We know s(0)=96. So, we know from the family of position functions (-16t^2-16t+C) that we're pinpointing the -16t^2-16t+96 function.

b) When the rock strikes the ground, the stone would stop on the ground, so the stone's velocity would be zero. v(t)=0 at t =-3,2, which is determined by finding the roots of the velocity function. We reject t=-3 because we designated t=0 when we dropped the stone. At t=2, s(t)=0, which is determined by plugging t=2 into the s(t) function.

c) Plug t=2 into v(t) to get v(2)=-80. (Refer to part b.)


dy/dx = y, meaning that the derivative function is the parent function itself and vice-versa, That's e^x. So, the slope is 0 at all the points on y=0. The slope is 1 at all the points on y=1. The slope is -2 at all the points on y=-2. The slope is 5/pi at all the points on y=5/pi. Etc. We drew the tangent lines at lattice points to easily visualize this field of slopes.

So, if a bird were to be at coordinate (0,1) and the slope fields were the direction the wind is blowing, then we can see the bird flying in the same direction as the wind in such a way that its trail can be seen as the graph of a function, which is shown in green.


The rest of the slides are for homework.

Solution to 9.1 #9 is on the last slide.

The human rights assembly for The International Day for the Elimination of Racial Discrimination, hosted by The DMCI Human Rights Group, which we all enjoyed watching, is an assembly that occurs prior to Spirit Week. When Monday comes, Daniel McIntyre will enter the era of Spirit Week, a week of festivities hosted by Student Council, including Monday and Thursday period 2. So, Tuesday's scribe is .:. J + ME .:..

Spirit Week Themes
  1. Monday: Sports Day - Sports Obstacle Course
  2. Tuesday: Superhero Day - Save the Day Relay Race (you could dress up in a costume or wear a t-shirt that has a superhero logo on it)
  3. Wednesday: Twin Day - Talent Show
  4. Thursday: PJ/Crazy Hair Day - The Chris Frolic Comedy Hypnosis Show ($2/ticket)
  5. Friday: Colour Code Day - Grade War Gym Riot (gr.12s wear green; gr.11s wear blue)
In preparation for the comedy hypnosis show and to continue the tradition of a YouTube vid per scribe post, here's a YouTube video of THE INCREDIBLE BORIS! But, keep in mind that he's a Las Vegas performer, and Student Council has enough money to hire a Comedy Network performer.

Fortunately, this year's Student Council is carrying on the Daniel McIntyre tradition of having a hypnotist perform at our school. This only happens once every two years, and our graduating year just so happens to be that year! It's $2/ticket for Thursday, March 26 to see your friends get hypnotized. Proceeds may go to Grad Committee if enough fundraising is made.

This will also be Daniel McIntyre's last traditional 5-day Spirit Week. Let's make the best of it. =)

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