# Click to start! StudyPracticeQuizApplication Welcome to beautiful…

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Click to start! StudyPracticeQuizApplication Welcome to beautiful…

Hi! My name is Dale Derivative, and Ill be working with you today! Before we head into the city, lets catch you up on what goes on around here! Click to get into Dales Cardx

Let me introduce you to my car! Click to go forward Click to go back Click to go home dx Questions will Display Here

Fish0 411 The screens give solutions to my questions, just click them to select an answer! Lets calibrate: 2+2= dx

XX XX You may want to try that Again, Im not sure it worked… dx

OO OO Alright! I think were ready to head on out! Lets see what there is to know about this place! Its okay if you didnt read the brochure. dx

Here in Function Junction, were crazy about functions! Not just how they work or look, though, but how they MOVE dx

dx Just like the hills on the Function Junction roads, the functions have a SLOPE, and it CHANGES every time you make the slightest movement on it; unless its straight, of course.

But you probably read all about that in the brochure! Lets get into the nitty-gritty… dx

Function Junction maps work a lot like your coordinate planes! Each roads a function, and here we study the CHANGE dx

dx Just pull the screen out of the center console! Before you Got here, I prepared a little in-depth look at how we study the changes in our roads, and how you can help! Screen

x!xnxn nx n+1 f(x) Great! So now that you know our purpose here, lets see what you can do! What is the derivative of f(x)? dx

XX XX Think About it this way, if f(x) is the original function, what do we call the derivative? dx

OO OO Right! The Derivative of a function f(x) is f(x)! dx

nx n+1 x n+1 /n nx n-1 x n-1 Now for something a little more specific… What is the derivative of x n ?dx

XX XX If you need a little help on this one, you can flip the screen back up! dx Screen

OO OO Thats right! You multiply x by the exponent and subtract 1 from the exponent! dx

x 5 /44x 3 x3x3 4x 5 Alright, so lets try something new… What is the derivative of x 4 ? dx

XX XX Almost! You multiply by the exponent and subtract 1 from the exponent! dx

OO OO Thats right! I think youve gotten the hang of it. Lets move on to WHY we use derivatives! dx

dx Just as a flat road may have a slope – uphill, downhill, or straight – so, too, does a curved road!

dx In Function Junction, most of our roads arent flat, and they move like functions. They usually look like these!

dx Here in Function Junction, our cars only move on the roads if they know the slope of the road. Otherwise they could go right through! Lines are very thin!

dx Thats why we, the people of Function Junction, need derivatives!

dx I have a boring mathematicians explanation on the console screen so that you can better understand how it all works! Screen

These tangent lines are how the car positions itself to move along functions. dx

(2,12)(2,4) (2,16)(2,8) So lets guide you through creating a tangent line. If f(x)=x 3, and Im at x=2, what is my coordinate position?dx

XX XX Close. Think of it like this, if Im on f(x), and Im at x=2, what is my position, (x,f(x))? dx

3x 2 3x 4 x 2 /3x 4 /3 Correct! Now, if Im going to find the slope, I need the derivative! What is f(x) if f(x)=x 3 dx

XX XX We might need to change that answer around, or well crash on that function! The derivative of x n is nx n-1 ! dx

3612 49 Right! So we know where we are, (2,8), and our derivative is 3x 2, so… What is f(2)dx

XX XX Try that again, our slope will be defined by f(x)=3x 2. To find f(2), plug in 2 for x! dx

y=12xy=12x-16 y=12x+16y=12x-8 Great! So one last step, to define the line! dx Use the point- slope formula to define a line of slope 12 that goes through (2,8)

XX XX Close, the point-slope formula is (y-y 1 )=m(x-x 1 ), so with our values, simplify (y-8)=12(x-2)! dx

Correct! And that is our tangent line! Following these steps, you can get along any basic road in Function Junction! So now that were at the car rental store, go grab a car and get a move on! Rent a car!dx

dx Alright! Now that you have your own car, you can get yourself around this town – if you know how! Ill meet you at my house, my wife, Iggy Integral and I have a spare room for you while you vacation here! Enter your new car!

Your car is designed to drive itself. All you need to do as the pilot is navigate it by giving it a derivative and a tangent line for the point you begin at on the curve. You have five curves ahead before you reach Dales place! Your car will generate guesses of the approaching curves analytics. You only have to select the correct guess. Your car will deny incorrect guesses, and will remain stationary until it is certain of your answer. All piloting will be controlled from this onboard navigation screen. Good Luck!

f(x)= 2x y=0 f(x)= 2x 3 y=0 f(x)=2x 3 y=1 f(x)= x 3 /2 y=1 We are approaching a curve f(x)=x 2. We will enter at the point (0,0). I need the derivative and the initial tangent line.

ERROR Upon further analysis, the pilots judgment is found to be invalid. The car will be stopped until the pilots judgment is corrected.

f(x)= x 2 /3 y=x/3-2/3 f(x)= x 4 /3 y=x/3-2/3 f(x)=3x 2 y=3x+2 f(x)= 3x 4 y=3x+2 We are approaching a curve f(x)=x 3. We will enter at the point (-1,-1). I need the derivative and the initial tangent line.

ERROR Upon further analysis, the pilots judgment is found to be invalid. The car will be stopped until the pilots judgment is corrected.

f(x)= 0 y=0 f(x)= 1 y=x f(x)=x 2 y=4x-6 f(x)= x 2 y=4x+6 We are approaching a curve f(x)=x. We will enter at the point (2,2). I need the derivative and the initial tangent line.

ERROR Upon further analysis, the pilots judgment is found to be invalid. The car will be stopped until the pilots judgment is corrected.

f(x)= 4x 3 y=32x-56 f(x)= x y=2x+4 f(x)=2x y=4x f(x)= 4x y=8x-8 We are approaching a curve f(x)=2x 2. We will enter at the point (2,8). I need the derivative and the initial tangent line.

ERROR Upon further analysis, the pilots judgment is found to be invalid. The car will be stalled until the pilots judgment is corrected.

f(x)= 4x 2 /3 y=4x/3+8/3 f(x)= 12x 2 y=12x-8 f(x)=12x 4 y=12x+16 f(x)= 4x 2 y=4x We are approaching a curve f(x)=4x 3. We will enter at the point (1,4). I need the derivative and the initial tangent line.

ERROR Upon further analysis, the pilots judgment is found to be invalid. The car will be stopped until the pilots judgment is corrected.

We have arrived at the destination point! You may turn off the car and enjoy your day!

dx Welcome to our abode! Youre welcome to spend the rest of your day doing whatever you want to do around town! Just try to make it for dinner, Iggys got something delicious cooking!

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