hi drew, studying for the sat and stumbled across your vid. it's been helpful but i wanted clarification on for the first question: how do you know x>k is wrong if you dont know k is?
We know if there are 2 POI’s then the graphs have to look something like the sketch we drew. The exponential function will eventually grow faster than the linear one after the second POI, so when x>k we have g(x) < f(x) instead
x is referring to somewhere on our x-axis. For example, if we are saying x>k then we are looking at the part of the function when the values on the x-axis are greater than x=k
I used demos for the last one and it gave me -7/2 and -139/250 as roots. so obviously 139 times 250 is going to be bigger but should I worry about desmos not rounding up and giving me bigger numbers instead of the ones I should actually use?
On the second problem did anyone else overcomplicate things and use calculus by expanding the g(x) function, taking the derivative, setting it equal to 0, and solving algebraically?
Creating g(x) and finding its minimum do not complicate things. That is what the question is asking for. You complicate if you are doing something that is not being asked.
We are making a substitution for a new variable of our choice. As long as we define our variable then we can make our equation in terms of the new variable (A, b, c, theta, basically anything) because they are equivalent
In the last problem, we can apply the GUESS & CHECK approach to get the answer quickly. A. 47 + 47 + 47 rad(2) Wrong B. 47 rad(2) + 47 rad(2) + 47(2) CORRECT ANSWER C. 94 + 94 + 94 rad(2) Wrong D. 94 rad(2) + 94 rad (2) + 94(2) Wrong Once you get the correct answer, there is no need to test the other answer choices. Your answer is correct by definition. Again, save time whenever you can.
@@DrewWerbowski You’re welcome. My suggestion is you describe several methods (if the problem allows) and then provide details on the preferred method.
For the purpose of the SAT exam, it is essential to get the answer as quickly as possible. In the second problem, f(x+5) is a horizontal shift operation. We can take the vertex of f(x), and then shift it 5 units to the left. The vertex of f(x) can be found as V(x) = -b/2a = -64/8 = -8 The vertex of f(x+5) is then equal to V(x+5) = -8 -5 = -13 If you cannot remember the shortcut formula, you can always complete the square. But do not complete the whole thing. Save time whenever you can. Completing the square, 4x^2 + 64x 4 (x^2 + 16x) 4 (x + 8)^2 The vertex is x = -8
You can take short cuts if you understand the principle. The formula makes sense because of the method shown in the video. If you fully understand then by all means use the formula!
Question for the last one: are the roots of the equation in the quadratic not effected by K? meaning when you did the quadratic formula, you disregarded K as if it wouldnt affect the entire equations factored form. isnt this a problem?
The roots of the function will remain the same regardless of the value of k. If you wanted to find k for this question, you could factor out 3 from the original function, then find the roots. You can verify that the roots will remain the same.
@@DrewWerbowski It is a good habit to factor out k before applying the quadratic formula. You will be working with smaller numbers. And you may be able just to factor by hand.
at 20:10 why are you allowed to just add a k wouldn't it alter the value of the equation to get the K wouldn't you need to factor out something from the two solutions
My notation was a bit sloppy there. There’s no need to add k. Just showing how by finding the roots we can express the function in the factored form shown in the question
Good video,but can I ask something. Since the definition of a bases states that not only must the list be linearly independent but must also span the entire vector space. How do we prove that or is it not necessary for this proof?
Yes, you’re right that we need to prove both the vectors are linearly independent and that they span the entire vector space the vectors are a basis for. There are multiple ways to show this and it depends on the question. I believe I have a video on my channel highlighting an example
i have a question here basis of kernal is empty then.it has no dimension..means zero dimension.but dimension.of Kernal equals nulity so here we seee nulity is 1
Nice, succinct explanation. I just wanted to confirm what an Eigenspace is and this was the perfect video. Didn't need to wade through a 20 minute exposition on all things "Eigen" to get there. Cheers!
Nice. A tip for the future, start off with the most central, interesting and motivating information, namely drawing the shortest distance from point P( 3,3,1), and from the start telling that the closest distance must be a line perpendicular to line L. THEN you construct the vector equations that gives you that sought vector ( the perpendicular one ) and the distance along it from 3,3,1 to the line.
There are unknown way to visualize subspace, or vector spaces. You can stretching the width of the x axis, for example, in the right line of a 3d stereo image, and also get depth, as shown below. L R |____| |______| TIP: To get the 3d depth, close one eye and focus on either left or right line, and then open it. This because the z axis uses x to get depth. Which means that you can get double depth to the image.... 4d depth??? :O p.s You're good teacher!
Why all of you guys placing vectors in matrix column by column you just do smth strange not RREF I mean you placed x x x y y y z z z Instead of x y z x y z x y z when you RREF second one you substract and adding some vectors to find depended once But actually thing that you did here seems to me wrong because you adding and substracting whole xs or whole ys or whole zs of different vectors WTF math always worked by logic,am i wrong or your solution is wrong,can someone explain
