Summer AP Calculus Work

[CheyenneH]

Hello! I have been having trouble with my summer AP work which is my ticket into the class. :| So, I need some help obviously as to why I'm here. I will try to show as much work as possible, but some of the questions I need help getting starting, so showing the first couple of steps to those would be tremendous. Most of the time I am a self-taught person, but being that I am going to be a senior in high school that speaks for itself.

Anyways for now I'll only ask two questions, both I have tried to get through, but neither have much work to them since the biggest problems I have with them is actually getting started.

Solve the equation both algebraically and graphically: |4x-3|=5(x+4)^(1/2)

The steps are really troubling me though, but I do not know how to get rid of the absolute bars. I've thought about squaring it since a square of either a negative or positive number is always positive, so that will keep it true, but I found out that it isn't plausible. Could someone help me get started with this equation?

And for my second one: Determine the range of: f(x)=13-20x-x^(2)-3x^(4). Also, find the max. and min. values of f(x), and state where those values occur.

So, I started by grouping (13-20x) together, but I find I can't factor it at all and I grouped (-x^(2)-3x^(4)) together. I factored that one into: -x^(2)(3x^(2)+1).

So, then I have: (13-20x)-x^(2)(3x^(2)+1). From there I'm stumped though. So, I would appreciate the help with this one.

Thanks for taking the time to read through and I hope that someone will be able to help me. ^-^

 

[galactus]

Solve the equation both algebraically and graphically: \(\displaystyle |4x-3|=5\sqrt{x+4}\)

With the absolute value, use the two cases.

\(\displaystyle 4x-3=5\sqrt{x+4}\)

\(\displaystyle -4x+3=5\sqrt{x+4}\)

and solve those for x. Can you do that?.

To get rid of the radical, square both sides.

Determine the range of: \(\displaystyle f(x)=13-20x-x^{2}-3x^{4}\). Also, find the max. and min. values of f(x), and state where those values occur.

To find the max and min, since it's calculus, they are probably expecting you to differentiate f(x), set to 0 and solve for x.

A polynomial is among the easiest to differentiate. Have you covered derivatives?. Upon differentiating, you will get a cubic. That is what you can factor and find the zeros.

 

[CheyenneH]

Solve the equation both algebraically and graphically: \(\displaystyle |4x-3|=5\sqrt{x+4}\)

With the absolute value, use the two cases.

\(\displaystyle 4x-3=5\sqrt{x+4}\)

\(\displaystyle -4x+3=5\sqrt{x+4}\)

and solve those for x. Can you do that?.

To get rid of the radical, square both sides.

Thank you! Now I remember how to do that, for some reason it wasn't coming to my mind at all! Yes that helps tremendously. I'll work it out now and see if I need more help, but I probably shouldn't because from there I think I can do it!

Determine the range of: \(\displaystyle f(x)=13-20x-x^{2}-3x^{4}\). Also, find the max. and min. values of f(x), and state where those values occur.

To find the max and min, since it's calculus, they are probably expecting you to differentiate f(x), set to 0 and solve for x.

A polynomial is among the easiest to differentiate. Have you covered derivatives?. Upon differentiating, you will get a cubic. That is what you can factor and find the zeros.

I believe I have covered derivatives, but its not coming to mind, so I might need to look over my past Pre-Calculus notes. Let me see if I can find those because I normally keep all my math notes, but that might be one of the subjects we went over some in Algebra 2 and unfortunately I had a terrible teacher for that course, so I have absolutely no notes from that course since the teacher never lectured, but just gave us questions to do everyday.

Thank you for helping me, hopefully once I come across some other problems or run into more trouble with the second question I'll be back for a bit more help.

 

[BigGlenntheHeavy]

\(\displaystyle First \ one: \ |4x-3| \ = \ 5(x+4)^{1/2}, \ 16x^{2}-24x+9 \ = \ 25(x+4) \ = \ 25x+100, \ squaring \ both \ sides.\)

\(\displaystyle Hence, \ 16x^{2}-49x-91 \ = \ 0, \ x \ = \ \frac{49 \pm 5\sqrt 329}{32} \ = \ 4.36536830425 \ or \ -1.30286830425\)

Since we are squaring a radical. one or both of our solutions may be false, however both ring true.

\(\displaystyle Second \ one: \ f(x) \ = \ 13-20x-x^{2}-3x^4\)

f ' (x ) = -20-2x-12x³, it doesn't factor, so using my trusty TI-89, I get Max = the point (-1.138799,29.433555).

This is a polynominal so its domain is always (-?,?) and its range is (-?,29.433555], There is no MIN.