1. Application of Derivatives (lengthy)

I will explain my thought process on each one....but I struggle heavily on Calculus.

1. Two opposite sides of a rectangle are each of length x. The perimeter of the rectangle is 12 feet. 1) Find the area as a function of x and 2) use the first derivative to find the maximum area (so here, i drew a rectangle, and labeled the sides, setting 12 equal to this...i multiplied the sides to get 6x -x^2, and that's where i am stuck...if i can get this, i think i may know what to do with the case of a triangle...but can you explain, what i must do different...different formula i presume??)

2. Given: 2x^3-8x^2+3x+3 has a relative max at point A and a relative min at point B 1) Find the coordinates of A and B and 2) Find the coordinates of the inflection point? (So here, I know that since the formula is given, you need to find the first derivative and set it equal to zero, to get the x values, do you just plug in the first derivative equation or the actual function to get the max and min? and for inflection point, it's just 2nd derivative right??)

3. An open box is constructed from a 12 by 18 inch rectangular piece of metal by cutting squares of lenth x inches from each corner and bending the sides. 1) Write the volume as a function of x and 2) Find the maximum possible volume of this box? (i drew a box and labeled the sides then drew the cut out squares, then placed it in the lwh eqn in terms of x...then i got the volume...but im stuck on finding the maximum possible value...i know that you cant have any number, seeing that then the cut out pieces wouldn't make sense)

4. Find the dimensions of the rectangle of maximum area that can be inscribed in an equilateral triangle of side length of 10 in, if two vertices of the rectangle lie on one of the sides of the triangle (im completely stuck on this problem...i tried labeling the base like 5-(x/2), x, and 5 - (x/2), and plug it into 1/2bh...but i doubth that works...im stuck..)

i'm struggling..i understand that...but i think if i get back on track..i will do good...

2. Originally Posted by vstexas09
I will explain my thought process on each one....but I struggle heavily on Calculus.

1. Two opposite sides of a rectangle are each of length x. The perimeter of the rectangle is 12 feet. 1) Find the area as a function of x and 2) use the first derivative to find the maximum area (so here, i drew a rectangle, and labeled the sides, setting 12 equal to this...i multiplied the sides to get 6x -x^2, and that's where i am stuck...if i can get this, i think i may know what to do with the case of a triangle...but can you explain, what i must do different...different formula i presume??)

2. Given: 2x^3-8x^2+3x+3 has a relative max at point A and a relative min at point B 1) Find the coordinates of A and B and 2) Find the coordinates of the inflection point? (So here, I know that since the formula is given, you need to find the first derivative and set it equal to zero, to get the x values, do you just plug in the first derivative equation or the actual function to get the max and min? and for inflection point, it's just 2nd derivative right??)

3. An open box is constructed from a 12 by 18 inch rectangular piece of metal by cutting squares of lenth x inches from each corner and bending the sides. 1) Write the volume as a function of x and 2) Find the maximum possible volume of this box? (i drew a box and labeled the sides then drew the cut out squares, then placed it in the lwh eqn in terms of x...then i got the volume...but im stuck on finding the maximum possible value...i know that you cant have any number, seeing that then the cut out pieces wouldn't make sense)

4. Find the dimensions of the rectangle of maximum area that can be inscribed in an equilateral triangle of side length of 10 in, if two vertices of the rectangle lie on one of the sides of the triangle (im completely stuck on this problem...i tried labeling the base like 5-(x/2), x, and 5 - (x/2), and plug it into 1/2bh...but i doubth that works...im stuck..)

i'm struggling..i understand that...but i think if i get back on track..i will do good...
1. The length of the rectangle is $\displaystyle x$. Let's call the width $\displaystyle y$.

The perimeter is 12 feet, so $\displaystyle P = 12 = 2x + 2y$

From this we get $\displaystyle y = 6 - x$.

The Area is given by $\displaystyle A = xy = x(6 - x) = 6x - x^2$.

Being a quadratic with a negative coefficient of $\displaystyle x^2$, this function has a maximum as its Turning Point. To find the maximum area, take the derivative and solve for x when the derivative is 0.

$\displaystyle \frac{dA}{dx} = 6 - 2x = 0$

So $\displaystyle x = 3$ and thus $\displaystyle y = 6 - 3 = 3$.

So the dimensions of the rectangle with maximum area are 3ft x 3ft, and thus the maximum area is $\displaystyle 9\textrm{ft}^2$.

3. thank you...can someone help with the a triangle....what would be different...do you use 1/2 b times h ???

4. Originally Posted by vstexas09
thank you...can someone help with the a triangle....what would be different...do you use 1/2 b times h ???
Yep, it's the exact same reasoning except the formula for area changes to 1/2 bh or 1/2xy depending what you wanna use.

5. does anyone know how to do 2222+2222*2222^2222, i know you have to use log...and the addition part does not matter....how do u do it?

6. Originally Posted by vstexas09
does anyone know how to do 2222+2222*2222^2222, i know you have to use log...and the addition part does not matter....how do u do it?
Unless I'm interpretting the question incorrectly... 2222+2222*2222^2222 should simplify to 2222+2222^2223 because for a general example, x*x^n=x^(n+1)

7. but then don't you have to use log to simplify it??

8. can someone show how log can be used...you have to rearrange right??

9. im really struggling with this

2222+2222*2222^2222....so the addition part doesnt make a difference and when u multiply...u just raise the exponent to one...

but how do u use log with that kind of problem??