Homework Statement
Homework Equations
Rolle's Theorem:
If f(a)=f(b)=0 then there is at least one a<c<b such that f'(c)=0
The Attempt at a Solution
$$y=2x^3-3x^2-12x-6~\rightarrow~y'=6x^2-6x-12$$
The function:
y':
How do i know y' isn't 0 somewhere? if it's continuously descending, so i...
Homework Statement
Homework Equations
Minimum/Maximum occurs when the first derivative=0
The Attempt at a Solution
$$y=\sin{x}+\cos{x}~\rightarrow~y'=\cos{x}-\sin{x}$$
$$y'=0:~\rightarrow~\cos{x}=\sin{x}~\rightarrow~x=\frac{\pi}{4}+n\cdot \pi$$
It's not correct
Homework Statement
Homework Equations
Minimum/Maximum occurs when the first derivative=0
The Attempt at a Solution
$$R'=2D\left( \frac{C}{2}-\frac{D}{3} \right)-\frac{1}{3}D^2$$
$$R'=0~\rightarrow~D=C$$
Homework Statement
Homework Equations
Minimum/Maximum occurs when the first derivative=0
The Attempt at a Solution
$$Q=\sqrt{\frac{2(K+pQ)}{h}}~\rightarrow~Q=\frac{2}{h}(KM+pM)$$
##Q'=0~## gives no sense result
Homework Statement
Homework Equations
Minimum/Maximum occurs when the first derivative=0
The Attempt at a Solution
$$y=20,000+60x,~~P=200-Ax$$
$$N=xP-Y=200x-Ax^2-20,000-60x,~~N'=140-2Ax$$
Two variables
Homework Statement
Homework Equations
marginal revenue[/B] (R') is the additional revenue that will be generated by increasing product sales by one unit
The Attempt at a Solution
I don't know how to start. Q is the number of items sold at price x. y is the marginal cost, the cost of...
Homework Statement
Homework Equations
Minimum/Maximum occurs when the first derivative=0
GM≤AM: ##~\sqrt{xy}\leq\frac{x+y}{2}##
The Attempt at a Solution
[/B]
If the sum of squares of the distances (setup 2) in an arbitrary point is bigger than the sum of the squares of the shortest...
Homework Statement
Homework Equations
Minimum/Maximum occurs when the first derivative=0
The Attempt at a Solution
$$l_1^2+l_2^2=(a^2+x^2)+[b^2+(d-x)^2]=a^2+b^2+x^2+\left[ \sqrt{c^2-(b-a)^2}-x \right]^2$$
$$(l_1^2+l_2^2)'=2x+2\left[ \sqrt{c^2-(b-a)^2}-x \right](-1)$$...
Homework Statement
Homework Equations
Volue of a sphere: ##~\displaystyle V=\frac{4}{3}\pi r^3##
Area of a sphere: ##~\displaystyle A=4\pi r^2##
Minimum/Maximum occurs when the first derivative=0
The Attempt at a Solution
The fixed area is k, the edge is a:
$$6a+4\pi...
Homework Statement
In a physics problem where V is the volume i have ##\displaystyle~3V-\frac{3}{4}V~##. i get 2 different answers when i calculate.
Homework Equations
$$a(b-c)=ab-ac$$
The Attempt at a Solution
I can:
$$3V-\frac{3}{4}V=3\left( 1-\frac{1}{4} \right)V=3\frac{3}{4}V$$
And if i...
Homework Statement
Homework Equations
Maxima/minima are where the first derivative is 0
Volue of a hemisphere: ##~\displaystyle V=\frac{2}{3}\pi r^3##
Area of a hemisphere: ##~\displaystyle A=2\pi r^2##
The Attempt at a Solution
$$A=2\pi rh+2\pi r^2=2\pi[h+r],~~V=\pi r^2h+\frac{2}{3}\pi...
Homework Statement
Homework Equations
Maxima/minima are where the first derivative is 0
The Attempt at a Solution
Surface area of the whole metal for creating the can (the end pieces i take as squares):
$$A=2\pi h+8r^2,~~V=\pi r^2 h~~\rightarrow~h=\frac{V}{\pi r^2}$$
$$A=2\pi \frac{V}{\pi...
Homework Statement
Homework Equations
Maxima/minima are where the first derivative is 0
The Attempt at a Solution
$$r^2=\frac{x^2}{4}+h^2~\rightarrow~h^2=r^2=\frac{x^2}{4}$$
S will be the strength of the light through the whole opening: the semicircle and the rectangle inside.
I take the...
Homework Statement
Homework Equations
Pitagora's: ##~a^2+b^2=c^2##
Maxima/minima are where the first derivative is 0
The Attempt at a Solution
$$\left( \frac{a}{2} \right)^2+\left( \frac{b}{2} \right)^2=r^2~\rightarrow~b^2=\frac{16r^2}{a^2}$$
The strength S has the proportion coefficient k...
Homework Statement
Homework Equations
First derivative=maxima/minima/vertical tangent/rising/falling
When f'(x)>0 ? the function rises
The Attempt at a Solution
Deriving relative to x:
$$10x-6(yy'+x)+10yy'=0~\rightarrow~y'=-\frac{x}{y}$$
What do i do with that?
Homework Statement
Homework Equations
Area of triangle=(base x height)/2
The Attempt at a Solution
a is the side of the triangle.
Area of an equilateral triangle: ##~\displaystyle \frac{a}{2h}=\tan 30^0=\frac{\sqrt{3}}{3}##
$$\rightarrow A_t=\frac{\sqrt{3}}{4}a^2$$
Side of the rectangle...
Homework Statement
Homework Equations
At local minimum or maximum the first derivative equals zero
The Attempt at a Solution
a) $$f'=2x-\frac{a}{x^2},~~2\cdot 2-\frac{a}{4}=0~\rightarrow a=16$$
Near 0 from the left a/x gets large negative values, smaller than for a=16. that's my proof for...
Homework Statement
Homework Equations
First derivative=maxima/minima/vertical tangent/rising/falling
The Attempt at a Solution
The profit S:
$$S=nx=\left[ \frac{a}{x-c}+\frac{b}{100-x} \right]x$$
$$S'=\frac{a}{x-c}+\frac{b}{100-x}+\left[ \frac{-a}{(x-c)^2}+\frac{-b}{(100-x)^2} \right]x$$...
Homework Statement
Homework Equations
First derivative=maxima/minima/vertical tangent/rising/falling
Volume of a cone: ##~\displaystyle V=\frac{\pi}{3}r^2h##
The Attempt at a Solution
$$L=a^2+b^2~\rightarrow b^2=L-a^2$$
$$V=\frac{\pi}{3}a(L-a^2)$$...
Homework Statement
Homework Equations
First derivative=maxima/minima/vertical tangent/rising/falling
The Attempt at a Solution
a are the sides of the base and b is the height
$$A=4ab+a^2,~~V=a^2b=a^2\frac{A-a^2}{4a}=...=\frac{1}{4}a(A-a^2)$$...
Homework Statement
Homework Equations
First derivative=maxima/minima/vertical tangent/rising/falling
When f'(x)>0 → the function rises
The Attempt at a Solution
$$V=(15-2a)(8-2a)=4a^2-46a+120$$
$$V'=8a-46,~~V'=0\rightarrow a=5.75$$
But: ##~2a<8,~~V(a=4)=0##
So 5.75>4
And: ##~V''=8## so it...
Homework Statement
The first derivative of the area ##~\displaystyle A(a)=a\sqrt{r^2-\frac{a^2}{4}}## is positive everywhere
Homework Equations
When f'(x)>0 → the function rises
The Attempt at a Solution
$$A'=a\frac{1}{2}\left( r^2-\frac{a^2}{4} \right)^{-1/2}\cdot\left( \frac{1}{4}...
Homework Statement
Only 15
Homework Equations
First derivative=maxima/minima/vertical tangent/rising/falling
Second derivative=points of inflection/concave upward-downward
The Attempt at a Solution
$$x=y^3+3y^2+3y+2~\rightarrow~1=3(y^2+2y+1)y'$$
$$y'=\frac{1}{3(y+1)^2}>0,~y\neq...
Homework Statement
2. Homework Equations
Similar triangles
The Attempt at a Solution
$$\frac{y}{30}=\frac{50}{x}~\rightarrow~x=\frac{1500}{y}$$
$$\frac{dx}{dt}=-\frac{1500}{y^2}$$
$$s=16t^2=16\frac{1}{4}=4$$
$$\frac{dx}{dt}=-\frac{1500}{16}$$
The answer should be ##~\displaystyle...
Homework Statement
Prove ##~\displaystyle \lim_{x \to \infty}\left(x\sin\left(\frac{\pi}{x}\right)\right)=1##
Homework Equations
$$\lim_{x \to 0} \frac{\sin\pi}{x}=0$$
The Attempt at a Solution
If i could multiply ##~x\sin\left(\frac{\pi}{x}\right)~## with something that would cancel the sin...
Homework Statement
Homework Equations
Newton's binomial's: ##(a+b)^n=C^0_n a^n+C^1_n a^{n-1}b+...+C^n_n b^n##
The Attempt at a Solution
I use induction and i try to prove for n+1, whilst the formula for n is given:
$$\frac{d^{n+1}(uv)}{dx^{n+1}}=\frac{d}{dx}\frac{d^{n}(uv)}{dx^n}=$$
The...
Homework Statement
Only the second part
Homework Equations
Second derivative:
$$\frac{d^2y}{dx^2}=\frac{d}{dx}\frac{dy}{dx}$$
The Attempt at a Solution
$$dx=(1-2t)\,dt,~~dy=(1-3t^2)\,dt$$
Do i differentiate the differential dt?
$$d^2x=(-2)\,dt^2,~~d^2y=(-6)t\,dt^2$$...