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  1. K

    Rolle's theorem, to show there's only one root

    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...
  2. K

    Min max: y=sin x+cos x

    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
  3. K

    Min max: optimal quantity of medicine

    Homework Statement Homework Equations Minimum/Maximum occurs when the first derivative=0 The Attempt at a Solution $$R'=2D\left( \frac{C}{2}-\frac{D}{3} \right)-\frac{1}{3}D^2$$ $$R'=0~\rightarrow~D=C$$
  4. K

    Minimization problem: Economics: quantity to order

    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
  5. K

    Min max: how much of the tax to absorb

    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
  6. K

    Min max problem

    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...
  7. K

    Min Max problem: the shortest distance for a light ray

    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...
  8. K

    Min max problem: length of pipe from 2 towns to the river

    Homework Statement Homework Equations Minimum/Maximum occurs when the first derivative=0 The Attempt at a Solution $$l_1^2+l_2^2=(a^2+x^2)+[b^2+(d-x)^2]=a^2+b^2+x^2+\left[ \sqrt{c^2-(b-a)^2}-x \right]^2$$ $$(l_1^2+l_2^2)'=2x+2\left[ \sqrt{c^2-(b-a)^2}-x \right](-1)$$...
  9. K

    Finding the ratio of the volumes of a cube and a sphere

    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...
  10. K

    Disassembling a product to it's factors

    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...
  11. K

    How to minimize the cost of a silo

    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...
  12. K

    Minimum/Maximum problem

    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...
  13. K

    Maximum/minimum problem

    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...
  14. K

    Maximization problem — Stiffest beam that can be cut from a log

    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...
  15. K

    Graph drawing—Finding the points on a curve that are nearest to the origin

    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?
  16. K

    Maximum-minimum area from a fixed length rope

    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...
  17. K

    Local minimum and maximum

    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...
  18. K

    Price for maximum profit

    Homework Statement Homework Equations First derivative=maxima/minima/vertical tangent/rising/falling The Attempt at a Solution The profit S: $$S=nx=\left[ \frac{a}{x-c}+\frac{b}{100-x} \right]x$$ $$S'=\frac{a}{x-c}+\frac{b}{100-x}+\left[ \frac{-a}{(x-c)^2}+\frac{-b}{(100-x)^2} \right]x$$...
  19. K

    Maximum volume of a cone

    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 $$L=a^2+b^2~\rightarrow b^2=L-a^2$$ $$V=\frac{\pi}{3}a(L-a^2)$$...
  20. K

    Maximum area from fixed length fence

    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 $$A=4ab+a^2,~~V=a^2b=a^2\frac{A-a^2}{4a}=...=\frac{1}{4}a(A-a^2)$$...
  21. K

    Maximum volume from a rectangular cardboard

    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...
  22. K

    Minimum-maximum problem with graph plotting

    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}...
  23. K

    Drawing graphs by derivatives

    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...
  24. K

    Related rates shadow speed

    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...
  25. K

    Complex uniform continuity

    Homework Statement Homework Equations $$a^2-b^2=(a-b)(a+b)$$ The Attempt at a Solution $$a^2=\sqrt{1-x_2^2}\,\,\, ,\ \ b^2=\sqrt{1-x_1^2}$$ $$|a^2-b^2|=\left| \sqrt{1-x_2^2}-\sqrt{1-x_1^2} \right|=\left| \sqrt[4]{1-x_2^2} - \sqrt[4]{1-x_1^2} \right|\cdot\left| \sqrt[4]{1-x_2^2} +...
  26. K

    Derivative as definition

    Homework Statement Homework Equations Derivative as a limit: $$y'=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ The Attempt at a Solution $$f'(x)=\lim{\Delta x\to 0}\frac{f(x)f(\Delta x)-(1+xg(x))}{\Delta x}=\bigstar$$ $$\left\{ \begin{array}{l} f(\Delta x)=1+\Delta x...
  27. K

    Limit problem

    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...
  28. K

    Prove the Leibnitz rule of derivatives

    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...
  29. K

    Second derivative in parametric equations

    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? $$d^2x=(-2)\,dt^2,~~d^2y=(-6)t\,dt^2$$...
  30. K

    Prove sin(x)<x and more

    Homework Statement Homework Equations $$\sin\,x=a/r$$ The Attempt at a Solution $$\sin\,x=a/r, ~~x=\mbox{arc}(a)/r$$ $$\frac{a}{r}<\frac{\mbox{arc}(a)}{r}~\rightarrow~\sin\,x<x$$ $$\frac{2x}{\pi}=\frac{2\,\mbox{arc}(a)}{\pi r}>\frac{2a}{\pi r}$$ $$\sin\,x=a/r>\frac{2}{\pi}\frac{a}{r}$$...
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