Category Archives: applications of maths

More questions on applications of derivatives: IITJEE mains maths tutorial

  1. Prove that the minimum value of (a+x)(b+x)/(c+x) for x>-c, is (\sqrt{a-c}+\sqrt{b-c})^{2}.
  2. A cylindrical vessel of volume 25\frac{1}{7} cubic meters, open at the top, is to be manufactured from a sheet of metal. Find the dimensions of the vessel so that the amount of metal used is the least possible.
  3. Assuming that the petrol burnt in driving a motor boat varies as the cube of its velocity, show that the most economical speed when going against a current of c kmph is 3c/2 kmph.
  4. Determine the altitude of a cone with the greatest possible volume inscribed in a sphere of radius R.
  5. Find the sides of a rectangle with the greatest possible perimeter inscribed in a semicircle of radius R.
  6. Prove that in an ellipse, the distance between the centre and any normal does not exceed the difference between the semi-axes.
  7. Determine the altitude of a cylinder of the greatest possible volume which can be inscribed in a sphere of radius R.
  8. Break up the number 8 into two parts such that the sum of their cubes is the least possible. (Use calculus techniques, not intelligent guessing).
  9. Find the greatest and least possible values of the following functions on the given interval: (i) y=x+2\sqrt{x} on [0,4]. (ii) y=\sqrt{100-x^{2}} on [-6,8] (iii) y=\frac{a^{2}}{x}+\frac{b^{2}}{1-x} on (0,1) with a>0 and b>0 (iv) y=2\tan{x}-\tan^{2}{x} on [0,\pi/2) (iv) y=\arctan{\frac{1-x}{1+x}} on [0,1].
  10. Prove the following inequalities: (in these subset of questions, you can try to use pure algebraic methods, apart from calculus techniques to derive alternate solutions): (i) 2\sqrt{x} >3-\frac{1}{x} for x>1 (ii) 2x\arctan{x} \geq \log{(1+x^{2})} (iii) \sin{x} < x-\frac{x^{3}}{6}+\frac{x^{5}}{120} for x>0. (iv) \log{(1+x)}>\frac{\arctan{x}}{1+x} for x>0 (iv) e^{x}+e^{-x} > 2+x^{2} for x \neq 0.
  11. Find the interval of monotonicity of the following functions: (i) y=x-e^{x} (ii) y=\log{(x+\sqrt{1+x^{2}})} (iii) y=x\sqrt{ax-x^{2}} (iv) y=\frac{10}{4x^{3}-9x^{2}+6x}
  12. Prove that if 0<x_{1}<x_{2}<\frac{\pi}{2}, then \frac{\tan{x_{2}}}{\tan{x_{1}}} > \frac{x_{2}}{x_{1}}
  13. On the graph of the function y=\frac{3}{\sqrt{2}}x\log{x} where x \in [e^{-1.5}, \infty), find the point M(x,y) such that the segment of the tangent to the graph of the function at the point intercepted between the point M and the y-axis, is the shortest.
  14. Prove that for 0 \leq p \leq 1 and for any positive a and b, the inequality (a+b)^{p} \leq a^{p}+b^{p} is valid.
  15. Given that f^{'}(x)>g^{'}(x) for all real x, and f(0)=g(0), prove that f(x)>g(x) for all x \in (0,\infty), and that f(x)<g(x) for all x \in (-\infty, 0).
  16. If f^{''}(x)<0 for all x \in (a,b), prove that f^{'}(x)=0 at most once in (a,b).
  17. Suppose that a function f has a continuous second derivative, f(0)=0, f^{'}(0)=0, f^{''}(x)<1 for all x. Show that |f(x)|<(1/2)x^{2} for all x.
  18. Show that x=\cos{x} has exactly one root in [0,\frac{\pi}{2}].
  19. Find a polynomial P(x) such that P^{'}(x)-3P(x)=4-5x+3x^{2}. Prove that there is only one solution.
  20. Find a function, if possible whose domain is [-3,3], f(-3)=f(3)=0, f(x) \neq 0 for all x \in (-3,3), f^{'}(-1)=f^{'}(1)=0, f^{'}(x)>0 if |x|>1 and f^{'}(x)<0, if |x|<1.
  21. Suppose that f is a continuous function on its domain [a,b] and f(a)=f(b). Prove that f has at least one critical point in (a,b).
  22. A right circular cone is inscribed in a sphere of radius R. Find the dimensions of the cone, if its volume is to be maximum.
  23. Estimate the change in volume of right circular cylinder of radius R and height h when the radius changes from r_{0} to r_{0}+dr and the height does not change.
  24. For what values of a, m and b does the function: f(x)=3, when x=0; f(x)=-x^{2}+3x+a, when 0<x<1; and f(x)=mx+b, when 1 \leq x \leq 2 satisfy the hypothesis of the Lagrange’s Mean Value Theorem.
  25. Let f be differentiable for all x, and suppose that f(1)=1, and that f^{'}<0 on (-\infty, 1) and that f^{'}>0 on (1,\infty). Show that f(x) \geq 1 for all x.
  26. If b, c and d are constants, for what value of b will the curve y=x^{3}+bx^{2}+cx+d have a point of inflection at x=1?
  27. Let f(x)=1+4x-x^{2} \forall x \in \Re and g(x)= \left\{  \begin{array}{ll} max \{ f(x): x\leq t\leq x+3\} & 0 \leq x \leq 3\\ min (x+3) & 3 \leq x \leq 5 \end{array} \right. Find the critical points of g on [0,5]
  28. Find a point P on the curve x^{2}+4y^{2}-4=0 so that the area of the triangle formed by the tangent at P and the co-ordinate axes is minimum.
  29. Let f(x) = \left \{ \begin{array}{ll} -x^{3}+\frac{b^{3}-b^{2}+b-1}{b^{2}+3b+2} & 0 \leq x <1 \\ 2x-3 & 1 \leq x \leq 3 \end{array}\right. Find all possible real values of b such that f(x) has the smallest value at x=1.
  30. The circle x^{2}+y^{2}=1 cuts the x-axis at P and Q. Another circle with centre Q and variable radius intersects the first circle at R above the x-axis and line segment PQ at x. Find the maximum area of the triangle PQR.
  31. A straight line L with negative slope passes through the point (8,2) and cuts the positive co-ordinate axes at points P and Q. Find the absolute minimum value of OP+PQ, as L varies, where O is the origin.
  32. Determine the points of maxima and minima of the function f(x)=(1/8)\log{x}-bx+x^{2}, with x>0, where b \geq 0 is a constant.
  33. Let (h,k) be a fixed point, where h>0, k>0. A straight line passing through this point cuts the positive direction of the co-ordinate axes at points P and Q. Find the minimum area of the triangle OPQ, O being the origin.
  34. Let -1 \leq p \leq 1. Show that the equations 4x^{3}-3x-p=0 has a unique root in the interval [1/2,1] and identify it.
  35. Show that the following functions have at least one zero in the given interval: (i) f(x)=x^{4}+3x+1, with [-2,-1] (ii) f(x)=x^{3}+\frac{4}{x^{2}}+7 with (-\infty,0) (iii) r(\theta)=\theta + \sin^{2}({\theta}/3)-8, with (-\infty, \infty)
  36. Show that all points of the curve y^{2}=4a(x+\sin{(x+a)}) at which the tangent is parallel to axis of x lie on a parabola.
  37. Show that the function f defined by f(x)=|x|^{m}|x-1|^{n}, with x \in \Re has a maximum value \frac{m^{m}n^{n}}{(m+n)^{m+n}} with m,n >0.
  38. Show that the function f defined by f(x)=\sin^{m}(x)\sin(mx)+\cos^{m}(x)\cos(mx) with x \in \Re has a minimum value at x=\pi/4 which m=2 and a maximum at x=\pi/4 when m=4,6.
  39. If f^{''}(x)>0 for all x \in \Re, then show that f(\frac{x_{1}+x_{2}}{2}) \leq (1/2)[f(x_{1})+f(x_{2})] for all x_{1}, x_{2}.
  40. Prove that (e^{x}-1)>(1+x)\log(1+x), if x \in (0,\infty).

Happy problem solving ! Practice makes man perfect.

Cheers,

Nalin Pithwa.

 

 

 

Maths and the Bomb: Sir Michael Atiyah at 80

Just paying yet another tribute to Sir Michael Atiyah (re-sharing one of the articles I have collected about him):

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Maths and the bomb Sir Michael Atiyah at 80

The Times

April 21 2009

When, five years ago, he shared the £480,000 Abel Prize, the equivalent of a Nobel prize in the world of mathematics, Sir Michael Atiyah might have listened to his wife’s urgings to put his feet up and settle into a comfortable life. But that would not have been his style. “Some mathematicians retire,” he concedes with a smile. “I don’t think I have.”

This week, Sir Michael’s 80th birthday and a life dedicated to science and political activism is celebrated in a series of events. A three-day conference celebrating his contribution to geometry and physics, at the University of Edinburgh Informatics Forum, ends today, his birthday. Tomorrow and on Friday his Sir Michael’s role in promoting disarmament is recognised with readings and lectures dedicated to exposing the folly of nuclear weapons.

Much has been achieved at an age when contemporaries might have settled for a quiet life. In 1995, as president of the Royal Society and aged 67, Sir Michael made a stinging attack on Britain’s nuclear weapons policy.

Subsequently he accepted the presidency of the influential Pugwash disarmament conferences, which unite scientists in opposition to the arms race.

He still believes passionately in the cause, which, he says, is more important to the world than maths, “because if we blow ourselves up, there will be no mathematics anyway”.

Sir Michael discovered his aptitude for mathematics during his boyhood in the Sudan. His Lebanese father was an Oxford graduate and a civil servant, his mother was Scottish and he grew up regarding himself as British, studying at Manchester Grammar School and Cambridge University.

The key professional encounters in his life came in the United States in the 1950s, when he joined the Institute for Advanced Study, at Princeton University, a gathering place for the world’s most brilliant mathematical minds. Here he forged relationships which have endured, and much of his greatest work has come from what he calls the “dialogues of ideas” established there.

His greatest achievement has been the Atiyah-Singer theorem, which secured his fame and prize money, shared with his collaborator, Isadore Singer, of the US. At the time, he said he couldn’t think what to do with his share; the sporty red Lexus parked outside the Informatics building suggests he has since given it more thought.

In simple terms, the theorem provided a kind of analytical bridge which could be shifted between disciplines. “The theorem technique enables you to get to an answer by-passing all the intervening calculations,” he says. The idea “was something where you could calculate numbers of solutions by very indirect methods which applied in a very wide range of situations: geometry, algebra, physics…”

Maths, he says, is something he plays out in his mind as he walks around his flat and his garden, and he jots things down – “the dull stuff” – only when he has to check something.

“Walking helps the physiological process. You have to maintain a very high pitch of concentration when you do mathematics. It’s illumination – shining the mind’s eye on a problem and really seeing through it.

“The old clichés about the beauty of maths are true. It has beauty within it, but not all parts are equally beautiful. Beauty in mathematics is the thing that helps you in the search for truth.”

Some people, he believes, are born with mathematical brains, although they might choose other careers. One former student won the Nobel Prize for Economics, another is the best-paid hedge fund manager in the US. So was Sir Michael never tempted to use his mathematical skill in a wider world? Could he have solved the global financial crisis?

“Economics is a combination of gambling, psychology and who knows what,” he says. “The current crisis? I think people made a bloody mess. You can foretell that the bubble will burst – the question is when. If you gambled on it you might win or lose a lot of money. I just didn’t gamble.”

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Regards,

Nalin Pithwa.

 

Sir Michael Atiyah passes away; tribute by The New York Times

Permutations and Combinations: A primer only

Math moments: uses of mathematics in today’s world

Mathematics versus Physics

The object of pure Physics is the unfolding of the laws of the intelligible world; the object of pure Mathematic that of unfolding the laws of human intelligence. — J. J. Sylvester.

In my opinion, for example, Boole’s Laws of (Human) Thought. 

Applications of Derivatives: Training for IITJEE Maths: Part V

Question I:

Show that the equation of the tangent to the curve x=a \frac{f(t)}{h(t)} and y=a \frac{g(t)}{h(t)} can be represented in the form:

\left| \begin{array}{ccc} x & y & a \\ f(t) & g(t) & h(t) \\ f^{'}(t) & g^{'}(t) & h^{'}(t) \end{array} \right|=0

Question 2:

Show that the derivative of the function f(x) = x \sin{\frac{\pi}{x}}, when x>0 and f(x)=0, when x=0 vanishes on an infinite set of points of the interval $latex (0,1), Hint: Use Rolle’s theorem.

Question 3:

Prove that \frac{1}{1+x}<\log (1+x) < x for x>0. Use Lagrange’s theorem.

Question 4:

Find the largest term in the sequence a_{n}=\frac{n^{2}}{n^{3}+200}. Hint: Consider the function f(x)=\frac{x^{2}}{x^{3}+200} in the interval [1,\infty).

Question 5:

A point P is given on the circumference of a circle with radius r. Chords QR are drawn parallel to the tangent at P. Determine the maximum possible area of the triangle PQR.

Question 6:

Find the polynomial f(x) of degree 6, which satisfies \lim_{x \rightarrow 0} (1+\frac{f(x)}{x^{3}})=e^{2} and has a local maximum at x=1 and local minimum at x=0 and 2.

Question 7:

For the circle x^{2}+y^{2}=r^{2}, find the value of r for which the area enclosed by the tangents drawn from the point (6,8) to the circle and the chord of contact is maximum.

Question 8:

Suppose that f has a continuous derivative for all values of x and f(0)=0, with |f^{'}(x)| <1 for all x. Prove that |f(x)| \leq |x|.

Question 9:

Show that (e^{x}-1)>(1+x)\log {1+x}, if x \in (0,\infty).

Question 10:

Let -1 \leq p \leq 1. Show that the equations 4x^{3}-3x-p=0 has a unique root in the interval [1/2,1] and identify it.

 

Applications of Derivatives IITJEE Maths tutorial: practice problems part IV

Question 1.

If the point on y = x \tan {\alpha} - \frac{ax^{2}}{2u^{2}\cos^{2}{\alpha}}, where \alpha>0, where the tangent is parallel to y=x has an ordinate \frac{u^{2}}{4a}, then what is the value of \alpha?

Question 2:

Prove that the segment of the tangent to the curve y=c/x, which is contained between the coordinate axes is bisected at the point of tangency.

Question 3:

Find all the tangents to the curve y = \cos{(x+y)} for -\pi \leq x \leq \pi that are parallel to the line x+2y=0.

Question 4:

Prove that the curves y=f(x), where f(x)>0, and y=f(x)\sin{x}, where f(x) is a differentiable function have common tangents at common points.

Question 5:

Find the condition that the lines x \cos{\alpha} + y \sin{\alpha} = p may touch the curve (\frac{x}{a})^{m} + (\frac{y}{b})^{m}=1.

Question 6:

Find the equation of a straight line which is tangent to one point and normal to the point on the curve y=8t^{3}-1, and x=4t^{2}+3.

Question 7:

Three normals are drawn from the point (c,0) to the curve y^{2}=x. Show that c must be greater than 1/2. One normal is always the x-axis. Find c for which the two other normals are perpendicular to each other.

Question 8:

If p_{1} and p_{2} are lengths of the perpendiculars from origin on the tangent and normal to the curve x^{2/3} + y^{2/3}=a^{2/3} respectively, prove that 4p_{1}^{2} + p_{2}^{2}=a^{2}.

Question 9:

Show that the curve x=1-3t^{2}, and y=t-3t^{3} is symmetrical about x-axis and has no real points for x>1. If the tangent at the point t is inclined at an angle \psi to OX, prove that 3t= \tan {\psi} +\sec {\psi}. If the tangent at P(-2,2) meets the curve again at Q, prove that the tangents at P and Q are at right angles.

Question 10:

Find the condition that the curves ax^{2}+by^{2}=1 and a^{'}x^{2} + b^{'}y^{2}=1 intersect orthogonality and hence show that the curves \frac{x^{2}}{(a^{2}+b_{1})} + \frac{y^{2}}{(b^{2}+b_{1})} = 1 and \frac{x^{2}}{a^{2}+b_{2}} + \frac{y^{2}}{(b^{2}+b_{2})} =1 also intersect orthogonally.

More later,

Nalin Pithwa.

Applications of Derivatives: Tutorial: IITJEE Maths: Part II

Another set of “easy to moderately difficult” questions:

  1. The function y = \frac{}x{1+x^{2}} decreases in the interval (a) (-1,1) (b) [1, \infty) (c) (-\infty, -1] (d) (-\infty, \infty). There are more than one correct choices. Which are those?
  2. The function f(x) = \arctan (x) - x decreases in the interval (a) (1,\infty) (b) (-1, \infty) (c) (-\infty, -\infty) (d) (0, \infty). There is more than one correct choice. Which are those?
  3. For x>1, y = \log(x) satisfies the inequality: (a) x-1>y (b) x^{2}-1>y (c) y>x-1 (d) \frac{x-1}{x}<y. There is more than one correct choice. Which are those?
  4. Suppose f^{'}(x) exists for each x and h(x) = f(x) - (f(x))^{2} + (f(x))^{3} for every real number x. Then, (a) h is increasing whenever f is increasing (b) h is increasing whenever f is decreasing (c) h is decreasing whenever f is decreasing (d) nothing can be said in general. Find the correct choice(s).
  5. If f(x)=3x^{2}+12x-1, when -1 \leq x \leq 2, and f(x)=37-x, when 2<x\leq 3. Then, (a) f(x) is increasing on [-1,2] (b) f(x) is continuous on [-1,3] (c) f^{'}(2) doesn’t exist (d) f(x) has the maximum value at x=2. Find all the correct choice(s).
  6. In which interval does the function y=\frac{x}{\log(x)} increase?
  7. Which is the larger of the functions \sin(x) + \tan(x) and f(x)=2x in the interval (0<x<\pi/2)?
  8. Find the set of all x for which \log {(1+x)} \leq x.
  9. Let f(x) = |x-1| + a, if x \leq 1; and, f(x)=2x+3, if x>1. If f(x) has local minimum at x=1, then a \leq ?
  10. There are exactly two distinct linear functions (find them), such that they map [-1,1] and [0,2].

more later, cheers,

Nalin Pithwa.

Applications of Derivatives: Tutorial Set 1: IITJEE Mains Maths

“Easy” questions:

Question 1:

Find the slope of the tangent to the curve represented by the curve x=t^{2}+3t-8 and y=2t^{2}-2t-5 at the point (2,-1).

Question 2:

Find the co-ordinates of the point P on the curve y^{2}=2x^{3}, the tangent at which is perpendicular to the line 4x-3y+2=0.

Question 3:

Find the co-ordinates of the point P(x,y) lying in the first quadrant on the ellipse x^{2}/8 + y^{2}/18=1 so that the area of the triangle formed by the tangent at P and the co-ordinate axes is the smallest.

Question 4:

The function f(x) = \frac{\log (\pi+x)}{\log (e+x)}, where x \geq 0 is

(a) increasing on (-\infty, \infty)

(b) decreasing on [0, \infty)

(c) increasing on [0, \pi/e) and decreasing on [\pi/e, \infty)

(d) decreasing on [0, \pi/e) and increasing on [\pi/e, \infty).

Fill in the correct multiple choice. Only one of the choices is correct.

Question 5:

Find the length of a longest interval in which the function 3\sin(x) -4\sin^{3}(x) is increasing.

Question 6:

Let f(x)=x e^{x(1-x)}, then f(x) is

(a) increasing on [-1/2, 1]

(b) decreasing on \Re

(c) increasing on \Re

(d) decreasing on [-1/2, 1].

Fill in the correct choice above. Only one choice holds true.

Question 7:

Consider the following statements S and R:

S: Both \sin(x) and \cos (x) are decreasing functions in the interval (\pi/2, \pi).

R: If a differentiable function decreases in the interval (a,b), then its derivative also decreases in (a,b).

Which of the following is true?

(i) Both S and R are wrong.

(ii) Both S and R are correct, but R is not the correct explanation for S.

(iii) S is correct and R is the correct explanation for S.

(iv) S is correct and R is wrong.

Indicate the correct choice. Only one choice is correct.

Question 8:

For which of the following functions on [0,1], the Lagrange’s Mean Value theorem is not applicable:

(i) f(x) = 1/2 -x, when x<1/2; and f(x) = (1/2-x)^{2}, when x \geq 1/2.

(ii) f(x) = \frac{\sin(x)}{x}, when x \neq 0; and f(x)=1, when x=0.

(iii) f(x)=x |x|

(iv) f(x)=|x|.

Only one choice is correct. Which one?

Question 9:

How many real roots does the equation e^{x-1}+x-2=0 have?

Question 10:

What is the difference between the greatest and least values of the function f(x) = \cos(x) + \frac{1}{2}\cos(2x) -\frac{1}{3}\cos(3x)?

More later,

Nalin Pithwa.