IITJEE Mains « Mathematics Hothouse

Light travels along straight lines. In fact, the shortest distance between any two points is the path taken by a light wave to travel from the initial point to the final point. In other words, it is a straight line. (A slight detour: using this elementary fact, can you prove the triangle inequality?)
Bodies falling from rest in a planet’s gravitational field do so in a straight line.
Bodies coasting under their own momentum (like a hockey puck gliding across the ice) do so in a straight line. (Think of Newton’s First Law of Motion).

So we often use the equations of lines (called linear equations) to study such motions.

Many important quantities are related by linear equations. Once we know that a relationship between two variables is linear, we can find it from any two pairs of corresponding values just as we find the equation of a line from the coordinates of two points.

Slope is important because it gives us a way to say how steep something is (roadbeds, roofs, stairs, banking of railway tracks). The notion of a slope also enables us to describe how rapidly things are changing. (To philosophize, everything in the observable universe is changing). For this reason, slope plays an important role in calculus.

More later,

Nalin Pithwa.

PS: Ref: Calculus and Analytic Geometry by G B Thomas and Finney; or any other book on calculus.

PS: I strongly recommend the Thomas and Finney book : You can get it from Amazon India or Flipkart:

https://www.amazon.in/Thomas-Calculus-George-B/dp/9353060419/ref=sr_1_1?crid=36S3685TG7OYF&keywords=thomas+calculus&qid=1561503390&s=books&sprefix=Thomas+%2Caps%2C259&sr=1-1

or Flipkart:

https://www.flipkart.com/thomas-calculus-1/p/itmebug5kzrnttfj?pid=9789332547278&lid=LSTBOK9789332547278CHN4GH&marketplace=FLIPKART&srno=s_1_23&otracker=AS_Query_OrganicAutoSuggest_2_9&otracker1=AS_Query_OrganicAutoSuggest_2_9&fm=SEARCH&iid=fdc8327b-756c-4f6d-aa10-b45acc900e12.9789332547278.SEARCH&ppt=sp&ppn=sp&ssid=uz7zckp71c0000001561503474614&qH=2488f76736a10369

Prove that the minimum value of $(a+x)(b+x)/(c+x)$ for $x>-c$ , is $(\sqrt{a-c}+\sqrt{b-c})^{2}$ .
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.
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.
Determine the altitude of a cone with the greatest possible volume inscribed in a sphere of radius R.
Find the sides of a rectangle with the greatest possible perimeter inscribed in a semicircle of radius R.
Prove that in an ellipse, the distance between the centre and any normal does not exceed the difference between the semi-axes.
Determine the altitude of a cylinder of the greatest possible volume which can be inscribed in a sphere of radius R.
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).
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 $[0,1]$ .
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 $x \neq 0$ .
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) $y=\frac{10}{4x^{3}-9x^{2}+6x}$
Prove that if $0<x_{1}<x_{2}<\frac{\pi}{2}$ , then $\frac{\tan{x_{2}}}{\tan{x_{1}}} > \frac{x_{2}}{x_{1}}$
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.
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.
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 $x \in (-\infty, 0)$ .
If $f^{''}(x)<0$ for all $x \in (a,b)$ , prove that $f^{'}(x)=0$ at most once in $(a,b)$ .
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.
Show that $x=\cos{x}$ has exactly one root in $[0,\frac{\pi}{2}]$ .
Find a polynomial $P(x)$ such that $P^{'}(x)-3P(x)=4-5x+3x^{2}$ . Prove that there is only one solution.
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 $|x|<1$ .
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 $(a,b)$ .
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.
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.
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.
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.
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 $x=1$ ?
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 $[0,5]$
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.
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 $x=1$ .
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.
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.
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.
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.
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.
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 $(-\infty, \infty)$
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.
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 $m,n >0$ .
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 $m=4,6$ .
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 $x_{1}, x_{2}$ .
Prove that $(e^{x}-1)>(1+x)\log(1+x)$ , if $x \in (0,\infty)$ .